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# Feedback Interconnections of Passive Linear Cone Complementarity Systems

Bernard Brogliato, Aneel Tanwani

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## To cite this version:

Bernard Brogliato, Aneel Tanwani. Feedback Interconnections of Passive Linear Cone Complementarity Systems. 2024. hal-04137144v2

HAL Id: hal-04137144

https://inria.hal.science/hal-04137144v2

Preprint submitted on 2 Apr 2024

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# Passivity Preservation in Interconnections of Linear Cone Complementarity Systems with State Jumps

Bernard Brogliato, Aneel Tanwani

<figure><img src="image_1.png" /></figure>

To cite this version:

Bernard Brogliato, Aneel Tanwani. Passivity Preservation in Interconnections of Linear Cone Complementarity Systems with State Jumps. Nonlinear Analysis: Hybrid Systems, 2026. hal-04137144v4

HAL Id: hal-04137144

https://inria.hal.science/hal-04137144v4

Submitted on 22 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Feedback Interconnections of Passive Linear Cone Complementarity Systems

Bernard Brogliato*

Aneel Tanwani†

April 2, 2024

## Abstract

This paper is largely concerned with generic interconnections of a class of passive nonsmooth nonlinear dynamical systems, namely linear cone complementarity systems (LCCS). Criteria which guarantee the passivity of the interconnection are given. Asymptotic stability is studied in a particular case. Application examples from nonsmooth circuits, state observer synthesis, and switching DAEs (Differential Algebraic Equations) are presented.

## 1 Introduction

We consider a particular class of nonlinear nonsmooth dynamical systems called Linear Cone Complementarity Systems (LCCS), which has found applications in circuits, mechanics, economics, etc, [17, 1], and are computational friendly [2]. The dynamics of such systems are described by

$$ \dot{x} = Ax + B\lambda + Lu+f \quad (1a) $$

$$ K \ni \lambda \perp w = Cx + D\lambda + Eu+g \in K^*, \quad (1b) $$

with state $x \in \mathbb{R}^n$ and the complementarity variables $\lambda, w \in \mathbb{R}^m$, $K \subseteq \mathbb{R}^m$ a constant closed convex nonempty cone, $K^*$ its dual cone, $u$ is an input, all matrices $A, B, C, D, E, L$ are constant, $f$ and $g$ are exogenous signals. The notation $K \ni \lambda \perp w \in K^*$ means:

<footnote>*Univ. Grenoble Alpes, INRIA, CNRS, LJK, Grenoble INP, 38000 Grenoble, France; (bernard.brogliato@inria.fr)</footnote>

<footnote>†LAAS CNRS, 7, avenue du Colonel Roche, BP 54200, 31031 Toulouse cedex 4, France (aneel.tanwani@cnrs.fr)</footnote>
# Passivity Preservation in Interconnections of Linear Cone Complementarity Systems with State Jumps

Bernard Brogliato<sup>a</sup>, Aneel Tanwani<sup>b</sup>

<sup>a</sup>Univ. Grenoble-Alpes, Inria, CNRS, LJK, Grenoble INP, Grenoble, 38000, France

<sup>b</sup>LAAS-CNRS, University of Toulouse, Toulouse, 31400, France

## Abstract

This article is largely concerned with generic interconnections of a class of passive nonsmooth nonlinear dynamical systems, namely linear cone complementarity systems (LCCS). We stipulate that each subsystem admits a positive definite storage function that characterizes the passivity of an underlying nonsmooth mapping. We provide algebraic criteria in terms of these individual storage functions to find the storage function which guarantees passivity of the overall interconnected system. State jumps in the interconnections are studied in detail. Examples from dynamic feedback control, switching DAEs, interconnected sweeping processes, and nonsmooth circuits are included as an illustration of the theoretical developments.

MSC: 15A39, 93D30, 93D25, 34A06, 34A34, 34A37, 34A60, 90C33.

Keywords: Linear complementarity systems, passivity, incremental passivity, feedback interconnections, maximal monotonicity, state jumps, Moreau's sweeping process

## 1. Introduction

A class of finite-dimensional nonlinear nonsmooth dynamical systems called Linear Cone Complementarity Systems (LCCS) is considered in this article. They are largely described by an ordinary differential equation and complementarity constraints or Linear Complementarity Problems (LCP) [34, 38]. Several modeling frameworks can be associated with this class of systems, which have been covered in different research monographs [10, 49, 47, 51, 66, 13, 3, 18, 39], and find applications in circuits, mechanics, economics, biology, networks, traffic, etc. Over the past three decades, various properties of such systems have been investigated and used for design purposes. Some earlier work [72, 44, 22, 26, 61, 19, 54] has focused on defining the modeling framework and the notion of solutions for such systems, where we observe that the notion of passivity for an underlying mapping plays an important role. In the later works, [20, 25, 23], the connections with passivity were studied in more depth, which has led to the study of existence and uniqueness of solutions for a broader class of systems. From a control point of view, the stability of complementarity systems has been studied in [40, 15, 63, 24, 55]. The questions related to controllability and control design have been addressed in [17, 21, 43, 68] along with some recent works on optimal control [12, 73, 30]. On the numerical side, we observe that the nonsmooth mapping in these systems can be efficiently implemented using some optimization algorithms, and there are several simulation techniques which combine the algorithms used for solving ODEs and the underlying optimization problem related to inequalities [2, 1, 42, 41].

For our purposes, we consider the dynamics of such systems described by the

$$ \dot{x} = Ax + B\lambda + Lu + f \quad (1a) $$

$$ K \ni \lambda \perp w = Cx + D\lambda + Eu + g \in K^*, \quad (1b) $$

Email addresses: bernard.brogliato@inria.fr (Bernard Brogliato), aneel.tanwani@laas.fr (Aneel Tanwani)
$w \in K^*$, $\lambda \in K$ and $\langle \lambda, w \rangle = 0$. When $K$ is the non-negative orthant of $\mathbb{R}^m$, that is, $K = \mathbb{R}_+^m$, then we also have $K^* = \mathbb{R}_+^m$. In that case, the inclusions in (1b) take the form of inequalities, commonly written as $0 \le \lambda \perp w \ge 0$, which are to be interpreted componentwise and (1) is commonly called a Linear Complementarity System (LCS) [20] in that case. The solutions of (1) are to be interpreted in the sense that we seek an absolutely continuous function of time $t \mapsto x(t)$ and the multiplier $t \mapsto \lambda(t) \in K$ such that (1a) holds for almost all $t \ge 0$ and the constraints in (1b) are satisfied for all $t \ge 0$. The variable $\lambda$ may be seen as a Lagrange multiplier associated with the constraint $w \in K^*$. One sees that, depending upon the value of $D$, the regularity of the multiplier $\lambda$ as a function of time, or as a function of the state varies and it is possibly discontinuous. Such issues have been well-documented in the literature, see for example [17] and references therein. Moreover, when the functions $u$ and $g$ are possibly discontinuous, this may lead to discontinuities in the state trajectory which are more formally studied under the framework of functions with bounded variation which is studied later in Section 3.

For these reasons, the system class (1) is inherently nonsmooth and several studies have been carried out in the literature to study the well-posedness or existence of solutions for such systems. In many of these works, the considered passive input/output operator associated with (1) is $\lambda \mapsto w$, because it is this pair of variables which plays a crucial role in the LCS well-posedness [20, 18, 15, 19, 23, 42, 21], and passivity of this input/output map naturally holds in applications like circuits or mechanics.

In this article, we are interested in studying the interconnections of LCS, where the coupling terms depend on the state and multiplier $\lambda$. To provide a motivation for such class of interconnected systems, let us consider the following setup which arises in output-based feedback control of LCS.

**Motivating example:** Consider an LCS described as in (1) with $f \equiv 0$, $g \equiv 0$, and an output $y = Hx$. A state observer for this system is proposed as:

$$ \begin{aligned} \dot{\hat{x}} &= (A - K_1C)\hat{x} + B\hat{\lambda} + K_1y + Lu \\ 0 \le \hat{\lambda} \perp \hat{w} &= (C - K_2H)\hat{x} + D\hat{\lambda} + Eu + K_2y \ge 0. \end{aligned} $$

This observer is used for feedback stabilization by choosing the control input to be $u = M\hat{x}$. Letting $e \triangleq x - \hat{x}$, the closed-loop system is an interaction of the $x$-dynamics:

$$ \Sigma_1: \begin{cases} \dot{x} = (A + LM)x - LMe + B\lambda \\ 0 \le \lambda \perp w = (C + EM)x + D\lambda - EMe \ge 0, \end{cases} \quad (2) $$

and the $e$-dynamics:

$$ \Sigma_2: \begin{cases} \dot{e} = (A - K_1C)e + B\lambda - B\hat{\lambda} \\ 0 \le \hat{\lambda} \perp \hat{w} = -(C - K_2H + EM)e + D\hat{\lambda} + (C + EM)x \ge 0. \end{cases} \quad (3) $$

We are primarily interested in addressing the question if the interconnection of $\Sigma_1$ and $\Sigma_2$ described in the aforementioned manner is well-posed and stable. The results of this
with state $x \in \mathbb{R}^n$ and the complementarity variables $\lambda, w \in \mathbb{R}^m$, $K \subseteq \mathbb{R}^m$ a constant closed convex nonempty cone, $K^*$ its dual cone, $u$ is an input, all matrices $A, B, C, D, E, L$ are constant, $f$ and $g$ are exogenous signals. The notation $K \ni \lambda \perp w \in K^*$ means: $w \in K^*$, $\lambda \in K$ and $\langle \lambda, w \rangle = 0$. When $K$ is the non-negative orthant of $\mathbb{R}^m$, that is, $K = \mathbb{R}_+^m$, then we also have $K^* = \mathbb{R}_+^m$. In that case, the inclusions in (1b) take the form of inequalities, commonly written as $0 \le \lambda \perp w \ge 0$, which are to be interpreted componentwise and (1) is commonly called a Linear Complementarity System (LCS) [22]. For the most part, we assume that the exogenous inputs $g$ and $u$ are absolutely continuous. With such regularity assumptions on the inputs, the solution of (1) comprises an absolutely continuous function of time $t \mapsto x(t)$ and a multiplier $t \mapsto \lambda(t) \in K$ such that (1a) holds for almost all $t \ge 0$ and the constraints in (1b) are satisfied for all $t \ge 0$. The variable $\lambda$ may be seen as a Lagrange multiplier associated with the constraint $w \in K^*$. One sees that, depending on the value of $D$, the regularity of the multiplier $\lambda$ as a function of time or as a function of the state varies and is possibly discontinuous. Such issues have been well documented in the literature; see, for example, [18] and references therein. When the functions $u$ and $g$ are discontinuous, it becomes necessary to define solutions with discontinuities in the state trajectory to satisfy the constraints in (1b). Such solutions are formally studied as functions with bounded variation, and we provide some details of this framework in Section 5.

For these reasons, the system class (1) is inherently nonsmooth and several studies have been carried out in the literature to study the well-posedness (existence and uniqueness of solutions) for such systems, as recalled above. In many of these works, the considered passive input/output operator associated with (1) is $\lambda \mapsto w$, because this pair of variables plays a crucial role in the well-posedness of LCCS [22, 19, 15, 20, 25, 68, 23], and the passivity of this input/output map naturally holds in applications such as circuits and mechanics.

In this article, we are interested in studying the interconnections of LCCS, where the coupling terms depend on the state and multiplier $\lambda$. To provide a motivation for such class of interconnected systems, let us consider the following setup which arises in output-based feedback control of LCS.

**Motivating example:** Consider an LCS described as in (1) with $f \equiv 0$, $g \equiv 0$, and an output $y = Hx$. A state observer for this system is proposed as:

$$ \begin{aligned} \hat{x} &= (A - K_1 H)\hat{x} + B\hat{\lambda} + K_1 y + Lu \\ 0 &\le \hat{\lambda} \perp \hat{w} = (C - K_2 H)\hat{x} + D\hat{\lambda} + Eu + K_2 y \ge 0. \end{aligned} $$

This observer is used for feedback stabilization by choosing the control input to be $u = M\hat{x}$. Letting $e \stackrel{\Delta}{=} x - \hat{x}$, the closed-loop system is an interaction of the x-dynamics:

$$ \Sigma_1: \left\{ \begin{array}{l} \dot{x} = (A + LM)x - LMe + B\lambda \\ 0 \le \lambda \perp w = (C + EM)x + D\lambda - EMe \ge 0, \end{array} \right. \qquad (2) $$

and the e-dynamics:

$$ \Sigma_2: \left\{ \begin{array}{l} \dot{e} = (A - K_1 H)e + B\lambda - B\hat{\lambda} \\ 0 \le \hat{\lambda} \perp \hat{w} = -(C - K_2 H + EM)e + D\hat{\lambda} + (C + EM)x \ge 0. \end{array} \right. \qquad (3) $$

We are interested in addressing the question if the interconnection of $\Sigma_1$ and $\Sigma_2$ described in the aforementioned manner has unique solutions and some stability properties. A quite powerful approach that allows us to bring an answer to those issues, is passivity. This is the reason why the first part of this article is dedicated to analyse conditions under which interconnections are passive. The results of this paper allow us to address such questions and provide guidelines for choosing the output injection gains $K_1, K_2$, and the state feedback matrix $M$ such that the closed-loop system has some desired properties (more details are given in section 6).

In the setting of this paper, we consider the nonsmooth systems of form (1) with an output described by a linear function of the state $x$ and the multiplier $\lambda$. The interconnections are described by taking the input of one subsystem as the output of another subsystem, see Fig. 1. Such interconnections of dynamical systems in nonsmooth setting have not received much attention in the literature. In a recent work [28], authors analyze series and parallel interconnections of one-port circuits to study the monotonicity of the overall system. The paper [60] analyzes certain classes of interconnections with monotone mappings to study the convergence of optimization algorithms. However, building upon our work in [67], the focus of this article is on analyzing the passivity of the mapping between the augmented state and the
paper allow us to address such questions and provide guidelines for choosing the output injection gains $K_1, K_2$, and the state feedback matrix $M$ such that the closed-loop system has some desired properties (see section 6).

In the setting of this paper, we consider the nonsmooth systems of form (1) with an output described by a linear function of the state $x$ and the multiplier $\lambda$; and the interconnections are described by taking the input of one subsystem as the output of another subsystem. Such interconnections of dynamical systems in nonsmooth setting have not received much attention in the literature. In a recent work [25], authors analyze series and parallel interconnections of one-port circuits to study the monotonicity of the overall system. However, building upon our work in [41], the focus of this article is on analyzing the passivity of the mapping between the augmented state and the augmented multiplier using the conditions on individual subsystem. In particular, we derive conditions which allow us to establish the existence of a quadratic storage function for the interconnected system. When looking at the literature on interconnection of (smooth) passive systems, one of the commonly asked questions is about the stability of the interconnected systems [44, 28, 16, 43] and such questions are addressed directly by computing bounds on the signal norms using passivity definitions without necessarily constructing the storage function for the interconnection. In [4], the authors address the stability question by associating a storage function with the interconnected system which is a linear combination of the storage functions associated with individual subsystems. In contrast to these references, the major difference in this work arises due to the fact that we interconnect the state and the nonsmooth multiplier through the input and output channels. Systems of the form (1) are not necessarily passive with respect to the input $u$ and the output. The first focus is basically the construction of a quadratic storage function for the interconnected system without changing the supply rate, because this property is essential for the subsequent analysis. In addressing such questions, we consider a more generic form of the storage functions which allows coupling between two subsystems and the resulting conditions are therefore more involved. We consider different passivity notions and study conditions for asymptotic stability with nontrivial equilibrium when the interconnection is not strictly passive.

This article is organized as follows: in section 2 generic interconnections of two passive LCCS are analyzed. Criteria such that passivity (section 2.1), strong passivity (section 2.2) and strict state passivity (section 2.3) are preserved after interconnection are given. Section 3 is devoted to state jumps analysis after interconnection. In section 4 the link with maximal monotonicity and incremental passivity is established. In section 5 the asymptotic stability of the interconnected system is studied in several cases. Section 6 is dedicated to present examples. Conclusions are given in section 7. Some technical results are provided in the appendix (sections A, B and C).

**Notation** The scalar product in $\mathbb{R}^n$ is denoted $\langle \cdot, \cdot \rangle$, the associated norm is $||x|| = \sqrt{\langle x, x \rangle}$. The unit ball in $\mathbb{R}^n$ is denoted $\mathbb{B}_n$. Let $M \in \mathbb{R}^{n \times m}$, $\sigma_i(M)$ are its singular
augmented multiplier using the conditions on individual subsystem. In particular, we derive conditions which allow us to establish the existence of a quadratic storage function for the interconnected system. When looking at the literature on interconnection of (smooth) passive systems, one of the commonly asked questions is about the stability of the interconnected systems [74, 35, 16, 71] and such questions are addressed directly by computing bounds on the signal norms using passivity definitions without necessarily constructing the storage function for the interconnection. In [5], the authors address the stability question by associating a storage function with the interconnected system which is a linear combination of the storage functions associated with individual subsystems. In contrast to these references, the major difference in this work arises due to the fact that we interconnect the state and the nonsmooth multiplier through the input and output channels. Systems of the form (1) are not necessarily passive with respect to the input u and the measured output. The first focus is basically the construction of a quadratic storage function for the interconnected system without changing the supply rate, because this property is essential for the subsequent analysis. In addressing such questions, we consider a more generic form of the storage functions which allows coupling between two subsystems and the resulting conditions are therefore more involved. We consider different passivity notions and study conditions for asymptotic stability with nontrivial equilibrium when the interconnection is not strictly passive.

This article is organized as follows: in Section 2 generic interconnections of two passive LCCS are analyzed. In Section 3, criteria such that passivity (Section 3.1), strong passivity (Section 3.2) and strict state passivity (Section 3.3) are preserved after interconnection are given. In Section 4, the link with maximal monotonicity and incremental passivity is established, and well-posedness of interconnections is established under passivity. Section 5 is devoted to state jumps analysis for the interconnected system. Section 6 is dedicated to present examples. Conclusions are given in Section 7. Some definitions and technical results are provided in Appendix A through Appendix E.

**Notation.** The scalar product in $\mathbb{R}^n$ is denoted $\langle \cdot, \cdot \rangle$, the associated norm is $\|x\| = \sqrt{\langle x, x \rangle}$. The unit ball in $\mathbb{R}^n$ is denoted by $\mathbb{B}_n$. Let $M \in \mathbb{R}^{n \times m}$, its entries are denoted $M^{ij}$, $1 \le i \le n$, $1 \le j \le m$, $\sigma_i(M)$ are its singular values, $\sigma_1(M) \ge \sigma_2(M) \ge \dots \ge \sigma_{\min\{n,m\}}(M) \ge 0$, $\sigma_{\max}(M)$ is its largest singular value. Let $m = n$ and $M = M^\top$, its eigenvalues are denoted by $\lambda_i(M)$, $1 \le i \le n$, $\lambda_{\max}(M)$ is its largest eigenvalue, $\lambda_{\min}(M)$ is its smallest eigenvalue. Positive definite matrix: $M > 0$ if and only if $x^\top M x > 0$ for all $x \ne 0$, positive semidefinite matrix: $M \ge 0$ if and only if $x^\top M x \ge 0$ for all $x$ ($M$ may be nonsymmetric). $M$ is a P-matrix if all its principal minors are positive. It is a copositive matrix if $x^\top M x \ge 0$ for all $x \in \mathbb{R}_+^n$. Let $M = M^\top \ge 0$, then $\lambda_i(M) \ge 0$. The Moore-Penrose pseudo-inverse of $M \in \mathbb{R}^{n \times m}$ is denoted by $M^\dagger \in \mathbb{R}^{m \times n}$. The range of $M$ is denoted by $\text{Im}(M)$, its null space by $\text{Ker}(M)$. The $m \times m$ identity matrix is denoted by $I_m$. Let $M_1$ and $M_2$ be two square matrices, we denote by $\text{diag}(M_1, M_2)$ the corresponding block-diagonal matrix. The normal cone to a nonempty closed convex set $\Phi \subseteq \mathbb{R}^n$ at $x \in \mathbb{R}^n$ is $\mathcal{N}_\Phi(x) = \{z \in \mathbb{R}^n | \langle z, y - x \rangle \le 0, \forall y \in \Phi\}$. Its inverse mapping is denoted by $\mathcal{N}_\Phi^{-1}(\cdot)$ (this is the subdifferential of the support function $\sigma_\Phi(\cdot)$, i.e., $\mathcal{N}_\Phi^{-1}(\cdot) = \partial\sigma_\Phi(\cdot)$). The dual cone of $\Phi$ is $\Phi^* = \{z \in \mathbb{R}^n | \langle z, w \rangle \ge 0, \forall w \in \Phi\}$. The tangent cone to $\Phi$ at $x \in \mathbb{R}^n$ is $\mathcal{T}_\Phi(x) = -(\mathcal{N}_\Phi(x))^*$. The relative interior of a set $\Phi$ is denoted by $\text{rint}(\Phi)$, its interior by $\text{Int}(\Phi)$. Let $\mathcal{M}: \mathbb{R}^n \to \mathbb{R}^m$ be a set-valued mapping, its range is denoted by $\text{Im}(\mathcal{M})$, its domain by $\text{Dom}(\mathcal{M})$. Interested readers may consult a standard reference, like [6] or any standard text on topology and set-theory, for precise definitions of the mathematical terms from set theory that are being used here. Id denotes the identity mapping $x \mapsto x$. Let $x, q \in \mathbb{R}^m$, $M \in \mathbb{R}^{m \times m}$, the linear complementarity problem LCP($M, q$) is defined as: $x \ge 0$, $w = Mx + q \ge 0$, $w^\top x = 0$, compactly $0 \le x \perp w \ge 0$, and SOLLCP($M, q$) is the set of solutions to LCP($M, q$). A comprehensive reference for LCP is [34]. The abbreviation AC is used for absolutely continuous, and RCLBV is used for right-continuous of locally bounded variation.

## 2. Generic Interconnections of Passive LCCS

Building upon our preliminary work in [67, Section 4], we consider interconnections of two LCCS. The dynamics of each of these systems are described by:

$$ \dot{x}_i = A_i x_i + B_i \lambda_i + L_i u_i + f_i \quad (4a) $$

$$ K_i \ni \lambda_i \perp w_i = C_i x_i + D_i \lambda_i + E_i u_i + g_i \in K_i^*, \quad (4b) $$

for $i \in \{1, 2\}$, with $x_i \in \mathbb{R}^{n_i}$, $u_1, y_2 \in \mathbb{R}^{n_{u_1}}$, $u_2, y_1 \in \mathbb{R}^{n_{u_2}}$, $w_i$, $\lambda_i \in \mathbb{R}^{m_i}$, $n = n_1 + n_2$, $m = m_1 + m_2$, $K_i \subseteq \mathbb{R}^{m_i}$ are closed convex nonempty cones, $f_i, g_i$ are exogenous signals. The interconnection between both subsystems in (4) is defined
values, $\sigma_1(M) \ge \sigma_2(M) \ge \dots \ge \sigma_{\min\{n,m\}}(M) \ge 0$, $\sigma_{\max}(M)$ is its largest singular value. Let $m=n$ and $M = M^\top$, its eigenvalues are denoted $\lambda_i(M)$, $1 \le i \le n$, $\lambda_{\max}(M)$ is its largest eigenvalue, $\lambda_{\min}(M)$ is its smallest eigenvalue. Positive definite matrix: $M > 0$ if and only if $x^\top M x > 0$ for all $x \ne 0$, positive semidefinite matrix: $M \ge 0$ if and only if $x^\top M x \ge 0$ for all $x$ (they may be nonsymmetric). $M$ is a P-matrix if all its principal minors are positive. It is a copositive matrix if $x^\top M x \ge 0$ for all $x \in \mathbb{R}_+^n$. Let $M = M^\top \ge 0$, then $\lambda_i(M) \ge 0$. The Moore-Penrose pseudo-inverse of $M \in \mathbb{R}^{n \times m}$ is denoted $M^\dagger$. The range of $M$ is denoted $\mathrm{Im}(M)$, its null space as $\mathrm{Ker}(M)$. The $m \times m$ identity matrix is denoted $I_m$. The normal cone to a nonempty closed convex set $\Phi \subseteq \mathbb{R}^n$ at $x \in \mathbb{R}^n$ is denoted as $\mathcal{N}_\Phi(x) = \{z \in \mathbb{R}^n | \langle z, y - x \rangle \le 0, \forall y \in \Phi\}$. Its inverse mapping is denoted as $\mathcal{N}_\Phi^{-1}(\cdot)$ (this is the subdifferential of the support function $\sigma_\Phi(\cdot)$, i.e., $\mathcal{N}_\Phi^{-1}(\cdot) = \partial\sigma_\Phi(\cdot)$). The dual cone of $\Phi$ is $\Phi^* = \{z \in \mathbb{R}^n | \langle z, w \rangle \ge 0, \forall w \in \Phi\}$. AC is for absolutely continuous functions. The relative interior of a set $\Phi$ is denoted $\mathrm{rint}(\Phi)$, its interior $\mathrm{Int}(\Phi)$. Let $\mathcal{M}: \mathbb{R}^n \to \mathbb{R}^m$ be a set-valued mapping, its range is denoted $\mathrm{Im}(\mathcal{M})$, its domain $\mathrm{Dom}(\mathcal{M})$. Interested readers may consult a standard reference, like [6] or any standard text on topology and set-theory, for precise definitions of the mathematical terms from set theory that are being used here. Let $x, q \in \mathbb{R}^m$, $M \in \mathbb{R}^{m \times m}$, the linear complementarity problem LCP($M, q$) is defined as: $x \ge 0$, $w = Mx + q \ge 0$, $w^\top x = 0$, compactly $0 \le x \perp w \ge 0$, and SOLLCP($M, q$) is the set of solutions to LCP($M, q$). A comprehensive reference for LCP is [27].

## 2 Generic Interconnections of Passive LCCS

In this section we consider interconnections of two LCCS. The dynamics of both systems are described by:

$$ \dot{x}_i = A_i x_i + B_i \lambda_i + L_i u_i \quad (4a) $$

$$ K_i \ni \lambda_i \perp w_i = C_i x_i + D_i \lambda_i + E_i u_i \in K_i^*, \quad (4b) $$

for $i \in \{1, 2\}$, with $x_i \in \mathbb{R}^{n_i}$, $u_1, y_2 \in \mathbb{R}^{n_{u_1}}$, $u_2, y_1 \in \mathbb{R}^{n_{u_2}}$, $w_i$, $\lambda_i \in \mathbb{R}^{m_i}$, $n = n_1 + n_2$, $m = m_1 + m_2$, $K_i \subseteq \mathbb{R}^{m_i}$ are closed convex nonempty cones. The interconnection between both subsystems in (4) is defined as:

$$ \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} y_2 \\ y_1 \end{pmatrix} = \begin{pmatrix} H_2 x_2 + G_2 \lambda_2 \\ H_1 x_1 + G_1 \lambda_1 \end{pmatrix} \quad (5) $$

The outputs $y_1$ and $y_2$ can be seen as the measured outputs for each subsystem, while $w_1$ and $w_2$ are internal variables (which can be considered as outputs of a passive internal
by the following equations:

$$ \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} y_2 \\ y_1 \end{pmatrix} = \begin{pmatrix} H_2 x_2 + G_2 \lambda_2 \\ H_1 x_1 + G_1 \lambda_1 \end{pmatrix}. \qquad (5) $$

For an illustration, see Fig. 1. The outputs $y_1$ and $y_2$ can be seen as the measured outputs for each subsystem, while

<figure><img src="image_3.png" /><figcaption>Figure 1: LCCS interconnection.</figcaption></figure>

$w_1$ and $w_2$ are internal variables (which can be considered as outputs of a passive internal subsystem as seen below). This yields the following interconnected system:

$$ \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \underbrace{\begin{pmatrix} A_1 & L_1 H_2 \\ L_2 H_1 & A_2 \end{pmatrix}}_{\triangleq \tilde{A} \in \mathbb{R}^{n \times n}} \underbrace{\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}}_{\triangleq x} + \underbrace{\begin{pmatrix} B_1 & L_1 G_2 \\ L_2 G_1 & B_2 \end{pmatrix}}_{\triangleq \tilde{B} \in \mathbb{R}^{n \times m}} \underbrace{\begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix}}_{\triangleq \lambda} + \begin{pmatrix} f_1 \\ f_2 \end{pmatrix} \qquad (6a) $$

$$ K \ni \begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix} \perp w = \underbrace{\begin{pmatrix} C_1 & E_1 H_2 \\ E_2 H_1 & C_2 \end{pmatrix}}_{\triangleq \tilde{C} \in \mathbb{R}^{m \times n}} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \underbrace{\begin{pmatrix} D_1 & E_1 G_2 \\ E_2 G_1 & D_2 \end{pmatrix}}_{\triangleq \tilde{D} \in \mathbb{R}^{m \times m}} \begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix} + \begin{pmatrix} g_1 \\ g_2 \end{pmatrix} \in K^*, \qquad (6b) $$

with $K \triangleq K_1 \times K_2$, $K^* \triangleq K_1^* \times K_2^*$. The coupling between the states $x_1$ and $x_2$ in the differential equation is defined through $L_1 H_2$ and $L_2 H_1$, and between the states and the multiplier through $L_1 G_2$ and $L_2 G_1$. Similarly, the terms $E_1 H_2$ and $E_1 G_2$ show the dependence of $\lambda_1$ on $x_2$ and $\lambda_2$, respectively, while the terms $E_2 H_1$ and $E_2 G_1$ show the dependence of $\lambda_2$ on $x_1$ and $\lambda_1$, respectively. We also notice that $E_1 G_2 \in \mathbb{R}^{m_1 \times m_2}$ and $E_2 G_1 \in \mathbb{R}^{m_2 \times m_1}$.

Passivity (or positive realness, its linear time-invariant systems analog) is known to be a fundamental property in Systems and Control; see, e.g., [35, 16, 5, 71]. For the interconnected system (6), we are interested in studying the passivity of the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$. In the first place, the passivity property allows us to establish the existence and uniqueness of solutions for the interconnection using Theorem C.1 in Appendix C. Moreover, by establishing various variants of the passivity for the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$, we can study several properties of the trajectories of the interconnected system. In what follows, the passivity of the interconnection is studied under the following basic assumption:

**Assumption 2.1.** The quadruples $(A_i, B_i, C_i, D_i)$, $i = 1, 2$, define a passive system. Equivalently, there exist $P_i = P_i^\top \succ 0$ such that:

$$ \begin{pmatrix} -A_i^\top P_i - P_i A_i & P_i B_i - C_i^\top \\ (P_i B_i - C_i^\top)^\top & D_i + D_i^\top \end{pmatrix} \succ 0. \qquad (7) $$

This means that the passivity of each LCS in (4) with $u_i \equiv 0$ holds with supply rates $\lambda_i^\top w_i$, and not $u_i^\top y_i$ as usual in the Passivity Theorem (compare Fig. 1 and [16, sections 5.1.1 and 5.1.2]). This is the reason why the interconnection
<figure><img src="image_4.png" /><figcaption>Figure 1: LCCS interconnection.</figcaption></figure>

subsystem as seen below). This yields the following interconnected system:

$$ 
\begin{gathered}
\begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \underbrace{\begin{pmatrix} A_1 & L_1 H_2 \\ L_2 H_1 & A_2 \end{pmatrix}}_{\triangleq \bar{A} \in \mathbb{R}^{n \times n}} \underbrace{\begin{pmatrix} x_1 \\ x_2 \end{pmatrix}}_{\triangleq x} + \underbrace{\begin{pmatrix} B_1 & L_1 G_2 \\ L_2 G_1 & B_2 \end{pmatrix}}_{\triangleq \bar{B} \in \mathbb{R}^{n \times m}} \underbrace{\begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix}}_{\triangleq \lambda} \\
K \ni \begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix} \perp w = \underbrace{\begin{pmatrix} C_1 & E_1 H_2 \\ E_2 H_1 & C_2 \end{pmatrix}}_{\triangleq \bar{C} \in \mathbb{R}^{m \times n}} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \underbrace{\begin{pmatrix} D_1 & E_1 G_2 \\ E_2 G_1 & D_2 \end{pmatrix}}_{\triangleq \bar{D} \in \mathbb{R}^{m \times m}} \begin{pmatrix} \lambda_1 \\ \lambda_2 \end{pmatrix} \in K^*,
\end{gathered}
\tag{6}
$$

with $K \triangleq \begin{pmatrix} K_1 \\ K_2 \end{pmatrix}$, $K^* \triangleq \begin{pmatrix} K_1^* \\ K_2^* \end{pmatrix}$. The interconnection matrices are $L_1$, $L_2$, $E_1$ and $E_2$. Notice that $E_1 G_2 \in \mathbb{R}^{m_1 \times m_2}$ and $E_2 G_1 \in \mathbb{R}^{m_2 \times m_1}$.

**Assumption 1.** The quadruples $(A_i, B_i, C_i, D_i)$, $i = 1, 2$, define a passive system. Equiv- alently, there exist $P_i = P_i^\top \succcurlyeq 0$ such that:

$$ \begin{pmatrix} -A_i^\top P_i - P_i A_i & P_i B_i - C_i^\top \\ (P_i B_i - C_i^\top)^\top & D_i + D_i^\top \end{pmatrix} \succcurlyeq 0. \tag{7} $$

This means that the passivity of each LCS in (4) with $u_i \equiv 0$ holds with supply rates $\lambda_i^\top w_i$, and not $u_i^\top y_i$ as usual in the Passivity Theorem [16] (notice that only AC solutions are considered throughout this article). This is the reason why the interconnection in (5) imposes $u_1 = y_2$ and not $u_1 = -y_2$ as in the usual passivity interconnections: the signum of the feedback is irrelevant in this context. The passivity is preserved under the
in (5) imposes $u_1 = y_2$ and not $u_1 = -y_2$ as in the usual passivity interconnections: the signum of the feedback is irrelevant in this context. The passivity is preserved under the interconnection constraint, if and only if there exists an $n \times n$ matrix $P = P^\top \succcurlyeq 0$ such that:

$$ \tilde{Q} \triangleq \begin{pmatrix} -\tilde{A}^{\top} P - P \tilde{A} & P \tilde{B} - \tilde{C}^{\top} \\ (P \tilde{B} - \tilde{C}^{\top})^{\top} & \tilde{D} + \tilde{D}^{\top} \end{pmatrix} \succcurlyeq 0. \quad (8) $$

Then the interconnected system is passive with the supply rate $w^\top \lambda$ and storage function $V(x) = \frac{1}{2} x^\top P x$. To establish the passivity of the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ using a quadratic storage function $\frac{1}{2} x^\top P x$, let us take

$$ P \triangleq \begin{pmatrix} \alpha_1 P_1 & P_{12} \\ P_{12}^{\top} & \alpha_2 P_2 \end{pmatrix}, \quad (9) $$

where $P_1 \in \mathbb{R}^{n_1 \times n_1}$, $P_2 \in \mathbb{R}^{n_2 \times n_2}$, $P_{12} \in \mathbb{R}^{n_1 \times n_2}$, $\alpha_1 > 0$, $\alpha_2 > 0$. Then,

$$ -\tilde{A}^{\top} P - P \tilde{A} = \operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q} $$

with $Q_1 \triangleq -A_1^\top P_1 - P_1 A_1$, $Q_2 \triangleq -A_2^\top P_2 - P_2 A_2$, and

$$ \tilde{Q} = \begin{pmatrix} P_{12}L_2H_1 + H_1^\top L_2^\top P_{12} & \alpha_1 P_1 L_1 H_2 + P_{12} A_2 + A_1^\top P_{12}^\top + \alpha_2 H_1^\top L_2^\top P_2 \\ \tilde{Q}_{12}^\top & P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12} \end{pmatrix} \stackrel{\Delta \tilde{Q}_{12}}{=} \begin{pmatrix} E_1 G_2 + G_1^\top E_2^\top \\ D_2 + D_2^\top \end{pmatrix}, \quad (10) $$

We also define $\tilde{Q}_1 \triangleq \alpha_1 Q_1 - P_{12} L_2 H_1 - H_1^\top L_2^\top P_{12}^\top$, $\tilde{Q}_2 \triangleq \alpha_2 Q_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12}$,

$$ \tilde{D} + \tilde{D}^{\top} = \begin{pmatrix} D_1 + D_1^{\top} & E_1 G_2 + G_1^{\top} E_2^{\top} \\ E_2 G_1 + G_2^{\top} E_1^{\top} & D_2 + D_2^{\top} \end{pmatrix}, \quad (11) $$

$$ \begin{aligned} P \tilde{B} - \tilde{C}^{\top} &= \begin{pmatrix} \alpha_1 P_1 B_1 - C_1^{\top} + P_{12} L_2 G_1 & \alpha_1 P_1 L_1 G_2 + P_{12} B_2 - H_1^{\top} E_2^{\top} \\ P_{12}^{\top} B_1 + \alpha_2 P_2 L_2 G_1 - H_2^{\top} E_1^{\top} & \alpha_2 P_2 B_2 - C_2^{\top} + P_{12}^{\top} L_1 G_2 \end{pmatrix} \\ \triangleq \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} &\in \mathbb{R}^{n \times n}. \end{aligned} \quad (12) $$

Finally, the matrix inequality in (8) is equivalent to:

$$ P \tilde{B} - \tilde{C}^{\top} = (P \tilde{B} - \tilde{C}^{\top})(\tilde{D} + \tilde{D}^{\top})^{\dagger}(\tilde{D} + \tilde{D}^{\top}) \quad (13a) $$

$$ \tilde{D} + \tilde{D}^{\top} \succcurlyeq 0 \quad (13b) $$

$$ \tilde{A}^{\top} P + P \tilde{A} \preceq 0 \quad (13c) $$

$$ -\tilde{A}^{\top} P - P \tilde{A} \succcurlyeq (P \tilde{B} - \tilde{C}^{\top})(\tilde{D} + \tilde{D}^{\top})^{\dagger}(\tilde{B}^{\top} P - \tilde{C}). \quad (13d) $$

The conditions (13) follow from [9, Proposition 10.2.5]. As is well-known, in case $\tilde{D} + \tilde{D}^{\top} = 0$, the conditions (13) reduce to the inequalities (13a) and (13c). Solving the matrix equalities/inequalities in (13) with unknowns $P_1$, $\alpha_1$, $P_2$, $\alpha_2$, $P_{12}$, $H_1$, $H_2$, $G_1$, $G_2$ (and possibly $L_1$, $L_2$, $E_1$, $E_2$ in some cases) is difficult in general. In the following sections, paths which facilitate their analysis are studied. The results in sections 3.1, 3.2, 3.3, provide conditions such that the interconnection quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ possesses required passivity properties. It is noteworthy that the matrices' structure is tightly related to LCCS structure.

### 3. Passivity Properties for the Interconnected System

In this section, it is analysed how the passivity, strict state passivity, and strong passivity of the two quadruples $(A_i, B_i, C_i, D_i)$ in (4), may be preserved for the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ of the interconnected system (6). Issues related to existence and uniqueness of solutions will be tackled later in Section 4.
interconnection constraint, if and only if there exists an $n \times n$ matrix $P = P^\top \succcurlyeq 0$ such that:

$$ \tilde{Q} \triangleq \begin{pmatrix} -\tilde{A}^\top P - P \tilde{A} & P \tilde{B} - \tilde{C}^\top \\ (P \tilde{B} - \tilde{C}^\top)^\top & \tilde{D} + \tilde{D}^\top \end{pmatrix} \succcurlyeq 0. \quad (8) $$

Then the interconnected system is passive with the supply rate $w^\top \lambda$ and storage functions $V(x) = x^\top P x$. Using (4) (5) (6), we obtain the differential inclusion (DI):

$$ \begin{align} \dot{x}_1 &\in A_1 x_1 - B_1 (D_1 + \mathcal{N}_{K_1}^{-1})^{-1} (C_1 x_1 + E_1 H_2 x_2 + E_1 G_2 \lambda_2) + L_1 (H_2 x_2 + G_2 \lambda_2) \nonumber \\ \dot{x}_2 &\in A_2 x_2 - B_2 (D_2 + \mathcal{N}_{K_2}^{-1})^{-1} (C_2 x_2 + E_2 H_1 x_1 + E_2 G_1 \lambda_1) + L_2 (H_1 x_1 + G_1 \lambda_1) \tag{9} \\ \lambda &\in -(\tilde{D} + \mathcal{N}_{K_*}^{-1})^{-1} (\tilde{C} x) \nonumber \end{align} $$

Let $P \triangleq \begin{pmatrix} P_1 & P_{12} \\ P_{12}^\top & P_2 \end{pmatrix}$, $P_1 \in \mathbb{R}^{n_1 \times n_1}$, $P_2 \in \mathbb{R}^{n_2 \times n_2}$, $P_{12} \in \mathbb{R}^{n_1 \times n_2}$. Then: $-\tilde{A}^\top P - P \tilde{A} = \text{diag}(Q_1, Q_2) + \bar{Q}$ with $Q_1 \triangleq -A_1^\top P_1 - P_1 A_1$, $Q_2 \triangleq -A_2^\top P_2 - P_2 A_2$, and

$$ \bar{Q} = \begin{pmatrix} P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top & \underbrace{P_1 L_1 H_2 + P_{12} A_2 + A_1^\top P_{12}^\top + H_1^\top L_2^\top P_2}_{\triangleq \tilde{Q}_{12}} \\ \bar{Q}_{12}^\top & P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12} \end{pmatrix}. \quad (10) $$

Also $\bar{Q}_1 \triangleq Q_1 + P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top$, $\bar{Q}_2 \triangleq Q_2 + P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12}$,

$$ \tilde{D} + \tilde{D}^\top = \begin{pmatrix} D_1 + D_1^\top & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & D_2 + D_2^\top \end{pmatrix}, \quad (11) $$

$$ \begin{align} P \tilde{B} - \tilde{C}^\top &= \begin{pmatrix} P_1 B_1 - C_1^\top + P_{12} L_2 G_1 & P_1 L_1 G_2 + P_{12} B_2 - H_1^\top E_2^\top \\ P_{12}^\top B_1 + P_2 L_2 G_1 - H_2^\top E_1^\top & P_2 B_2 - C_2^\top + P_{12}^\top L_1 G_2 \end{pmatrix} \tag{12} \\ \triangleq \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \in \mathbb{R}^{n \times n}. \nonumber \end{align} $$

Finally the matrix inequality in (8) is equivalent to:

$$ \begin{align} \text{(a)} \quad & P \tilde{B} - \tilde{C}^\top = (P \tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger(\tilde{D} + \tilde{D}^\top) \nonumber \\ \text{(b)} \quad & \tilde{D} + \tilde{D}^\top \succcurlyeq 0 \nonumber \\ \text{(c)} \quad & \tilde{A}^\top P + P \tilde{A} \preccurlyeq 0 \nonumber \\ \text{(d)} \quad & - \tilde{A}^\top P - P \tilde{A} \succcurlyeq (P \tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger(\tilde{B}^\top P - \tilde{C}) \tag{13} \end{align} $$

The conditions (13) follow from [9, Proposition 8.2.4]. As is well-known, in case $\tilde{D}+\tilde{D}^\top = 0$, the conditions (13) reduce to (a) and (c). Solving the matrix equalities/inequalities in (13) with unknowns $P_1, P_2, P_{12}, H_1, H_2, G_1, G_2$ (and possibly $L_1, L_2, E_1, E_2$ in some cases) is difficult in general. In the following sections paths which facilitate their analysis are studied. The results in sections 2.1, 2.2, 2.3, provide conditions such that the interconnection quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ possesses required passivity properties. It is noteworthy that the matrices' structure is tightly related to LCCS structure.
### 3.1. Passivity Preservation after Interconnection

In this section, we will provide conditions that allow us to study passivity of the interconnected system under Assumption 2.1. In Section 3.1.1, we provide sufficient conditions under which the interconnected system is passive with storage function $x \mapsto \frac{1}{2}x^\top P x$ with $P$ being of the form (9). In Section 3.1.2, we take $P$ to be of the form $P = \text{diag}(P_1, P_2)$ which is less generic, yet useful to investigate conditions using only the subsystems storage functions without cross-terms. Then, in Section 3.1.3, we provide necessary conditions for the interconnection to be passive.

#### 3.1.1. Sufficient conditions for quadratic storage function

Let us start with several lemmata which state sufficient conditions for (13) to hold. In particular, Lemma 3.1 is for (13a); Lemma 3.2 is for (13b); Lemma 3.3 is for (13c); and Lemma 3.5 is for (13d). Throughout this section, it is assumed that $P$ is of the form (9), Assumption 2.1 holds and $P_i$ is a solution of the LMI (7), $i = 1, 2$.

**Lemma 3.1.** Inequality (13a) holds if and only if:

$$ \mathrm{Im}\left( \begin{array}{c} \alpha_1 B_1^\top P_1 - C_1 + G_1^\top L_2^\top P_{12} \\ \alpha_1 G_2^\top L_1^\top P_1 + B_2^\top P_{12}^\top - E_2 H_1 \end{array} \right) + \mathrm{Im}\left( \begin{array}{c} B_1^\top P_{12} + \alpha_2 G_1^\top L_2^\top P_2 - E_1 H_2 \\ \alpha_2 B_2^\top P_2 - C_2 + G_2^\top L_1^\top P_{12} \end{array} \right) \\ \subseteq \mathrm{Im}\left( \begin{array}{c} D_1 + D_1^\top \\ E_2 G_1 + G_2^\top E_1^\top \end{array} \right) + \mathrm{Im}\left( \begin{array}{c} E_1 G_2 + G_1^\top E_2^\top \\ D_2 + D_2^\top \end{array} \right). \quad (14) $$

In addition, (14) holds if:

(i) $\mathrm{Im}(G_1^\top L_2^\top P_{12}^\top) \subseteq \mathrm{Im}(\alpha_1 B_1^\top P_1 - C_1)$

(ii) $\mathrm{Im}(B_1^\top P_{12} + \alpha_2 G_1^\top L_2^\top P_2 - E_1 H_2) \subseteq \mathrm{Im}(\alpha_1 B_1^\top P_1 - C_1)$

(iii) $\mathrm{Im}(G_2^\top L_1^\top P_{12}) \subseteq \mathrm{Im}(\alpha_2 B_2^\top P_2 - C_2)$

(iv) $\mathrm{Im}(\alpha_1 G_2^\top L_1^\top P_1 + B_2^\top P_{12}^\top - E_2 H_1) \subseteq \mathrm{Im}(\alpha_2 B_2^\top P_2 - C_2)$

(v) $\mathrm{Im}(E_1 G_2 + G_1^\top E_2^\top) \subseteq \mathrm{Im}(D_1 + D_1^\top)$

(vi) $\mathrm{Im}(E_2 G_1 + G_2^\top E_1^\top) \subseteq \mathrm{Im}(D_2 + D_2^\top)$

(vii) $\mathrm{Im}(\alpha_1 B_1^\top P_1 - C_1) \subseteq \mathrm{Im}(D_1 + D_1^\top)$

(viii) $\mathrm{Im}(\alpha_2 B_2^\top P_2 - C_2) \subseteq \mathrm{Im}(D_2 + D_2^\top)$.

The last two inclusions in (15) hold if $\alpha_1 = \alpha_2 = 1$.

*Proof.* Using [9, Fact 8.4.3], (13a) holds true if and only if $\mathrm{Im}(\tilde{B}^\top P - \tilde{C}) \subseteq \mathrm{Im}(\tilde{D} + \tilde{D}^\top)$. Equivalently (14) holds, from [9, Fact 3.14.8] (i.e., $\mathrm{Im}(A) + \mathrm{Im}(B) = \mathrm{Im}([A \ B])$) for any $A$ and $B$ with same number of rows). To see that (15) implies (14), we recall that, for two matrices $A$ and $B$, if $\mathrm{Im}(B) \subseteq \mathrm{Im}(A)$, then $\mathrm{Im}(A+B) \subseteq \mathrm{Im}(A)$, and $\mathrm{Im}(A)+\mathrm{Im}(B) = \mathrm{Im}(A)$.

Also $\mathrm{Im}(A/B) \subseteq (\mathrm{Im}(A)/\mathrm{Im}(B))$. The last assertion stems from $\mathrm{Im}(B_1^\top P_1 - C_1) \subseteq \mathrm{Im}(D_1 + D_1^\top)$ and $\mathrm{Im}(B_2^\top P_2 - C_2) \subseteq \mathrm{Im}(D_2 + D_2^\top)$ due to passivity assumption on the quadruples $(A_i, B_i, C_i, D_i)$, $i = 1, 2$, stated in Assumption 2.1. $\square$

It happens that checking the sufficient conditions (15), which are derived from (14), requires elementary operations on the involved matrices. Such conditions are therefore useful in checking (13a), simplifying required calculations. For instance, first and third inclusions in (15) hold trivially if $P_{12} = 0$. The last four inclusions hold trivially when $D_1 + D_1^\top$ and $D_2 + D_2^\top$ are full rank matrices. If on the contrary $D_1 + D_1^\top = 0$ and $D_2 + D_2^\top = 0$ (as is the case in many applications with relative degree 1 systems), then the fifth and sixth inclusions directly show how coupling matrices have to be chosen.

**Lemma 3.2.** Inequality (13b) holds if any of the following three conditions hold:

1. $\mathrm{Im}(E_1 G_2 + G_1^\top E_2^\top) \subseteq \mathrm{Im}(D_1 + D_1^\top)$, $\sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top) < \sigma_{r_1}(D_1 + D_1^\top)\lambda_{\max}(D_2 + D_2^\top)$ when: $D_2 + D_2^\top > 0$, $\mathrm{rank}(D_1 + D_1^\top) = r_1 \le m_1$, $D_1 + D_1^\top \ne 0$.
## 2.1 Passivity Preservation after Interconnection

Let us start with several lemmata which state under which conditions the conditions in (13) hold. It is assumed that $P$ is as above and $P_i$ is a solution of the LMI (7), $i = 1, 2$.

**Lemma 1.** (13) (a) holds if and only if:

$$ \operatorname{Im} \begin{pmatrix} B_1^\top P_1 - C_1 + G_1^\top L_2^\top P_{12}^\top \\ G_2^\top L_1^\top P_1 + B_2^\top P_{12}^\top - E_2 H_1 \end{pmatrix} + \operatorname{Im} \begin{pmatrix} B_1^\top P_{12} + G_1^\top L_2^\top P_2 - E_1 H_2 \\ B_2^\top P_2 - C_2 + G_2^\top L_1^\top P_{12} \end{pmatrix} \\ \subseteq \operatorname{Im} \begin{pmatrix} D_1 + D_1^\top \\ E_2 G_1 + G_2^\top E_1^\top \end{pmatrix} + \operatorname{Im} \begin{pmatrix} E_1 G_2 + G_1^\top E_2^\top \\ D_2 + D_2^\top \end{pmatrix}. \quad (14) $$

In addition, (14) holds if:

$$ \begin{align} \operatorname{Im}(G_1^\top L_2^\top P_{12}^\top) &\subseteq \operatorname{Im}(B_1^\top P_1 - C_1) \nonumber \\ \operatorname{Im}(B_1^\top P_{12} + G_1^\top L_2^\top P_2 - E_1 H_2) &\subseteq \operatorname{Im}(B_1^\top P_1 - C_1) \nonumber \\ \operatorname{Im}(G_2^\top L_1^\top P_{12}^\top) &\subseteq \operatorname{Im}(B_2^\top P_2 - C_2) \nonumber \\ \operatorname{Im}(G_2^\top L_1^\top P_1 + B_2^\top P_{12}^\top - E_2 H_1) &\subseteq \operatorname{Im}(B_2^\top P_2 - C_2) \tag{15} \\ \operatorname{Im}(E_1 G_2 + G_1^\top E_2^\top) &\subseteq \operatorname{Im}(D_1 + D_1^\top) \nonumber \\ \operatorname{Im}(E_2 G_1 + G_2^\top E_1^\top) &\subseteq \operatorname{Im}(D_2 + D_2^\top). \nonumber \end{align} $$

*Proof.* Using [10, Fact 8.4.3], (13) (a) holds true if and only if $\operatorname{Im}(\tilde{B}^\top P - \tilde{C}) \subseteq \operatorname{Im}(\tilde{D} + \tilde{D}^\top)$, equivalently (14) holds, from [10, Fact 3.14.8] (i.e., $\operatorname{Im}(A) + \operatorname{Im}(B) = \operatorname{Im}(A \ B)$ for any $A$ and $B$ with same number of rows). Let us now prove (15). For two matrices $A$ and $B$, let $\operatorname{Im}(B) \subseteq \operatorname{Im}(A)$, then $\operatorname{Im}(A + B) \subseteq \operatorname{Im}(A)$, and $\operatorname{Im}(A) + \operatorname{Im}(B) = \operatorname{Im}(A)$. Also $\operatorname{Im}(\begin{pmatrix} A \\ B \end{pmatrix}) \subseteq \begin{pmatrix} \operatorname{Im}(A) \\ \operatorname{Im}(B) \end{pmatrix}$. Using these inclusions it follows that (15) implies (14), using $\operatorname{Im}(B_1^\top P_1 + C_1) \subseteq \operatorname{Im}(D_1 + D_1^\top)$ and $\operatorname{Im}(B_2^\top P_2 + C_2) \subseteq \operatorname{Im}(D_2 + D_2^\top)$ from Assumption 1. $\square$

**Corollary 1.** Assume that $P_{12} = 0$, then (14) holds if:

$$ \begin{align} P_1 L_1 G_2 - H_1^\top E_2^\top &= 0 \nonumber \\ P_2 L_2 G_1 - H_2^\top E_1^\top &= 0 \tag{16} \end{align} $$

*Proof.* Follows from Assumption 1, since $\operatorname{Im}(B_1^\top P_1 + C_1) \subseteq \operatorname{Im}(D_1 + D_1^\top)$ and $\operatorname{Im}(B_2^\top P_2 + C_2) \subseteq \operatorname{Im}(D_2 + D_2^\top)$ (see Appendix B). $\square$

**Lemma 2.** Inequality (13) (b) holds if: $\operatorname{Im}(E_1 G_2 + G_1^\top E_2^\top) \subseteq \operatorname{Im}(D_1 + D_1^\top)$, $\sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top) < \sigma_{r_1}(D_1 + D_1^\top)\lambda_{\max}(D_2 + D_2^\top)$ when $D_2 + D_2^\top \succ 0$, $\operatorname{rank}(D_1 + D_1^\top) = r_1 \le m_1$, $D_1 + D_1^\top \ne 0$. Or if: $\operatorname{Im}(E_2 G_1 + G_2^\top E_1^\top) \subseteq \operatorname{Im}(D_2 + D_2^\top)$, $\sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top) < \sigma_{r_2}(D_2 + D_2^\top)\lambda_{\max}(D_1 + D_1^\top)$ when $D_1 + D_1^\top \succ 0$, $\operatorname{rank}(D_2 + D_2^\top) = r_2 \le m_1$, $D_2 + D_2^\top \ne 0$. Or if: $(D_1 + D_1^\top = 0$ or $D_2 + D_2^\top = 0)$ and $E_1 G_2 + G_1^\top E_2^\top = 0$.
2. Im($E_2G_1 + G_2^\top E_1^\top$) $\subseteq$ Im($D_2 + D_2^\top$), $\sigma_{\max}^2(E_1G_2 + G_1^\top E_2^\top) < \sigma_{r_2}(D_2 + D_2^\top)\lambda_{\max}(D_1 + D_1^\top)$ when: $D_1 + D_1^\top > 0$, $\text{rank}(D_2 + D_2^\top) = r_2 \le m_1$, $D_2 + D_2^\top \ne 0$.

3. ($D_1 + D_1^\top = 0$ or $D_2 + D_2^\top = 0$) and $E_1G_2 + G_1^\top E_2^\top = 0$.

*Proof.* Let us first consider (11). From Assumption 2.1, $D_1 + D_1^\top \ge 0$ and $D_2 + D_2^\top \ge 0$. It follows that $\tilde{D} + \tilde{D}^\top \ge 0$ if and only if [9, Proposition 10.2.5]:

$$ (\alpha) \quad D_2 + D_2^\top \ge (E_2G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger(E_1G_2 + G_1^\top E_2^\top) \quad (16) $$

$$ (\beta) \quad E_1G_2 + G_1^\top E_2^\top = (D_1 + D_1^\top)(D_1 + D_1^\top)^\dagger(E_1G_2 + G_1^\top E_2^\top), $$

equivalently:

$$ (\alpha') \quad D_1 + D_1^\top \ge (E_1G_2 + G_1^\top E_2^\top)(D_2 + D_2^\top)^\dagger(E_2G_1 + G_2^\top E_1^\top) \quad (17) $$

$$ (\beta') \quad E_1G_2 + G_1^\top E_2^\top = (E_1G_2 + G_1^\top E_2^\top)(D_2 + D_2^\top)^\dagger(D_2 + D_2^\top). $$

Let us examine the above conditions:

(α) Assume that $D_2 + D_2^\top > 0$, $\text{rank}(D_1 + D_1^\top) = r_1 \le m_1$, $D_1 + D_1^\top \ne 0$. Using Corollary A.1 and [8, Corollary 9.6.5], the inequality (α) holds if $\sigma_{\max}((D_2 + D_2^\top) > \sigma_{\max}((E_2G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger(E_1G_2 + G_1^\top E_2^\top))$. From [8, Corollary 9.6.5, Fact 6.3.28], it follows that $\sigma_{\max}((E_2G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger(E_1G_2 + G_1^\top E_2^\top)) \le \sigma_{\max}^2(E_1G_2 + G_1^\top E_2^\top)\sigma_{\max}((D_1 + D_1^\top)^\dagger) = \sigma_{\max}^2(E_1G_2 + G_1^\top E_2^\top)\sigma_{r_1}^{-1}(D_1 + D_1^\top)$ and $\sigma_{\max}(D_2 + D_2^\top) = \lambda_{\max}(D_2 + D_2^\top)$.

(α') Assume that $D_1 + D_1^\top > 0$, $\text{rank}(D_2 + D_2^\top) = r_2 \le m_1$, $D_2 + D_2^\top \ne 0$. Then for same reasons, the inequality (α) holds if $\sigma_{\max}^2(E_1G_2 + G_1^\top E_2^\top)\sigma_{r_2}^{-1}((D_2 + D_2^\top) < \lambda_{\max}(D_1 + D_1^\top)$.

(β) Using [9, Fact 8.4.3], this is equivalent to $\text{Im}(E_1G_2 + G_1^\top E_2^\top) \subseteq \text{Im}(D_1 + D_1^\top)$.

(β') Similarly, the equality is equivalent to $\text{Im}(E_2G_1 + G_2^\top E_1^\top) \subseteq \text{Im}(D_2 + D_2^\top)$.

Therefore, conditions 1 and 2 in the statement of Lemma imply that (16) and (17) are satisfied. To end the proof, notice that if $D_1 + D_1^\top = 0$ or if $D_2 + D_2^\top = 0$ then [(α) and (β)], or [(α') and (β')] $\Rightarrow E_1G_2 = -G_1^\top E_2^\top$. Also $E_1G_2 = -G_1^\top E_2^\top$ guarantees that (α), (β), (α') and (β') are satisfied. Thus if $D_1 + D_1^\top = 0$ or $D_2 + D_2^\top = 0$, we have $\tilde{D} + \tilde{D}^\top \ge 0 \Leftrightarrow E_1G_2 = -G_1^\top E_2^\top$. Thus condition 3 guarantees (13b). $\square$

To give some intuition about Lemma 3.2, each of the three conditions ensures positive semi-definiteness of the matrix $\tilde{D} + \tilde{D}^\top$. These conditions essentially require the diagonal terms to dominate the off-diagonal coupling terms, or in case one of the diagonal terms is identically zero, then the off-diagonal terms must be identically zero as well. In the matrix $\tilde{D} + \tilde{D}^\top$, the off-diagonal terms basically relate the complementarity variable of one subsystem with the multiplier of another subsystem ($w_1$ with $\lambda_2$, and vice versa). A similar methodology is used in the following lemma to provide conditions for non-negative definiteness of $\tilde{A}^\top P + P\tilde{A}$, or equivalently for $\text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q}$ to be positive semidefinite as required in inequality (13c).

**Lemma 3.3.** *Inequality (13c) holds if and only if the following three conditions hold:*

(i) $\tilde{Q}_1 \ge 0, \tilde{Q}_2 \ge 0$

(ii) $\text{Im}(\tilde{Q}_{12}) \subseteq \text{Im}(\tilde{Q}_1), \text{Im}(\tilde{Q}_{12}^\top) \subseteq \text{Im}(\tilde{Q}_2)$

(iii) There exists $\mathcal{K}$ with $\sigma_{\max}(\mathcal{K}) \le 1$ such that $\tilde{Q}_{12} = \tilde{Q}_1^{1/2}\mathcal{K}\tilde{Q}_2^{1/2}$, or equivalently, $\sigma_{\max}((\tilde{Q}_1^\dagger)^{1/2}\tilde{Q}_{12}(\tilde{Q}_2^\dagger)^{1/2}) \le 1$.

In addition, if any of the following conditions is satisfied, then inequality (13c) holds:

1. $\tilde{Q}_1 \ge 0$, $P_1\text{Im}(L_1H_2) + \text{Im}(P_{12}A_2) + \text{Im}(A_1^\top P_{12}^\top) + \text{Im}(H_1^\top L_2^\top P_2) \subseteq \text{Im}(\alpha_1 Q_1 - P_{12}L_2H_1 - H_1^\top L_2^\top P_{12}^\top)$, and $\sigma_{\max}^2(\tilde{Q}_{12}) < \sigma_{q_1}(\tilde{Q}_1)\sigma_{\min}(\tilde{Q}_2)$ when: $\tilde{Q}_2 > 0$, $\text{rank}(\tilde{Q}_1) = q_1$, $\tilde{Q}_1 \ne 0$.

2. $\tilde{Q}_2 \ge 0$, $\text{Im}(H_2^\top L_1^\top P_1) + \text{Im}(A_2^\top P_{12}^\top) + \text{Im}(P_{12}A_1) + P_2\text{Im}(L_2H_1) \subseteq \text{Im}(\alpha_2 Q_2 - P_{12}L_1H_2 - H_2^\top L_1^\top P_{12})$, and $\sigma_{\max}^2(\tilde{Q}_{12}) < \sigma_{q_2}(\tilde{Q}_2)\sigma_{\min}(\tilde{Q}_1)$ when: $\tilde{Q}_1 > 0$, $\text{rank}(\tilde{Q}_2) = q_2$, $\tilde{Q}_2 \ne 0$.
Proof. Consider (11). From Assumption 1, $D_1 + D_1^\top \succcurlyeq 0$ and $D_2 + D_2^\top \succcurlyeq 0$. It follows that $\tilde{D} + \tilde{D}^\top \succcurlyeq 0$ if and only if [9, Proposition 8.2.4]:

$$ (\alpha) \quad D_2 + D_2^\top \succcurlyeq (E_2 G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger (E_1 G_2 + G_1^\top E_2^\top) \quad (17) $$

$$ (\beta) \quad E_1 G_2 + G_1^\top E_2^\top = (D_1 + D_1^\top)(D_1 + D_1^\top)^\dagger (E_1 G_2 + G_1^\top E_2^\top), $$

equivalently:

$$ (\alpha') \quad D_1 + D_1^\top \succcurlyeq (E_1 G_2 + G_1^\top E_2^\top)(D_2 + D_2^\top)^\dagger (E_2 G_1 + G_2^\top E_1^\top) \quad (18) $$

$$ (\beta') \quad E_1 G_2 + G_1^\top E_2^\top = (E_1 G_2 + G_1^\top E_2^\top)(D_2 + D_2^\top)^\dagger (D_2 + D_2^\top), $$

Let us examine the above conditions:

(α) Assume that $D_2 + D_2^\top \succ 0$, $\mathrm{rank}(D_1 + D_1^\top) = r_1 \le m_1$, $D_1 + D_1^\top \ne 0$. Using Corollary 8 and [9, Corollary 9.6.5], the inequality (α) holds if $\sigma_{\max}(D_2 + D_2^\top) > \sigma_{\max}((E_2 G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger (E_1 G_2 + G_1^\top E_2^\top))$. From [9, Corollary 9.6.5, Fact 6.3.28], it follows that $\sigma_{\max}((E_2 G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^\dagger (E_1 G_2 + G_1^\top E_2^\top)) \le \sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top)\sigma_{\max}((D_1 + D_1^\top)^\dagger) = \sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top)\sigma_{r_1}^{-1}(D_1 + D_1^\top)$ and $\sigma_{\max}(D_2 + D_2^\top) = \lambda_{\max}(D_2 + D_2^\top)$.

(α') Assume that $D_1 + D_1^\top \succ 0$, $\mathrm{rank}(D_2 + D_2^\top) = r_2 \le m_2$, $D_2 + D_2^\top \ne 0$. Then for same reasons, the inequality (α) holds if $\sigma_{\max}^2(E_1 G_2 + G_1^\top E_2^\top)\sigma_{r_2}^{-1}((D_2 + D_2^\top) < \lambda_{\max}(D_1 + D_1^\top)$.

(β) Using [10, Fact 8.4.3], this is equivalent to $\mathrm{Im}(E_1 G_2 + G_1^\top E_2^\top) \subseteq \mathrm{Im}(D_1 + D_1^\top)$.

(β') Similarly, the equality is equivalent to $\mathrm{Im}(E_2 G_1 + G_2^\top E_1^\top) \subseteq \mathrm{Im}(D_2 + D_2^\top)$.

Therefore the Lemma's condition imply that (17) and (18) are satisfied. Notice that if $D_1 + D_1^\top = 0$ or if $D_2 + D_2^\top = 0$ then [(α) and (β)], or [(α') and (β')] $\Rightarrow E_1 G_2 = -G_1^\top E_2^\top$. Also $E_1 G_2 = -G_1^\top E_2^\top$ guarantees that (α), (β), (α') and (β') are satisfied. Thus if $D_1 + D_1^\top = 0$ or $D_2 + D_2^\top = 0$, we have $\tilde{D} + \tilde{D}^\top \succ 0 \Leftrightarrow E_1 G_2 = -G_1^\top E_2^\top$.

□

**Lemma 3.** (13) (c) holds if: $\bar{Q}_1 \succcurlyeq 0$, $\bar{Q}_2 \succcurlyeq 0$, and: $\mathrm{Im}(\bar{Q}_{12}) \subseteq \mathrm{Im}(P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top)$, $\sigma_{\max}^2(\bar{Q}_{12}) < \sigma_{q_1}(\bar{Q}_1)\sigma_{\max}(\bar{Q}_2)$ when $\bar{Q}_2 \succ 0$, $\mathrm{rank}(\bar{Q}_1) = q_1$, $\bar{Q}_1 \ne 0$, or $\mathrm{Im}(\bar{Q}_{12}^\top) \subseteq \mathrm{Im}(P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12}^\top)$, $\sigma_{\max}^2(\bar{Q}_{12}) < \sigma_{q_2}(\bar{Q}_2)\sigma_{\max}(\bar{Q}_1)$ when $\bar{Q}_1 \succ 0$, $\mathrm{rank}(\bar{Q}_2) = q_2$, $\bar{Q}_2 \ne 0$.

Proof. The conditions which guarantee that $\tilde{A}^\top P + P\tilde{A} \prec 0$ are studied from the fact that $-\tilde{A}^\top P - P\tilde{A} = \mathrm{diag}(Q_1, Q_2) + \bar{Q}$ with $\bar{Q}$ in (10). We have $\mathrm{diag}(Q_1, Q_2) + \bar{Q} \succ 0$ if and only if: $\bar{Q}_1 = Q_1 + P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top \succ 0$, $\bar{Q}_2 = Q_2 + P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12}^\top \succ 0$, $\bar{Q}_{12} = (P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top)(P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top)^\dagger \bar{Q}_{12} \iff \mathrm{Im}(\bar{Q}_{12}) \subseteq \mathrm{Im}(P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top)$, and $\bar{Q}_2 \succ \bar{Q}_{12}^\top \bar{Q}_1^\dagger \bar{Q}_{12}$, where we used [9, Proposition 8.2.4] [10, Fact 8.4.3]. The last inequality can be treated similarly as we did in case (b) (α) and (α'). Assume that $\bar{Q}_2 \succ 0$, $\mathrm{rank}(\bar{Q}_1) = q_1$, $\bar{Q}_1 \ne 0$. Then the last inequality holds if $\sigma_{\max}^2(\bar{Q}_{12}) < 0$.
**Proof.** The first part, that is, the equivalence between (13c) and conditions (i), (ii), and (iii), follows from [45, Theorem 7.7.9]. For the sufficient conditions, we proceed as follows:

1. We have $\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q} \ge 0$ if and only if: $\bar{Q}_1 = \alpha_1 Q_1 - P_{12} L_2 H_1 - H_1^\top L_2^\top P_{12}^\top \ge 0$, and $\bar{Q}_{12} = (\alpha_1 Q_1 - P_{12} L_2 H_1 - H_1^\top L_2^\top P_{12}^\top)(\alpha_1 Q_1 - P_{12} L_2 H_1 - H_1^\top L_2^\top P_{12}^\top)^\dagger \bar{Q}_{12} \iff \operatorname{Im}(\bar{Q}_{12}) \subseteq \operatorname{Im}(\alpha_1 Q_1 - P_{12} L_2 H_1 - H_1^\top L_2^\top P_{12}^\top)$, and the inequality $\bar{Q}_2 \ge \bar{Q}_{12}^\top \bar{Q}_2^\dagger \bar{Q}_{12}$. We used [9, Proposition 10.2.5, Fact 8.4.3], as well as [9, Fact 3.13.15, equation (3.5.7)] which implies $\operatorname{Im}(\bar{Q}_{12}) \subseteq P_1 \operatorname{Im}(L_1 H_2) + \operatorname{Im}(P_{12} A_2) + \operatorname{Im}(A_1^\top P_{12}^\top) + \operatorname{Im}(H_1^\top L_2^\top P_2)$. The inequality can be treated similarly as we did for $(\alpha)$ and $(\alpha')$ in the proof of Lemma 3.2. Assume that $\bar{Q}_2 > 0$, $\operatorname{rank}(\bar{Q}_1) = q_1$, $\bar{Q}_1 \ne 0$. Then the last inequality holds if $\sigma_{\max}^2(\bar{Q}_{12}) < \sigma_{q_1}(\bar{Q}_1)\sigma_{\min}(\bar{Q}_2)$.

2. We can use also the equivalent conditions assuring $\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q} \ge 0$: $\bar{Q}_2 = \alpha_2 Q_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12} \ge 0$, $\bar{Q}_{12} = \bar{Q}_{12}(\alpha_2 Q_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12})^\dagger(\alpha_2 Q_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12}) \iff \operatorname{Im}(\bar{Q}_{12}^\top) \subseteq \operatorname{Im}(\alpha_2 Q_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12})$, and $\bar{Q}_1 \ge \bar{Q}_{12} \bar{Q}_2^\dagger \bar{Q}_{12}^\top$. Same tools are used as in item 1. Assume that $\bar{Q}_1 > 0$, $\operatorname{rank}(\bar{Q}_2) = q_2$, $\bar{Q}_2 \ne 0$. Then the last inequality holds if $\sigma_{\max}^2(\bar{Q}_{12}) < \sigma_{q_2}(\bar{Q}_2)\sigma_{\min}(\bar{Q}_1)$.

Thus, the lemma is proved. $\square$

In the statement of Lemma 3.3, the two conditions in items 1 and 2 are symmetric to each other and they correspond to the two cases where $\bar{Q}_1 > 0$ or $\bar{Q}_2 > 0$. In either case, one of the diagonal terms in $\tilde{A}^\top P + P\tilde{A}$ is negative definite and the remaining conditions ensure that the other diagonal terms are at least negative semi-definite with certain other conditions involving the off-diagonal terms.

**Remark 3.4.** Sufficient conditions guaranteeing $\bar{Q} \ll 0$ ($\Rightarrow \operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q} \ge 0$) can be derived also using [9, Proposition 10.2.5, Fact 8.4.3]. A sufficient condition guaranteeing $\bar{Q}_1 \ge 0$ (resp. $\bar{Q}_2 \ge 0$) is $P_{12} L_2 H_1 \ll 0$ (resp. $P_{12}^\top L_1 H_2 \ll 0$).

Let us now examine sufficient conditions for (13d) to be satisfied, which not only requires $-(\tilde{A}^\top P + P\tilde{A})$ to be positive semidefinite but it should also dominate another positive semidefinite matrix.

**Lemma 3.5.** The following results hold:

1. The inequality (13d) holds true if: $P_{12} = -(L_2 H_1)^\top = -L_1 H_2$, $\operatorname{rank}(P_{12}) = n_1$, $\sigma_{\max}^2(\bar{Q}_{12}) < \lambda_{\min}(\bar{Q}_1)\lambda_{\min}(\bar{Q}_2)$, $\operatorname{rank}(\tilde{D} + \tilde{D}^\top) = d$, $\tilde{D} + \tilde{D}^\top \ne 0$, and

$$ \frac{\sigma_{\max}^2(p_{11}) + \sigma_{\max}^2(p_{12}) + \sigma_{\max}^2(p_{22}) + \sigma_{\max}^2(p_{21}) + \sqrt{\Delta}}{2} < \sigma_d(\tilde{D} + \tilde{D}^\top)\lambda_{\max}\begin{pmatrix} \bar{Q}_1 & \bar{Q}_{12} \\ \bar{Q}_{12}^\top & \bar{Q}_2 \end{pmatrix} \quad (18) $$

$$ \text{with } \Delta = (\sigma_{\max}^2(p_{11}) + \sigma_{\max}^2(p_{12}) - \sigma_{\max}^2(p_{22}) - \sigma_{\max}^2(p_{21})) + 4(\sigma_{\max}(p_{11})\sigma_{\max}(p_{21}) + \sigma_{\max}(p_{12})\sigma_{\max}(p_{22})). $$

2. Let both sides of (13d) be decomposed as $MM^\top$ and $NN^\top$, respectively, so that (13d) is rewritten as $MM^\top \ge NN^\top$.

Let $M = U_1 \begin{pmatrix} T & 0 \\ 0 & 0 \end{pmatrix} U_2$, where $U_1$ and $U_2$ are unitary matrices, $T = \operatorname{diag}(\sigma_1(M), \dots, \sigma_r(M))$, $r = \operatorname{rank}(M)$, and $U_1^\top N = \begin{pmatrix} N_1 \\ 0 \end{pmatrix}$. Assume that:

$$ \sigma_{\max}\begin{pmatrix} N_1 \\ 0 \end{pmatrix} \le |\lambda_r(\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q})|^{\frac{1}{2}}, \quad (19) $$

then (13d) is satisfied.

**Proof.** 1. Let $\bar{Q}_2 > 0$, $\bar{Q}_1 > \bar{Q}_{12} \bar{Q}_2^{-1} \bar{Q}_{12}^\top$, or $\bar{Q}_1 > 0$, $\bar{Q}_2 > \bar{Q}_{12}^\top \bar{Q}_1^{-1} \bar{Q}_{12}$. If $Q_1 \ge 0$ and $Q_2 \ge 0$ (both subsystems are just passive), then $\bar{Q}_1 > 0$ and $\bar{Q}_2 > 0$ hold if $P_{12} = -(L_2 H_1)^\top = -L_1 H_2$, and $\operatorname{rank}(P_{12}) = n_1$. The other two inequalities hold if $\sigma_{\max}^2(\bar{Q}_{12}) < \lambda_{\min}(\bar{Q}_2)\lambda_{\min}(\bar{Q}_1)$ [8, Fact 8.19.2] and Schur complement Theorem [16, Theorem A.65]. Then applying Corollary A.1, the inequality (13d) holds if $\sigma_{\max}^2(P\tilde{B} - \tilde{C}^\top)\sigma_{\max}((\tilde{D} + \tilde{D}^\top)^\dagger) < \sigma_{\max}(\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q})$. Let $\operatorname{rank}(\tilde{D} + \tilde{D}^\top) = d$, $\tilde{D} + \tilde{D}^\top \ne 0$, then the last inequality holds if

$$ \sigma_{\max}^2 \begin{pmatrix} \sigma_{\max}(p_{11}) & \sigma_{\max}(p_{12}) \\ \sigma_{\max}(p_{21}) & \sigma_{\max}(p_{22}) \end{pmatrix} < \sigma_d(\tilde{D} + \tilde{D}^\top) \lambda_{\max} \begin{pmatrix} \bar{Q}_1 & \bar{Q}_{12} \\ \bar{Q}_{12}^\top & \bar{Q}_2 \end{pmatrix}, \quad (20) $$

where we used [8, Facts 6.3.28, 9.14.12]. The left-hand side in (20) is calculated explicitly to get (18). Hence item 1 is proved.
$\sigma_{q_1}(\bar{Q}_1)\sigma_{\max}(\bar{Q}_2)$. We can use also the equivalent conditions $\bar{Q}_{12} = \bar{Q}_{12}(P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12})^\dagger(P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12}) \iff \mathrm{Im}(\bar{Q}_{12}^\top) \subseteq \mathrm{Im}(P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12})$, and $\bar{Q}_1 \asymp \bar{Q}_{12}\bar{Q}_2^\dagger\bar{Q}_{12}^\top$. Assume that $\bar{Q}_1 \succ 0$, $\mathrm{rank}(\bar{Q}_2) = q_2$, $\bar{Q}_2 \neq 0$. Then the last inequality holds if $\sigma_{\max}^2(\bar{Q}_{12}) < \sigma_{q_2}(\bar{Q}_2)\sigma_{\max}(\bar{Q}_1)$. Thus the lemma is proved. $\square$

Using [10, Fact 3.13.15, equation (3.5.7], $\mathrm{Im}(\bar{Q}_{12}) \subseteq P_1\mathrm{Im}(L_1H_2) + \mathrm{Im}(P_{12}A_2) + \mathrm{Im}(A_1^\top P_{12}^\top) + \mathrm{Im}(H_1^\top L_2^\top P_2)$, and similarity for $\mathrm{Im}(\bar{Q}_{12}^\top)$. This can be used as a sufficient condition in the lemma.

**Corollary 2.** When $P_{12} = 0$, (13) (c) holds provided that

$$P_1 L_1 H_2 = -(P_2 L_2 H_1)^\top, \qquad (19)$$

*Proof.* The conditions guarantee that $\bar{Q}_{12} = 0$, hence $\bar{Q} = 0$, hence $-\tilde{A}^\top P - P \tilde{A} = \mathrm{diag}(Q_1, Q_2)$. $\square$

Let us now provide conditions for (13) (d) to be satisfied.

**Lemma 4.** Let the conditions in Lemmata 2 and 3 hold (hence (13) (b) and (c) are satisfied). The following statements hold:

1. Let $N \triangleq \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \left( \begin{pmatrix} D_1 + D_1^\top & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & D_2 + D_2^\top \end{pmatrix}^\dagger \right)^{\frac{1}{2}}$. Then (13) (d) holds true only if:

$$\mathrm{Im}(\mathrm{diag}(Q_1, Q_2) + \bar{Q}) \supseteq \mathrm{Im}(N) \qquad (20)$$

$$\lambda_i(\mathrm{diag}(Q_1, Q_2) + \bar{Q}) \geq \lambda_i(NN^\top) \geq 0, \quad \text{for all } i \in \{1, \dots, m\}, \qquad (21)$$

$$\mathrm{tr}(\mathrm{diag}(Q_1, Q_2) + \bar{Q}) \geq \mathrm{tr}(NN^\top) \geq 0. \qquad (22)$$

2. Let $E_1G_2 + G_1^\top E_2^\top = 0$, $P_{12} = (L_2H_1)^\top = L_1H_2$. Let us denote $D_{s,1} \triangleq D_1 + D_1^\top$, $D_{s,2} \triangleq D_2 + D_2^\top$. Then (13) (d) holds true only if:

$$\begin{align} (i) \quad & \mathrm{Im}(\bar{Q}_1) \supseteq \mathrm{Im}(p_{11}D_{s,1}^\top p_{11}^\top + p_{12}D_{s,2}^\top p_{12}^\top) \nonumber \\ (ii) \quad & \mathrm{Im}(\bar{Q}_{12} - p_{11}D_{s,1}^\top p_{21}^\top - p_{12}D_{s,2}^\top p_{22}^\top) \subseteq \mathrm{Im}(\bar{Q}_1 - p_{11}D_{s,1}^\top p_{11}^\top - p_{12}D_{s,2}^\top p_{12}^\top) \nonumber \\ (iii) \quad & \mathrm{Im}(\bar{Q}_2 - p_{21}D_{s,1}^\top p_{21}^\top - p_{22}D_{s,2}^\top p_{22}^\top) \supseteq \mathrm{Im}((\bar{Q}_{12}^\top - p_{21}D_{s,1}^\top p_{11}^\top - p_{22}D_{s,2}^\top p_{12}^\top) \times \nonumber \\ & \qquad (\bar{Q}_1 - p_{11}D_{s,1}^\top p_{11}^\top - p_{12}D_{s,2}^\top p_{12}^\top)^\dagger(\bar{Q}_{12} - p_{11}D_{s,1}^\top p_{21}^\top - p_{12}D_{s,2}^\top p_{22}^\top)) \tag{23} \end{align}$$

3. Then (13) (d) holds true if: $P_{12} = (L_2H_1)^\top = L_1H_2$, $\mathrm{rank}(P_{12}) = n_1$, $\sigma_{\max}^2(\bar{Q}_{12}) < \lambda_{\min}(\bar{Q}_1)\lambda_{\min}(\bar{Q}_2)$, $\mathrm{rank}(\tilde{D} + \tilde{D}^\top) = d$, $\tilde{D} + \tilde{D}^\top \neq 0$, and

$$\frac{\sigma_{\max}^2(p_{11}) + \sigma_{\max}^2(p_{12}) + \sigma_{\max}^2(p_{22}) + \sigma_{\max}^2(p_{21}) + \sqrt{\Delta}}{2} < \sigma_d(\tilde{D} + \tilde{D}^\top)\lambda_{\max} \begin{pmatrix} \bar{Q}_1 & \bar{Q}_{12} \\ \bar{Q}_{12}^\top & \bar{Q}_2 \end{pmatrix} \quad (24)$$
2. The proof is based on the proof of [8, Theorem 8.6.2]. The factorization of M follows from [8, Theorem 5.6.3], where $0 < \sigma_1(M) = \sigma_{\max}(M)$, $\sigma_r(M) > 0$ [8, section 5.6]. Let $L \stackrel{\Delta}{=} U_2^\top \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} N_1 \\ 0 \end{pmatrix}$. Using the reasoning in the proof of [8, Theorem 8.6.2], it is inferred that $N = ML$, and $NN^\top \le \lambda_{\max}(LL^\top)MM^\top$. Using [8, Corollary 9.6.5], we have:

$$ \begin{aligned} \lambda_{\max}(LL^\top) &= \sigma_{\max}^2(L) \\ &\le \sigma_{\max}^2(U_2^\top)\sigma_{\max}^2(U_2)\sigma_{\max}^2 \begin{pmatrix} N_1 \\ 0 \end{pmatrix}\sigma_{\max}^2(N_1^\top 0)\sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \\ &= \sigma_{\max}^4(U_2)\sigma_{\max}^4 \begin{pmatrix} N_1 \\ 0 \end{pmatrix}\sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} = \sigma_{\max}^4 \begin{pmatrix} N_1 \\ 0 \end{pmatrix}\sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \end{aligned} \quad (21) $$

since $U_2$ is unitary. Notice that since $T = \text{diag}(\sigma_1(M), \dots, \sigma_r(M))$, with $\sigma_1(M) = \sigma_{\max}(M) \ge \sigma_2(M) \ge \dots \ge \sigma_r(M) > 0$, we have $\sigma_{\max}(T^{-1}) = \sigma_r^{-1}(M)$. Therefore, $\sigma_{\max} \begin{pmatrix} N_1 \\ 0 \end{pmatrix}\sigma_r^{-1}(M) \le 1$ implies that $\lambda_{\max}(LL^\top) \le 1$, so that $MM^\top \ge NN^\top$. Note that the equality $\sigma_r(M) = \lambda_r^{\frac{1}{2}}(MM^\top) = |\lambda_r(\text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q})|^{\frac{1}{2}}$ yields (19). Hence item 2 is proved. $\square$

As we shall see in section 6.1, item 2 encompasses cases where the right-hand side of inequality (19) vanishes. We can now use the above four lemmata to arrive at the following main result of this section:

**Theorem 3.6.** (Passivity preservation: Extended Passivity Theorem) Let Assumption 2.1 hold, and consider the quadruples $(A_i, B_i, C_i, D_i)$ in (4), $i = 1, 2$, with the interconnection in (6). The quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive with storage function matrix $P$ defined in (9) if the sufficient conditions stated in Lemma 3.1, Lemma 3.2, Lemma 3.3, and in Lemma 3.5 hold, and at least one of the following items hold:

(a) $\sigma_{\max}^2(P_{12}) < \alpha_1\alpha_2\lambda_{\min}(P_1)\lambda_{\min}(P_2)$, when $P_1 > 0, P_2 > 0$,

(b) $P_1 \succcurlyeq 0$ and $\text{Im}(P_{12}) \subseteq \text{Im}(P_1)$ and $\alpha_2 P_2 \succcurlyeq \frac{1}{\alpha_1} P_{12}^\top P_1^\top P_{12}$.

*Proof.* The conditions and assumptions which are stated in Lemma 3.1, Lemma 3.2, Lemma 3.3, and Lemma 3.5 guarantee that (13a)-(13d) hold, and item (a) (respectively (b)) secures that $P > 0$ (resp. $P \succcurlyeq 0$) using Corollary A.1, [9, Proposition 10.2.5], and (9). $\square$

### 3.1.2. Sufficient conditions for block diagonal storage function matrix

As a particular case of Theorem 3.6, we can provide sufficient conditions such that the interconnection is passive with a storage function matrix having the form $P = \text{diag}(P_1, P_2)$. The advantage is that these conditions are much simpler to write.

**Corollary 3.7.** Suppose that Assumption 2.1 holds, and consider the quadruples $(A_i, B_i, C_i, D_i)$ in (4), $i = 1, 2$, with the interconnection in (6). The quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive with storage function matrix $P = \text{diag}(P_1, P_2)$, if

$$ E_1 G_2 = -G_1^\top E_2^\top, \quad (22a) $$

$$ P_1 L_1 G_2 - H_2^\top E_2^\top = 0, \quad (22b) $$

$$ P_2 L_2 G_1 - H_1^\top E_1^\top = 0, \quad (22c) $$

$$ P_1 L_1 H_2 + (P_2 L_2 H_1)^\top = 0. \quad (22d) $$

*Proof.* We basically show once again that (13) holds with $P = \text{diag}(P_1, P_2)$, i.e., $\alpha_1 = \alpha_2 = 1$. Let us first examine the inclusion in (14), which is equivalent to the following inclusion under the conditions (22b)-(22c):

$$ \text{Im} \begin{pmatrix} \alpha_1 B_1^\top P_1 - C_1 \\ 0 \end{pmatrix} + \text{Im} \begin{pmatrix} 0 \\ \alpha_2 B_2^\top P_2 - C_2 \end{pmatrix} \subseteq \text{Im} \begin{pmatrix} D_1 + D_1^\top \\ E_2 G_1 + G_2^\top E_1^\top \end{pmatrix} + \text{Im} \begin{pmatrix} E_1 G_2 + G_1^\top E_2^\top \\ D_2 + D_2^\top \end{pmatrix}. \quad (23) $$

For $\alpha_1 = \alpha_2 = 1$, the inclusion in (23) holds under Assumption 2.1 and hence, (13a) holds. For showing (13b), it is seen that the conditions listed in Lemma 3.2 hold true when (22a) is satisfied. Next, the condition (22d) guarantees that $\tilde{Q}_{12} = 0$, hence $\tilde{Q} = 0$, which further yields $-\tilde{A}^\top P - P \tilde{A} = \text{diag}(\alpha_1 Q_1, \alpha_2 Q_2)$, so that (13c) holds. The conditions (22b)-(22c) imply $P \tilde{B} - \tilde{C}^\top = \text{diag}(\alpha_1 P_1 B_1 - C_1^\top, \alpha_2 P_2 B_2 - C_2^\top)$. The condition $E_1 G_2 = -(E_2 G_1)^\top$ implies $\tilde{D} + \tilde{D}^\top = \text{diag}(D_1 + D_1^\top, D_2 + D_2^\top)$, while (22d) guarantees that $\tilde{Q} = 0$ in (10), hence $-\tilde{A}^\top P - P \tilde{A} = \text{diag}(\alpha_1 Q_1, \alpha_2 Q_2)$. Thus the interconnection is decoupled, and since both LCS are passive, (13d) is true. $\square$
with $\Delta = (\sigma_{\max}^2(p_{11}) + \sigma_{\max}^2(p_{12}) - \sigma_{\max}^2(p_{22}) - \sigma_{\max}^2(p_{21})) + 4(\sigma_{\max}(p_{11})\sigma_{\max}(p_{21}) + \sigma_{\max}(p_{12})\sigma_{\max}(p_{22}))$.

4. Let both sides of (13) (d) be decomposed as $MM^\top$ and $NN^\top$, respectively, so that (13) (d) is rewritten as $MM^\top \succ NN^\top$. Let $M = U_1 \begin{pmatrix} T & 0 \\ 0 & 0 \end{pmatrix} U_2$, where $U_1$ and $U_2$ are unitary matrices, $T = \text{diag}(\sigma_1(M), \dots, \sigma_r(M))$, $r = \text{rank}(M)$, and $U_1^\top N = \begin{pmatrix} N_1 \\ 0 \end{pmatrix}$. Assume that:

$$ \sigma_{\max} \begin{pmatrix} N_1 \\ 0 \end{pmatrix} \le |\lambda_r(\text{diag}(Q_1, Q_2) + \bar{Q})|, \quad (25) $$

then (13) (d) is satisfied.

*Proof.* Let us examine (13) (d). We have $-\tilde{A}^\top P - P \tilde{A} = \text{diag}(Q_1, Q_2) + \bar{Q}$, with $\bar{Q}$ in (10), and $P \tilde{B} - \tilde{C}^\top$ in (12), $\tilde{D} + \tilde{D}^\top$ in (11). Obviously the conditions in (c) are implied by the conditions in this lemma.

1. If the conditions in Lemmata 2 and 3 hold, then (13) (b) and (c) are satisfied. Then applying [10, Fact 7.17.24], both sides of (13) (d) are positive semidefinite and symmetric, thus they can be decomposed as $MM^\top$ and $NN^\top$ for some matrices $M$ and $N$, respectively. Hence (13) (d) is equivalently rewritten as $MM^\top \succ NN^\top$. From [9, Theorems 8.6.2, 2.4.3] this implies $\text{Im}(N) \subseteq \text{Im}(M) = \text{Im}(MM^\top)$, which is (20), a necessary condition for (13) (d). The necessary conditions (21) and (22) follow from [9, Theorem 8.4.9, Corollary 8.4.10] since both matrices are symmetric positive semidefinite.

2. Let us assume that $E_1 G_2 + G_1^\top E_2^\top = 0$. Then $(\tilde{D} + \tilde{D}^\top)^\top = \text{diag}(D_{s,1}^\top, D_{s,2}^\top)$ [9, Fact 6.5.21]. Thus the inequality in (13) (d) is rewritten as:

$$ \begin{pmatrix} \bar{Q}_1 - p_{11} D_{s,1}^\top p_{11}^\top - p_{12} D_{s,2}^\top p_{12}^\top & \bar{Q}_{12} - p_{11} D_{s,1}^\top p_{21}^\top - p_{12} D_{s,2}^\top p_{22}^\top \\ \bar{Q}_{12}^\top - p_{21} D_{s,1}^\top p_{11}^\top - p_{22} D_{s,2}^\top p_{12}^\top & \bar{Q}_2 - p_{21} D_{s,1}^\top p_{21}^\top - p_{22} D_{s,2}^\top p_{22}^\top \end{pmatrix} \succ 0. \quad (26) $$

Let us apply [9, Proposition 8.2.4] [10, Fact 8.4.3] to infer that (26) is equivalent to:

$$ \begin{align} & (i) \quad \bar{Q}_1 - p_{11} D_{s,1}^\top p_{11}^\top - p_{12} D_{s,2}^\top p_{12}^\top \succ 0 \tag{27} \\ & (ii) \quad \text{Im}(\bar{Q}_{12} - p_{11} D_{s,1}^\top p_{21}^\top - p_{12} D_{s,2}^\top p_{22}^\top) \subseteq \text{Im}(\bar{Q}_1 - p_{11} D_{s,1}^\top p_{11}^\top - p_{12} D_{s,2}^\top p_{12}^\top) \nonumber \\ & (iii) \quad \bar{Q}_2 - p_{21} D_{s,1}^\top p_{21}^\top - p_{22} D_{s,2}^\top p_{22}^\top \succ (\bar{Q}_{12}^\top - p_{21} D_{s,1}^\top p_{11}^\top - p_{22} D_{s,2}^\top p_{12}^\top) \times \nonumber \\ & \qquad \times (\bar{Q}_1 - p_{11} D_{s,1}^\top p_{11}^\top - p_{12} D_{s,2}^\top p_{12}^\top)^\top (\bar{Q}_{12} - p_{11} D_{s,1}^\top p_{21}^\top - p_{12} D_{s,2}^\top p_{22}^\top) \nonumber \end{align} $$

Recall that $\bar{Q}_1 = \bar{Q}_1^\top = Q_1 + P_{12} L_2 H_1 + H_1^\top L_2^\top P_{12}^\top$, $\bar{Q}_2 = \bar{Q}_2^\top = Q_2 + P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12}$. If condition $P_{12} = (L_2 H_1)^\top = L_1 H_2$ holds, then $\bar{Q}_1 \succ 0$ and $\bar{Q}_2 \succ 0$. Also $p_{11} D_{s,1}^\top p_{11}^\top + p_{12} D_{s,2}^\top p_{12}^\top \succ 0$ [10, Proposition 8.1.7]. Using [10, Fact 7.17.24], there exists $M_1$ and $N_1$ such that the decompositions $\bar{Q}_1 = M_1 M_1^\top$ and $p_{11} D_{s,1}^\top p_{11}^\top + p_{12} D_{s,2}^\top p_{12}^\top = N_1 N_1^\top$ hold. The
### 3.1.3. Some necessary conditions for passivity

As stated earlier, passivity of the interconnected system is characterized by the expressions (13a)–(13d). Among these, Lemma 3.1 already provided necessary conditions for (13a). The inequalities in (13b) and (13c) correspond to positive semidefiniteness of the matrices $\tilde{D}+\tilde{D}^\top$ and $-\tilde{A}^\top P - P\tilde{A}$, respectively. In the proof of Lemma 3.2, expressions (16) and (17) provide necessary and sufficient conditions for $\tilde{D}+\tilde{D}^\top$ to be positive semidefinite, and in particular the necessary conditions. Similar expressions can be written for matrix $-\tilde{A}^\top P - P\tilde{A}$, which would also provide necessary conditions for (13c) to hold. In essence, what remains is to analyze the necessary conditions for (13d), and this is the topic of our next result.

**Proposition 3.8.** Suppose that the matrices $\tilde{D}+\tilde{D}^\top$ and $-\tilde{A}^\top P - P\tilde{A} = \text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q}$ are positive semidefinite, that is, (13b) and (13c) hold. The following two statements hold:

1. Let

$$N \triangleq \begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \left( \begin{pmatrix} D_1 + D_1^\top & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & D_2 + D_2^\top \end{pmatrix} \right)^{\frac{1}{2}},$$

then (13d) holds true only if:

$$\text{Im}(\text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q}) \supseteq \text{Im}(N) \quad (24a)$$

$$\lambda_i(\text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q}) \geq \lambda_i(NN^\top) \geq 0, \text{ for all } i \in \{1, \dots, m\}, \quad (24b)$$

$$\text{tr}(\text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q}) \geq \text{tr}(NN^\top) \geq 0. \quad (24c)$$

2. If $\tilde{D} + \tilde{D}^\top$ is block-diagonal and, in the description of $-\tilde{A}^\top P - P\tilde{A}$, the diagonal terms of $\bar{Q}$ are identically zero, then (13d) holds true only if:

$$\text{Im}(\bar{Q}_1) \supseteq \text{Im}(p_{11} D_{s,1}^\dagger p_{11}^\top + p_{12} D_{s,2}^\dagger p_{12}^\top) \quad (25a)$$

$$\text{Im}(\bar{Q}_{12} - p_{11} D_{s,1}^\dagger p_{21}^\top - p_{12} D_{s,2}^\dagger p_{22}^\top) \subseteq \text{Im}(\bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top) \quad (25b)$$

$$\text{Im}(\bar{Q}_2 - p_{21} D_{s,1}^\dagger p_{21}^\top - p_{22} D_{s,2}^\dagger p_{22}^\top) \supseteq \text{Im}((\bar{Q}_{12} - p_{21} D_{s,1}^\dagger p_{11}^\top - p_{22} D_{s,2}^\dagger p_{12}^\top) \times (\bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top)^\dagger (\bar{Q}_{12} - p_{11} D_{s,1}^\dagger p_{21}^\top - p_{12} D_{s,2}^\dagger p_{22}^\top)) \quad (25c)$$

where we used the notation $D_{s,1} \triangleq D_1 + D_1^\top$ and $D_{s,2} \triangleq D_2 + D_2^\top$.

*Proof.* Let us examine (13d). We have $-\tilde{A}^\top P - P\tilde{A} = \text{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \bar{Q}$, with $\bar{Q}$ in (10), and $P\tilde{B} - \tilde{C}^\top$ in (12), $\tilde{D} + \tilde{D}^\top$ in (11).

1. We start with the case when (13b) and (13c) are satisfied. Then applying [9, Fact 7.17.24], both sides of (13d) are positive semidefinite and symmetric, thus they can be decomposed as $MM^\top$ and $NN^\top$ for some matrices $M$ and $N$, respectively. Hence (13d) is equivalently rewritten as $MM^\top \succ NN^\top$. From [8, Theorems 8.6.2, 2.4.3] this implies $\text{Im}(N) \subseteq \text{Im}(M) = \text{Im}(MM^\top)$, which is (24a), a necessary condition for (13d). The necessary conditions (24b) and (24c) follow from [8, Theorem 8.4.9, Corollary 8.4.10] since both matrices are symmetric positive semidefinite.

2. If $\tilde{D} + \tilde{D}^\top$ is block-diagonal, then under Assumption 2.1, (13b) is true and $(\tilde{D} + \tilde{D}^\top)^\dagger = \text{diag}(D_{s,1}^\dagger, D_{s,2}^\dagger)$ [8, Fact 6.5.21]. The inequality in (13d) is rewritten as:

$$\begin{pmatrix} \bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top & \bar{Q}_{12} - p_{11} D_{s,1}^\dagger p_{21}^\top - p_{12} D_{s,2}^\dagger p_{22}^\top \\ \bar{Q}_{12}^\top - p_{21} D_{s,1}^\dagger p_{11}^\top - p_{22} D_{s,2}^\dagger p_{12}^\top & \bar{Q}_2 - p_{21} D_{s,1}^\dagger p_{21}^\top - p_{22} D_{s,2}^\dagger p_{22}^\top \end{pmatrix} \succ 0. \quad (26)$$

Let us apply [9, Proposition 10.2.5, Fact 8.4.3] to infer that (26) is equivalent to:

$$\bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top \succ 0 \quad (27a)$$

$$\text{Im}(\bar{Q}_{12} - p_{11} D_{s,1}^\dagger p_{21}^\top - p_{12} D_{s,2}^\dagger p_{22}^\top) \subseteq \text{Im}(\bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top) \quad (27b)$$

$$\bar{Q}_2 - p_{21} D_{s,1}^\dagger p_{21}^\top - p_{22} D_{s,2}^\dagger p_{22}^\top \succ (\bar{Q}_{12}^\top - p_{21} D_{s,1}^\dagger p_{11}^\top - p_{22} D_{s,2}^\dagger p_{12}^\top) \times (\bar{Q}_1 - p_{11} D_{s,1}^\dagger p_{11}^\top - p_{12} D_{s,2}^\dagger p_{12}^\top)^\dagger (\bar{Q}_{12} - p_{11} D_{s,1}^\dagger p_{21}^\top - p_{12} D_{s,2}^\dagger p_{22}^\top). \quad (27c)$$
inequality (i) is $M_1 M_1^\top \succcurlyeq N_1 N_1^\top \implies \mathrm{Im}(N_1) \subseteq \mathrm{Im}(M_1)$ [10, Theorem 10.6.2]. Therefore using [9, Theorem 2.4.3], it follows that condition (i) in (23) is necessary for (27) (i). Similarly, condition (iii) in (23) is necessary for in (27) (iii).

3. Let $\tilde{Q}_2 \succcurlyeq 0$, $\tilde{Q}_1 \succcurlyeq \tilde{Q}_{12} \tilde{Q}_2^{-1} \tilde{Q}_{12}^\top$, or $\tilde{Q}_1 \succcurlyeq 0$, $\tilde{Q}_2 \succcurlyeq \tilde{Q}_{12}^\top \tilde{Q}_1^{-1} \tilde{Q}_{12}$. If $Q_1 \succcurlyeq 0$ and $Q_2 \succcurlyeq 0$ (both subsystems are just passive), then $\tilde{Q}_1 \succcurlyeq 0$ and $\tilde{Q}_2 \succcurlyeq 0$ hold if $P_{12} = (L_2 H_1)^\top = L_1 H_2$, and $\mathrm{rank}(P_{12}) = n_1$. The other two inequalities hold if $\sigma_{\max}^2(\tilde{Q}_{12}) < \lambda_{\min}(\tilde{Q}_2) \lambda_{\min}(\tilde{Q}_1)$ [9, Fact 8.19.2]. Then applying Corollary 8, the inequality (13) (d) holds if $\sigma_{\max}^2(P \tilde{B} - \tilde{C}^\top) \sigma_{\max}((\tilde{D}+\tilde{D}^\top)^\dagger) < \sigma_{\max}(\mathrm{diag}(Q_1, Q_2)+\tilde{Q})$. Let $\mathrm{rank}(\tilde{D}+\tilde{D}^\top) = d$, $\tilde{D}+\tilde{D}^\top \neq 0$, then the last inequality holds if

$$ \sigma_{\max}^2 \begin{pmatrix} \sigma_{\max}(p_{11}) & \sigma_{\max}(p_{12}) \\ \sigma_{\max}(p_{21}) & \sigma_{\max}(p_{22}) \end{pmatrix} < \sigma_d(\tilde{D} + \tilde{D}^\top) \lambda_{\max} \begin{pmatrix} \tilde{Q}_1 & \tilde{Q}_{12} \\ \tilde{Q}_{12}^\top & \tilde{Q}_2 \end{pmatrix}, \quad (28) $$

where we used [9, Facts 6.3.28, 9.14.12]. The left-hand side in (28) is easily calculated explicitly to get (24). Hence item 4.3 is proved.

4. The proof is based on the proof of [9, Theorem 8.6.2]. The factorization of $M$ follows from [9, Theorem 5.6.3], where $0 < \sigma_1(M) = \sigma_{\max}(M)$, $\sigma_r(M) > 0$ [9, section 5.6]. Let $L \triangleq U_2^\top \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} N_1 \\ 0 \end{pmatrix}$. Using the reasoning in the proof of [9, Theorem 8.6.2], it is inferred that $N = ML$, and $NN^\top \le \lambda_{\max}(LL^\top)MM^\top$. Using [9, Corollary 9.6.5], we have:

$$ \begin{align} \lambda_{\max}(LL^\top) &= \sigma_{\max}^2(L) \nonumber \\ &\le \sigma_{\max}^2(U_2^\top) \sigma_{\max}^2(U_2) \sigma_{\max}^2 \begin{pmatrix} N_1 \\ 0 \end{pmatrix} \sigma_{\max}^2(N_1^\top \ 0) \sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \tag{29} \\ &= \sigma_{\max}^4(U_2) \sigma_{\max}^4 \begin{pmatrix} N_1 \\ 0 \end{pmatrix} \sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} = \sigma_{\max}^4 \begin{pmatrix} N_1 \\ 0 \end{pmatrix} \sigma_{\max}^4 \begin{pmatrix} T^{-1} & 0 \\ 0 & 0 \end{pmatrix} \nonumber \end{align} $$

since $U_2$ is unitary. Notice that since $T = \mathrm{diag}(\sigma_1(M), \dots, \sigma_r(M))$, with $\sigma_1(M) = \sigma_{\max}(M) \ge \sigma_2(M) \ge \dots \ge \sigma_r(M) > 0$, we have $\sigma_{\max}(T^{-1}) = \sigma_r^{-1}(M)$. Therefore, $\sigma_{\max} \begin{pmatrix} N_1 \\ 0 \end{pmatrix} \sigma_r^{-1}(M) \le 1$ implies that $\lambda_{\max}(LL^\top) \le 1$, so that $MM^\top \ge NN^\top$. The equality $\sigma_r(M) = \lambda_r^{\frac{1}{2}}(MM^\top) = |\lambda_r(\mathrm{diag}(Q_1, Q_2) + \tilde{Q})|$ yields (25). $\square$

Condition (21) involves the singular values of $N$. Sufficient conditions such that it holds can be derived, using [10, Corollary 11.6.8] to consider both matrices in $N$ separately. It is also worth recalling that most of these criteria simplify when $D_i = 0$ ($\Rightarrow P_i B_i - C_i^\top = 0$) which occurs often in practice. For instance $(\tilde{D} + \tilde{D}^\top)^\dagger$ can be calculated using [10, Fact 8.9.31], simplifying the expression for $N$.

**Lemma 5.** Let $P_{12} = 0$, assume that (16), (19) and $E_1 G_2 = -(E_2 G_1)^\top$ are satisfied. Then (13) (d) is true.
Recall that if the diagonal terms of $\tilde{Q}$ are zero, then $\tilde{Q}_1 = \tilde{Q}_1^\top = \alpha_1 Q_1$ and $\tilde{Q}_2 = \tilde{Q}_2^\top = \alpha_2 Q_2$. Using Assumption 2.1, we have $\tilde{Q}_1 \succcurlyeq 0$ and $\tilde{Q}_2 \succcurlyeq 0$. Also $p_{11}D_{s,1}^\dagger p_{11}^\top + p_{12}D_{s,2}^\dagger p_{12}^\top \succcurlyeq 0$ [9, Proposition 8.1.7 xxx]). Using [9, Fact 7.17.24], there exist $M_1$ and $N_1$ such that the decompositions $\tilde{Q}_1 = M_1 M_1^\top$ and $p_{11}D_{s,1}^\dagger p_{11}^\top + p_{12}D_{s,2}^\dagger p_{12}^\top = N_1 N_1^\top$ hold. The inequality (27a) is $M_1 M_1^\top \succcurlyeq N_1 N_1^\top \implies \mathrm{Im}(N_1) \subseteq \mathrm{Im}(M_1)$ [9, Theorem 10.6.2]. Therefore using [8, Theorem 2.4.3], it follows that condition in (25a) is necessary for (27a). Similarly, condition in (25c) is necessary for inequality (27c). $\square$

Condition (24a) is not practical, and it is of interest to analyse the influence of (15) on its satisfaction. We have $\mathrm{Im}((\tilde{D} + \tilde{D}^\top)^\dagger)^{\frac{1}{2}} = \mathrm{Im}(\tilde{D} + \tilde{D}^\top)$, using [8, Proposition 6.1.6, Theorem 2.4.3], and $\mathrm{Im}(N) \subseteq \mathrm{Im}\begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix}$ [8, Lemma 2.4.1].

Assume now that (15) holds, which implies that (14) holds, equivalently (see (12)) $\mathrm{Im}\begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \subseteq \mathrm{Im}(\tilde{D} + \tilde{D}^\top)$ [9, Fact 8.4.3]. Therefore, if we have $\mathrm{Im}(\tilde{D} + \tilde{D}^\top) = \mathrm{Im}\begin{pmatrix} D_1 + D_1^\top \\ E_2 G_1 + G_2^\top E_1^\top \end{pmatrix} + \mathrm{Im}\begin{pmatrix} E_1 G_2 + G_1^\top E_2^\top \\ D_2 + D_2^\top \end{pmatrix} \subseteq \mathrm{Im}(\mathrm{diag}(\alpha_1 Q_1, \alpha_2 Q_2) + \tilde{Q})$, it is inferred that the necessary condition (24a) holds. Condition (24b) involves the singular values of $N$. Sufficient conditions such that it holds can be derived, using [9, Corollary 11.6.8] to consider both matrices in $N$ (as defined in Proposition 3.8 item 1) separately. It is also worth recalling that most of these criteria simplify when $D_i = 0$ ($\Rightarrow P_i B_i - C_i^\top = 0$) which occurs often in practice. For instance $(\tilde{D} + \tilde{D}^\top)^\dagger$ can be calculated using [9, Fact 8.9.31], simplifying the expression for $N$.

## 3.2. Strong Passivity Preservation after Interconnection

Let us now state a result about the interconnection of strongly passive subsystems. Hence the matrix inequality (7) is modified as indicated in Appendix B, and it is assumed that $P_i = P_i^\top > 0$, $i = 1, 2$, are solutions of the the matrix inequality which describes strong passivity.

**Theorem 3.9.** (Preservation of strong passivity) Assume that both quadruples $(A_i, B_i, C_i, D_i)$ in (4) are strongly passive. Then the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is strongly passive with $P > 0$ in (9), if conditions (28), (29) and (30) are satisfied, where:

$$ \min(\lambda_{\min}(D_1 + D_1^\top), \lambda_{\min}(D_2 + D_2^\top)) \geq \frac{1}{2} \lambda_{\max}^{\frac{1}{2}}(E_1 G_2 G_2^\top E_1^\top) + \frac{1}{2} \lambda_{\max}^{\frac{1}{2}}(E_2 G_1 G_1^\top E_2^\top) \quad (28) $$

and:

$$ \sigma_{\max}(\tilde{Q}) \leq \min(\alpha_1 \lambda_{\min}(Q_1), \alpha_2 \lambda_{\min}(Q_2)), \quad (29) $$

with $\tilde{Q}$ in (10), $Q_1 > 0$ and $Q_2 > 0$ (these conditions hold if $P_{12} = 0$ and $\alpha_1 P_1 L_1 H_2 = -(\alpha_2 P_2 L_2 H_1)^\top$, since then $\tilde{Q} = 0$ hence $-\tilde{A}^\top P - P \tilde{A} > 0$), and:

$$ \sigma_{\max} \left\{ \mathrm{diag}(\alpha_1 P_1 B_1 - C_1^\top, \alpha_2 P_2 B_2 - C_2^\top) + \begin{pmatrix} P_{12} L_2 G_1 & \alpha_1 P_1 L_1 G_2 + P_{12} B_2 - H_1^\top E_2^\top \\ P_{12}^\top B_1 + \alpha_2 P_2 L_2 G_1 - H_2^\top E_1^\top & P_{12}^\top L_1 G_2 \end{pmatrix} \right\} \leq \\ \left( \lambda_{\min}(\mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top)) + \lambda_{\min} \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & 0 \end{pmatrix} \right)^{\frac{1}{2}} \left( \lambda_{\min}(\mathrm{diag}(\alpha_1 Q_1, \alpha_2 Q_2)) - \lambda_{\min}(\tilde{Q}) \right)^{\frac{1}{2}}, \quad (30) $$

and if, in addition, the conditions in item (a) of Theorem 3.6 hold.

**Proof.** Strong passivity of both quadruples implies that $D_1 > 0$ and $D_2 > 0$. Consider (11). We have

$$ \tilde{D} + \tilde{D}^\top = \mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top) + \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ G_2^\top E_1^\top + E_2 G_1 & 0 \end{pmatrix} $$

Using [8, Fact 9.14.15], it follows that

$$ \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ E_2 G_1 & 0 \end{pmatrix} < \frac{1}{2\sigma_{\max}(\mathrm{diag}((D_1+D_1^\top)^{-1}, (D_2+D_2^\top)^{-1}))} = \frac{1}{2}\lambda_{\min}(\mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top)) $$
Proof. The conditions (16) imply $P\tilde{B} - \tilde{C}^\top = \text{diag}(P_1B_1 - C_1^\top, P_2B_2 - C_2^\top)$. The condition $E_1G_2 = -(E_2G_1)^\top$ implies $\tilde{D} + \tilde{D}^\top = \text{diag}(D_1 + D_1^\top, D_2 + D_2^\top)$, while (19) guarantees $-\tilde{A}P - P\tilde{A} = \text{diag}(Q_1, Q_2)$. Thus the interconnection is decoupled, and since both LCS are passive, (13) (d) is true. $\square$

Some comments on Lemma 4: The necessary conditions (23) may not be easy to check. If $D_{s,1} = 0$ and $D_{s,2} = 0$, they simplify as follows: (i) always holds, (ii) implies $\text{Im}(\tilde{Q}_{12}) \subseteq \text{Im}(Q_1)+\text{Im}(P_{12}L_2H_1)+\text{Im}(P_{12}L_2H_1)^\top$, (iii) implies $\text{Im}(\tilde{Q}_2) \subseteq \text{Im}(\tilde{Q}_{12}^\top\tilde{Q}_1\tilde{Q}_{12}) \Rightarrow \text{Im}(\tilde{Q}_2) \subseteq \text{Im}(\tilde{Q}_{12}^\top)$, which furnish simpler necessary conditions. The conditions simplify also if $\tilde{Q}_1 > 0$ and $\tilde{Q}_2 > 0$, as it may occur (see section 6), using Corollary 8 to analyze (23) (ii) and (iii) (then (i) is always true).

Similarly, condition (20) in item 4.1 in Lemma 4, is not very practical, and it is of interest to analyse the influence of (15) on its satisfaction. We have $\text{Im}((\tilde{D}+\tilde{D}^\top)^\dagger)^{\frac{1}{2}} = \text{Im}(\tilde{D}+\tilde{D}^\top)$, using [9, Proposition 6.1.6, Theorem 2.4.3], and $\text{Im}(N) \subseteq \text{Im}\begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix}$ [9, Lemma 2.4.1].

Assume now that (15) holds, which implies that (14) holds, equivalently $\text{Im}\begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \subseteq \text{Im}(\tilde{D}+\tilde{D}^\top)$ [10, Fact 8.4.3]. Thus if $\text{Im}(\tilde{D}+\tilde{D}^\top) = \text{Im}\begin{pmatrix} D_1+D_1^\top & E_1G_2+E_1^\top E_2^\top \\ E_2G_1+G_2^\top E_1^\top \end{pmatrix} + \text{Im}\begin{pmatrix} D_2+D_2^\top & E_2G_2+G_2^\top E_1^\top \end{pmatrix} \subseteq \text{Im}(\text{diag}(Q_1, Q_2) + \tilde{Q})$, it is inferred that the necessary condition (20) holds. To summarize: (15) $\Rightarrow$ (14) $\Leftrightarrow$ $\text{Im}\begin{pmatrix} p_{11} & p_{12} \\ p_{21} & p_{22} \end{pmatrix} \subseteq \text{Im}(\tilde{D}+\tilde{D}^\top) \Rightarrow \text{Im}(N) \subseteq \text{Im}(\tilde{D}+\tilde{D}^\top) \Rightarrow [\text{Im}(\tilde{D}+\tilde{D}^\top) \subseteq \text{Im}(\text{diag}(Q_1, Q_2) + \tilde{Q}) \Rightarrow (20)]$, while (20) $\Leftarrow$ (13) (d). The next proposition concatenates Lemmata 1 through 4

**Proposition 1.** (passivity preservation) Let Assumption 1 hold, consider the quadruples $(A_i, B_i, C_i, D_i)$ in (4), $i = 1, 2$, and the interconnection in (6). Let $P = \begin{pmatrix} P_1 & P_{12} \\ P_{12}^\top & P_2 \end{pmatrix}$. (i) Assume that $\sigma_{\max}^2(P_{12}) < \lambda_{\min}(P_1)\lambda_{\min}(P_2)$ if $P_1 > 0$, $P_2 > 0$, or $P \ge 0$ if $P_1 \ge 0$ and $\text{Im}(P_{12}) \subseteq \text{Im}(P_1)$ and $P_2 \ge P_{12}^\top P_{12}$. It is inferred that if the conditions in Lemmata 1 through 4 hold, then the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive with storage function matrix $P = \begin{pmatrix} P_1 & P_{12} \\ P_{12}^\top & P_2 \end{pmatrix}$. (ii) If the conditions in items 4.1 or 4.2 in Lemma 4 do not hold, then it is not passive. (iii) It is passive with $P = \text{diag}(P_1, P_2)$, if $E_1G_2 = -G_1^\top E_2^\top$, $P_1L_1G_2 - H_1^\top E_2^\top = 0$, $P_2L_2G_1 - H_2^\top E_1^\top = 0$, and $P_1L_1H_2 + (P_2L_2H_1)^\top = 0$.

Proof. The conditions guarantee that (13) hold, and (i) secures that $P > 0$ or $P \ge 0$ using Corollary 8 and [9, Proposition 8.2.4]. Assertion (iii) follows from Lemma 5. It is also possible to check that (20), (21) and (22) hold under these conditions, due to the passivity of each subsystem. $\square$
guarantees from Corollary A.1 that $\tilde{D} > 0$. Still using [8, Fact 9.14.15] we have

$$ \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ E_2 G_1 & 0 \end{pmatrix} = \sigma_{\max} \left( \begin{pmatrix} 0 & E_1 G_2 \\ 0 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ E_2 G_1 & 0 \end{pmatrix} \right) \le \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ 0 & 0 \end{pmatrix} + \sigma_{\max} \begin{pmatrix} 0 & 0 \\ E_2 G_1 & 0 \end{pmatrix}. $$

Now using [8, Fact 4.10.12, Proposition 5.5.20 v)]:

$$ \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ 0 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}} \begin{pmatrix} E_1 G_2 G_2^\top E_1^\top & 0 \\ 0 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}} (E_1 G_2 G_2^\top E_1^\top). $$

Similarly, $\sigma_{\max} \begin{pmatrix} 0 & 0 \\ E_2 G_1 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}} (E_2 G_1 G_1^\top E_2^\top)$. Continuing:

$$ \lambda_{\min}(\mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top)) = \min(\lambda_{\min}(D_1 + D_1^\top), \lambda_{\min}(D_2 + D_2^\top)). \quad (31) $$

Hence (28) guarantees that $\tilde{D} > 0$ ($\Leftrightarrow \tilde{D} + \tilde{D}^\top > 0$), which guarantees in passing that (13a) holds trivially. Obviously this is a conservative criterion since $E_1 G_2 + G_1^\top E_2^\top = 0$ guarantees $\tilde{D} + \tilde{D}^\top > 0$ also.

The second condition is a consequence of $-\tilde{A}^\top P - P\tilde{A} = \mathrm{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q}$, and applying Corollary A.1. It follows that $-\tilde{A}^\top P - P\tilde{A} > 0$ if $\sigma_{\max}(\tilde{Q}) \le \frac{1}{\sigma_{\max}(\mathrm{diag}(\frac{1}{\alpha_1}Q_1^{-1}, \frac{1}{\alpha_2}Q_2^{-1}))}$. Since $Q_i = Q_i^\top > 0$, we have

$$ \sigma_{\max}(\mathrm{diag}(\frac{1}{\alpha_1}Q_1^{-1}, \frac{1}{\alpha_2}Q_2^{-1})) = \lambda_{\max}(\mathrm{diag}(\frac{1}{\alpha_1}Q_1^{-1}, \frac{1}{\alpha_2}Q_2^{-1})) = \lambda_{\min}^{-1}(\mathrm{diag}(\alpha_1 Q_1, \alpha_2 Q_2)). \quad (32) $$

Then (29) follows which guarantees that $-\tilde{A}^\top P - P\tilde{A} > 0$. Now, using [9, Proposition 10.2.5] the inequality in (8) is satisfied strictly if and only if

$$ - \tilde{A}^\top P - P\tilde{A} - (P\tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^{-1}(P\tilde{B} - \tilde{C}^\top)^\top > 0, \quad (33) $$

and $-\tilde{A}^\top P - P\tilde{A} > 0$ (which is assured by the previous step). Using Corollary A.1, it is inferred that (33) holds if

$$ \begin{aligned} \|P\tilde{B} - \tilde{C}^\top\|_{2,2}^2 \|(\tilde{D} + \tilde{D}^\top)^{-1}\|_{2,2} &\le \frac{1}{\sigma_{\max}((-\tilde{A}^\top P - P\tilde{A})^{-1})} \\ &= \frac{1}{\lambda_{\max}((-\tilde{A}^\top P - P\tilde{A})^{-1})} = \frac{1}{\lambda_{\min}^{-1}(-\tilde{A}^\top P - P\tilde{A})} \end{aligned} \quad (34) $$

(where it is used that the $\|\cdot\|_{2,2}$ norm is submultiplicative). From [9, Fact 10.12.17], this is equivalent to $\sigma_{\max}((\tilde{D} + \tilde{D}^\top)^{-1}) = \lambda_{\max}((\tilde{D} + \tilde{D}^\top)^{-1}) = \frac{1}{\lambda_{\min}(\tilde{D} + \tilde{D}^\top)} \le \frac{\lambda_{\min}(-\tilde{A}^\top P - P\tilde{A})}{\sigma_{\max}^2(P\tilde{B} - \tilde{C}^\top)}$, which is equivalently rewritten as $\lambda_{\min}(\tilde{D} + \tilde{D}^\top)\lambda_{\min}(-\tilde{A}^\top P - P\tilde{A}) \ge \sigma_{\max}^2(P\tilde{B} - \tilde{C}^\top)$. Using [8, Fact 5.12.2], (12), (11), inequality (34) is implied by (30). The last condition (item (a) in Theorem 3.6) guarantees $P > 0$ from [9, Fact 10.12.17]. $\square$

Next, we provide sufficient conditions for strong passivity with storage function matrix being block diagonal.

**Corollary 3.10.** Let both quadruples $(A_i, B_i, C_i, D_i)$ in (4) be strongly passive. Then the interconnection quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is strongly passive with storage function matrix $P = \mathrm{diag}(P_1, P_2)$, if the four equations in (22) hold: (i) $E_1 G_2 = -(E_2 G_1)^\top$, (ii) $P_1 L_1 H_2 = -(P_2 L_2 H_1)^\top$, (iii) $P_1 L_1 G_2 = (E_2 H_1)^\top$ and $P_2 L_2 G_1 = (E_1 H_2)^\top$.

*Proof.* Under the condition (22a), it follows from (31) that $\tilde{D} + \tilde{D}^\top > 0$. Equation (22d) implies that $\tilde{Q}_{12} = 0$ in (10) since $\alpha_1 = \alpha_2 = 1$ in (9), thus (13c) holds with strict inequality. Equations (22b)-(22c) imply that $P\tilde{B} - \tilde{C}^\top = \mathrm{diag}(P_1 B_1 - C_1^\top, P_2 B_2 - C_2^\top)$. From the strong passivity of both quadruples, it follows that $\begin{pmatrix} -\tilde{A}^\top P - P\tilde{A} & -P\tilde{B} + \tilde{C}^\top \\ (-P\tilde{B} + \tilde{C}^\top)^\top & \tilde{D} + \tilde{D}^\top \end{pmatrix} > 0$, which ends the proof (see Appendix B). $\square$

It is easily deduced from $\tilde{D} + \tilde{D}^\top > 0$ (strict version of (13b)) and [16, Theorem A.65] that strong passivity of the interconnection quadruple implies that $D_i + D_i^\top > 0 \Leftrightarrow D_i > 0$ for $i = 1, 2$. The extension of Corollary 3.10 with $\alpha_1 \ne 1$ and $\alpha_2 \ne 1$ is not tackled due to space limitations.
## 2.2 Strong Passivity Preservation after Interconnection

Let us now state a result about the interconnection of strongly passive subsystems.

**Proposition 2.** (preservation of strong passivity) Assume that both subsystems in (4) are strongly passive. Then the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is strongly passive with $P = \begin{pmatrix} P_1 & P_{12} \\ P_{12}^\top & P_2 \end{pmatrix} > 0$ where $P_1 > 0$ and $P_2 > 0$ are solutions of (7), if conditions (30), (31) and (32) are verified, where:

$$ \min(\lambda_{\min}(D_1 + D_1^\top), \lambda_{\min}(D_2 + D_2^\top)) \geq \frac{1}{2}\lambda_{\max}^{\frac{1}{2}}(E_1 G_2 G_2^\top E_1^\top) + \frac{1}{2}\lambda_{\max}^{\frac{1}{2}}(E_2 G_1 G_1^\top E_2^\top) \quad (30) $$

(then $\tilde{D} + \tilde{D}^\top > 0$), and:

$$ \sigma_{\max}(\bar{Q}) \leq \min(\lambda_{\min}(Q_1), \lambda_{\min}(Q_2)), \quad (31) $$

with $\bar{Q}$ in (10), $Q_1 > 0$ and $Q_2 > 0$ (these conditions hold if $P_{12} = 0$ and $P_1 L_1 H_2 = -(P_2 L_2 H_1)^\top$, since then $\bar{Q} = 0$ hence $-\tilde{A}^\top P - P \tilde{A} > 0$), and:

$$ \sigma_{\max} \left\{ \begin{array}{l} \mathrm{diag}(P_1 B_1 - C_1^\top, P_2 B_2 - C_2^\top) \\ \quad + \begin{pmatrix} P_{12} L_2 G_1 & P_1 L_1 G_2 + P_{12} B_2 - H_1^\top E_2^\top \\ P_{12}^\top B_1 + P_2 L_2 G_1 - H_2^\top E_1^\top & P_{12}^\top L_1 G_2 \end{pmatrix} \end{array} \right\} \\ \leq \left( \lambda_{\min}(\mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top)) + \lambda_{\min} \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & 0 \end{pmatrix} \right)^{\frac{1}{2}} \times \left( \lambda_{\min}(\mathrm{diag}(Q_1, Q_2)) + \lambda_{\min}(\bar{Q}) \right)^{\frac{1}{2}}, \quad (32) $$

and the conditions in item (i) in Proposition 1 hold.

*Proof.* Strong passivity of both subsystems implies that $D_1 > 0$ and $D_2 > 0$. Consider (11). We have

$$ \tilde{D} + \tilde{D}^\top = \mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top) + \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ G_2^\top E_1^\top + E_2 G_1 & 0 \end{pmatrix} $$

Using [9, Fact 9.14.15], it follows that

$$ \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ E_2 G_1 & 0 \end{pmatrix} < \frac{1}{2\sigma_{\max}(\mathrm{diag}((D_1+D_1^\top)^{-1}, (D_2+D_2^\top)^{-1}))} = \frac{1}{2}\lambda_{\min}(\mathrm{diag}(D_1+D_1^\top, D_2+D_2^\top)) $$

guarantees from Corollary 8 that $\tilde{D} > 0$. Still using [9, Fact 9.14.15] we have $\sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ E_2 G_1 & 0 \end{pmatrix} = \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ 0 & 0 \end{pmatrix} + \sigma_{\max} \begin{pmatrix} 0 & 0 \\ E_2 G_1 & 0 \end{pmatrix}$. Now using [9,
### 3.3. Strict State Passivity Preservation after Interconnection

The third result in this section concerns the interconnection of two LCS with strictly state passive quadruples $(A_i, B_i, C_i, D_i)$ (see Appendix B for the corresponding matrix inequality (7)). The conditions in (13c) and (13d) are replaced by conditions for strict state passivity, given below in (37a) and (37b), respectively. Conditions (13a) and (13b) remain unchanged.

**Theorem 3.11.** (Preservation of strict state passivity) Assume that both quadruples $(A_i, B_i, C_i, D_i)$ in (4) are strictly state passive with constants $\mu_1 > 0$, $\mu_2 > 0$, with $P_i > 0$ the solution of the strict state passivity LMI, $i = 1, 2$. Let P be defined as in (9). Suppose that:

(SS-1) the inclusion in (14) holds,

(SS-2) one of the conditions stated in Theorem 3.6 (item (a) or (b)), and the conditions listed in Lemma 3.2 (items 1, 2, and 3) hold,

(SS-3) the inequalities in (35) hold:

$$ (\mu_1 - \mu)\alpha_1 P_1 - P_{12}L_2H_1 - H_1^\top L_2^\top P_{12}^\top \geq 0, \quad (35a) $$

$$ (\mu_2 - \mu)\alpha_2 P_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12} \geq 0, \quad (35b) $$

$$ \mathrm{Im}(\bar{Q}_{12}^\top - \mu P_{12}^\top) \subseteq \mathrm{Im}((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2), \quad (35c) $$

$$ (\mu_1 - \mu)\alpha_1 P_1 - P_{12}L_2H_1 - H_1^\top L_2^\top P_{12}^\top \geq (\bar{Q}_{12} - \mu P_{12})((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2)^\dagger (\bar{Q}_{12}^\top - \mu P_{12}^\top) \quad (35d) $$

(SS-4) One of the next conditions is satisfied:

(SS-4a) $\mathrm{rank}(\tilde{D} + \tilde{D}^\top) = d, \tilde{D} + \tilde{D}^\top \neq 0$ and:

$$ \sigma_{\max} \left\{ \mathrm{diag}(\alpha_1 P_1 B_1 - C_1^\top, P_2 B_2 - C_2^\top) + \begin{pmatrix} P_{12} L_2 G_1 & \alpha_1 P_1 L_1 G_2 + P_{12} B_2 - H_1^\top E_2^\top \\ P_{12}^\top B_1 + \alpha_2 P_2 L_2 G_1 - H_2^\top E_1^\top & P_{12}^\top L_1 G_2 \end{pmatrix} \right\} < \lambda_d^{\frac{1}{2}} \left( \mathrm{diag}(D_1 + D_1^\top, D_2 + D_2^\top) + \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & 0 \end{pmatrix} \right) (\sigma_{\max}(\mathrm{diag}(\alpha_1 Q_1, \alpha_2 Q_2) + \bar{Q}) - \mu \lambda_{\max}(P))^{\frac{1}{2}} \quad (36) $$

with $\bar{Q}$ in (10).

(SS-4b) $\tilde{D} + \tilde{D}^\top = 0$, equivalently: $D_1 + D_1^\top = 0, D_2 + D_2^\top = 0$ and $E_2 G_1 + G_2^\top E_1^\top = 0$.

Then the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is strictly state passive with storage function matrix P and dissipation constant $\mu > 0$.

*Proof.* First let us notice that the LMI for strict state passivity is $P = P^\top > 0$ with P in (9), conditions (13a), (13b), and

$$ -\tilde{A}^\top P - P \tilde{A} - \mu P \geq 0 \quad (37a) $$

$$ -\tilde{A}^\top P - P \tilde{A} - \mu P \geq (P \tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger (\tilde{B}^\top P - \tilde{C}) \quad (37b) $$

for some $\mu > 0$ (see Appendix B). In the following, it is shown that (SS-1), (SS-2), (SS-3) and (SS-4) imply strict state passivity of the interconnection (6), that is, (13a), (13b), (37a) and (37b) are satisfied. Let us notice first that (SS-2) implies positive definiteness or semidefiniteness of the matrix P in (9).

* The inequality (13a) holds under (14), hence under (SS-1).

* The inequality (13b): the sufficient conditions such that condition (13b) holds are the same as in Lemma 3.2, whose conditions are assumed to hold by (SS-2).
Fact 4.10.12, Proposition 5.5.20 v)]:

$$ \sigma_{\max} \begin{pmatrix} 0 & E_1 G_2 \\ 0 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}} \begin{pmatrix} E_1 G_2 G_2^{\top} E_1^{\top} & 0 \\ 0 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}} (E_1 G_2 G_2^{\top} E_1^{\top}) $$

Similarly $\sigma_{\max} \begin{pmatrix} 0 & 0 \\ E_2 G_1 & 0 \end{pmatrix} = \lambda_{\max}^{\frac{1}{2}}(E_2 G_1 G_1^{\top} E_2^{\top})$. Continuing:

$$ \lambda_{\min}(\mathrm{diag}(D_1 + D_1^{\top}, D_2 + D_2^{\top})) = \min(\lambda_{\min}(D_1 + D_1^{\top}), \lambda_{\min}(D_2 + D_2^{\top})). \quad (33) $$

Hence (30) guarantees that $\tilde{D} \succ 0$ ($\Leftrightarrow \tilde{D} + \tilde{D}^{\top} \succ 0$), which guarantees in passing that (13) (a) holds trivially. Obviously this is a conservative criterion since $E_1 G_2 + G_1^{\top} E_2^{\top} = 0$ guarantees $\tilde{D} + \tilde{D}^{\top} \succ 0$ also.

The second condition is a consequence of partitioning $-\tilde{A}^{\top}P-P\tilde{A} = \mathrm{diag}(Q_1, Q_2)+\bar{Q}$, and applying Corollary 8. It follows that $\sigma_{\max}(\bar{Q}) \le \frac{1}{\sigma_{\max}(\mathrm{diag}(Q_1^{-1}, Q_2^{-1}))}$. Since $Q_i = Q_i^{\top} \succ 0$, we have

$$ \sigma_{\max}(\mathrm{diag}(Q_1^{-1}, Q_2^{-1})) = \lambda_{\max}(\mathrm{diag}(Q_1^{-1}, Q_2^{-1})) = \lambda_{\min}^{-1}(\mathrm{diag}(Q_1, Q_2)). $$

Then (31) follows which guarantees that $-\tilde{A}^{\top}P-P\tilde{A} \succ 0$.

Now using [9, Proposition 8.2.4] the inequality in (8) is satisfied strictly if and only if

$$ -\tilde{A}^{\top}P - P\tilde{A} - (P\tilde{B} - \tilde{C}^{\top})(\tilde{D} + \tilde{D}^{\top})^{-1}(P\tilde{B} - \tilde{C}^{\top})^{\top} \succ 0. \quad (34) $$

and $-\tilde{A}^{\top}P + P\tilde{A} \succ 0$ (which is assured by the previous step). Using Corollary 8 it is inferred that (34) holds if $||P\tilde{B} - \tilde{C}^{\top}||_{2,2}^2||(\tilde{D} + \tilde{D}^{\top})^{-1}||_{2,2} \le \frac{1}{\sigma_{\max}((-\tilde{A}^{\top}P-P\tilde{A})^{-1})} = \frac{1}{\lambda_{\max}((-\tilde{A}^{\top}P-P\tilde{A})^{-1})} = \frac{1}{\lambda_{\min}((-\tilde{A}^{\top}P-P\tilde{A})^{-1})}$ (where it is used that the $||.||_{2,2}$ norm is submulti- plicative). From [10, Fact 10.12.17] this is equivalent to $\sigma_{\max}((\tilde{D} + \tilde{D}^{\top})^{-1}) = \lambda_{\max}((\tilde{D} + \tilde{D}^{\top})^{-1}) = \frac{1}{\lambda_{\min}((-\tilde{A}^{\top}P-P\tilde{A})^{-1})} \le \frac{\lambda_{\min}(-\tilde{A}^{\top}P-P\tilde{A})}{\sigma_{\max}(P\tilde{B}-\tilde{C}^{\top})}$, which is equivalently rewritten as $\lambda_{\min}(\tilde{D} + \tilde{D}^{\top})\lambda_{\min}(-\tilde{A}^{\top}P-P\tilde{A}) \ge \sigma_{\max}^2(P\tilde{B}-\tilde{C}^{\top})$. Using [9, Fact 5.12.2], (12), (11), this in- equality is implied by (32). The last condition (item (i) in Proposition 1) guarantees $P \succ 0$ from [10, Fact 10.12.17]. $\square$

Notice that $D_1 + D_1^{\top} \succ 0$ and $D_2 + D_2^{\top} \succ 0$ imply that (15) is simplified since the last two inclusions hold trivially.

**Lemma 6.** Let both subsystems in (4) be strongly passive. The interconnection quadruple is strongly passive with storage function matrix $P = \mathrm{diag}(P_1, P_2)$, if: (i) $E_1G_2 = -(E_2G_1)^{\top}$, (ii) $P_1L_1H_2 = -(P_2L_2H_1)^{\top}$, (iii) $P_1L_1G_2 = (E_2H_1)^{\top}$ and $P_2L_2G_1 = (E_1H_2)^{\top}$.
* The inequality (37a) is analyzed by noting that

$$-\tilde{A}^{\top}P - P\tilde{A} = \operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q}$$

with $\tilde{Q}$ in (10), with $Q_1 - \mu_1 P_1 \ge 0$ and $Q_2 - \mu_2 P_2 \ge 0$ using the strict state passivity of each quadruple $(A_i, B_i, C_i, D_i)$. Thus $-\tilde{A}^{\top}P - P\tilde{A} \ge \operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2) - \tilde{Q}$. Let us study the conditions such that $\operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2) - \tilde{Q} \ge \mu P \Leftrightarrow \operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2)\tilde{Q} - \mu P \ge 0$ for some $\mu > 0$. To this aim let us analyse the inequality:

* The inequality (37a) is analyzed by noting that

$$-\tilde{A}^{\top}P - P\tilde{A} = \operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) - \tilde{Q}$$

with $\tilde{Q}$ in (10), with $Q_1 - \mu_1 P_1 \ge 0$ and $Q_2 - \mu_2 P_2 \ge 0$ using the strict state passivity of each quadruple $(A_i, B_i, C_i, D_i)$. Thus $-\tilde{A}^{\top}P - P\tilde{A} \ge \operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2) - \tilde{Q}$. Let us study the conditions such that $\operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2) - \tilde{Q} \ge \mu P \Leftrightarrow \operatorname{diag}(\mu_1\alpha_1 P_1, \mu_2\alpha_2 P_2) - \tilde{Q} - \mu P \ge 0$ for some $\mu > 0$. To this aim let us analyse the inequality:

$$\left( \begin{array}{cc} (\mu_1 - \mu)\alpha_1 P_1 - P_{12}L_2H_1 - H_1^\top L_2^\top P_{12} & \tilde{Q}_{12} - \mu P_{12} \\ \tilde{Q}_{12}^\top - \mu P_{12}^\top & (\mu_2 - \mu)\alpha_2 P_2 - P_{12}^\top L_1 H_2 - H_2^\top L_1^\top P_{12} \end{array} \right) \ge 0. \quad (38)$$

Recall that $P_{12}L_2H_1 \in \mathbb{R}^{n_1 \times n_1}$, and $P_{12}^\top L_1H_2 \in \mathbb{R}^{n_2 \times n_2}$. The necessary and sufficient conditions for (38) to hold are in [9, Proposition 10.2.5, Fact 8.4.3] and are given in item (SS-3), that is, (35) (these are the counterparts of (13) for (38)).

* For inequality (37b), it is seen that if (37b) holds for $\mu > 0$, then it holds strictly for some $\mu' < \mu$ since $P > 0$. Hence without loss of generality, we consider a strict inequality in (37b). Applying Corollary A.1, (37b) holds if $\sigma_{\max}((P\tilde{B} - \tilde{C}^{\top})(\tilde{D} + \tilde{D}^{\top})^{\dagger}(\tilde{B}^{\top}P - \tilde{C}) + \mu P) < \sigma_{\max}(-\tilde{A}^{\top}P - P\tilde{A})$. Using [8, Facts 9.14.15, Corollary 9.6.5], we get

$$\sigma_{\max}((P\tilde{B} - \tilde{C}^{\top})(\tilde{D} + \tilde{D}^{\top})^{\dagger}(\tilde{B}^{\top}P - \tilde{C}) + \mu P) \le \sigma_{\max}^2(P\tilde{B} - \tilde{C}^{\top})\sigma_{\max}((\tilde{D} + \tilde{D}^{\top})^{\dagger}) + \sigma_{\max}(\mu P). \quad (39)$$

Let us analyse both conditions (SS-4a) and (SS-4b):

(SS-4a): If $\operatorname{rank}(\tilde{D} + \tilde{D}^{\top}) = d$ and $\tilde{D} + \tilde{D}^{\top} \ne 0$, then $\sigma_{\max}((\tilde{D} + \tilde{D}^{\top})^{\dagger}) = \sigma_d^{-1}(\tilde{D} + \tilde{D}^{\top})$ [8, Fact 6.3.28]. Consequently (37b) holds if $\sigma_{\max}^2(P\tilde{B} - \tilde{C}^{\top})\sigma_d^{-1}(\tilde{D} + \tilde{D}^{\top}) + \mu\lambda_{\max}(P) < \sigma_{\max}(\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) + \tilde{Q})$. Since $\tilde{D} + \tilde{D}^{\top} \ge 0$ by item (b), $\sigma_d(\tilde{D} + \tilde{D}^{\top}) = \lambda_d(\tilde{D} + \tilde{D}^{\top})$. We get equivalently $\sigma_{\max}^2(P\tilde{B} - \tilde{C}^{\top}) < \lambda_d(\tilde{D} + \tilde{D}^{\top})\sigma_{\max}(\operatorname{diag}(\alpha_1 Q_1, \alpha_2 Q_2) + \tilde{Q}) - \mu\lambda_{\max}(P)$. Using (11) and (12) yields (36), which therefore guarantees that (37b) holds.

(SS-4b): We have $\tilde{D} + \tilde{D}^{\top} = 0 \Leftrightarrow D_1 + D_1^\top = 0, D_2 + D_2^\top = 0, E_1G_2 + G_1^\top E_2^\top = 0$. From (13a) and (14), it follows that $P\tilde{B} - \tilde{C}^{\top} = 0$. Thus, (37b) holds.

□

**Remark 3.12.** Let us examine the conditions given in (35) one by one. The matrix inequality in (35a) is satisfied strictly if $\mu_1 > \mu$, and $\sigma_{\max}(P_{12}L_2H_1) < \frac{(\mu_1-\mu)}{2}\lambda_{\max}(\alpha_1 P_1)$, because it yields $\sigma_{\max}(P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top) \le \sigma_{\max}(P_{12}L_2H_1) + \sigma_{\max}(H_1^\top L_2^\top P_{12}^\top) = 2\sigma_{\max}(P_{12}L_2H_1) < (\mu_1 - \mu)\sigma_{\max}(\alpha_1 P_1) = (\mu_1 - \mu)\lambda_{\max}(\alpha_1 P_1)$ (where we used [8, Fact 9.14.15] and Corollary A.1). Or (35a) is satisfied if: $\mu_1 \ge \mu$, $P_{12}L_2H_1 \ge 0$, which holds if $P_{12} = H_1^\top L_2^\top$ (this secures the positive semidefiniteness). Similarly, the matrix inequality (35b) is satisfied strictly if $\mu_2 > \mu$, and $\sigma_{\max}(P_{12}L_1H_2) < \frac{(\mu_2-\mu)}{2}\lambda_{\max}(\alpha_2 P_2)$, because it once again results in $\sigma_{\max}(P_{12}^\top L_1H_2 + H_2^\top L_1^\top P_{12}^\top) < (\mu_2 - \mu)\lambda_{\max}(\alpha_2 P_2)$. Or (35b) is satisfied if $\mu_2 \ge \mu$, $P_{12}^\top L_1H_2 \ge 0$, which holds if $P_{12} = -L_1H_2$ (this secures the positive semidefiniteness). The condition (35c) uses [9, Fact 8.4.3]; the first set of sufficient conditions given above for (35b) guarantees strict inequalities, consequently (35c) holds since the right-hand side matrix in the inclusion is full-rank. Inequality (35d)
Proof. (i) using the comment after (33): $\tilde{D}+\tilde{D}^\top > 0$, (ii) implies that $\bar{Q}_{12} = 0$ in (10), thus (13) (c) holds with strict inequality, (iii) implies that $P\tilde{B}-\tilde{C}^\top = \text{diag}(P_1B_1-C_1^\top, P_2B_2-C_2^\top)$. From the strong passivity of both subsystems, it follows that $\begin{pmatrix} -\tilde{A}^\top P - P\tilde{A} & -P\tilde{B} + \tilde{C}^\top \\ (-P\tilde{B} + \tilde{C}^\top)^\top & \tilde{D} + \tilde{D}^\top \end{pmatrix} > 0$, which ends the proof (see section B). $\square$

It is easily deduced from $\tilde{D} + \tilde{D}^\top > 0$ (strict version of (13)(a)) and [16, Theorem A.65] that strong passivity of the interconnection implies that $D_i + D_i^\top > 0 \Leftrightarrow D_i > 0$ for $i = 1, 2$.

## 2.3 Strict State Passivity Preservation after Interconnection

Finally a third result concerns the interconnection of two strictly state passive subsystems.

**Proposition 3. (preservation of strict state passivity)** Assume that both subsystems in (4) are strictly state passive with constants $\mu_1 > 0$, $\mu_2 > 0$, with $P_i > 0$ the solution of the strict state passivity LMI, $i = 1, 2$. Let $P = \begin{pmatrix} P_1 & P_{12} \\ P_{12}^\top & P_2 \end{pmatrix}$. Suppose that: (1) (14) holds, (2) conditions in Proposition 1 (i) and in Lemma 2 hold, and:

(3) the next conditions hold:

$$ \text{i)} (\mu_1 - \mu)P_1 + P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top \ge 0 \quad (\text{true if: i1 } \mu_1 > \mu, \sigma_{\max}(P_{12}L_2H_1) < \frac{(\mu_1-\mu)}{2} \lambda_{\max}(P_1) \text{ or if: i2 } \mu_1 \ge \mu, P_{12} = H_1^\top L_2^\top), $$

$$ \text{ii)} (\mu_2 - \mu)P_2 + P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12} \ge 0 \quad (\text{true if: ii1 } \mu_2 > \mu, \sigma_{\max}(P_{12}L_1H_2) < \frac{(\mu_2-\mu)}{2} \lambda_{\max}(P_2) \text{ or if: ii2 } \mu_2 \ge \mu, P_{12} = H_2^\top L_1^\top), $$

$$ \text{iii)} \operatorname{Im}(\bar{Q}_{12}^\top - \mu P_{12}^\top) \subseteq \operatorname{Im}((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2) \quad (\text{true if ii1 holds}), $$

$$ \text{iv)} (\mu_1 - \mu)P_1 + P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top \ge (\bar{Q}_{12} - \mu P_{12})((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2)^\dagger (\bar{Q}_{12}^\top - \mu P_{12}^\top) $$

(true if: i1, ii1 hold, and:

$$ \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12}) < \lambda_{\min}((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2) \times \lambda_{\max}((\mu_1 - \mu)P_1 + P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top), \quad (35) $$

(4) the next conditions hold:
stems from [9, Proposition 10.2.5, iii)], under the sufficient conditions discussed for (35a) and (35b), so that both matrices in the left-hand and the right-hand sides in inequality (35d) are > 0. We have:

$$
\begin{align*}
\sigma_{\max}((\bar{Q}_{12} - \mu P_{12})((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2)^{-1}(\bar{Q}_{12}^\top - \mu P_{12}^\top))
&\le \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\sigma_{\max}((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2)^{-1}) \\
&= \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\lambda_{\max}((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2)^{-1}) \\
&= \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\lambda_{\min}^{-1}((\mu_2 - \mu)\alpha_2 P_2 - H_2^\top L_1^\top P_{12} - P_{12}^\top L_1 H_2),
\end{align*}
$$

where [8, Corollary 9.6.5] is used. Then Corollary A.1 can be applied to obtain the sufficient condition stated in (35d).

3.4. Comments

Some comments and insights on the foregoing results are given:

* The weights $\alpha_1 > 0$ and $\alpha_2 > 0$ in (9) can be helpful to satisfy conditions involving upper bounds, see, e.g., conditions in Lemma 3.3, and (18), (19), (29), (35).

* The decomposition in Proposition 3.8 item 1 can be done with $N = (P\tilde{B} - \tilde{C}^T)((\tilde{D} + \tilde{D}^T)^\dagger)^{\frac{1}{2}}$. The Cholesky decomposition [8, Fact 8.9.38], or Gram matrix [8, Fact 8.9.37] are possible choices, also in the proof of Proposition 3.8 item 2, and in item 2 of Lemma 3.5. Similar condition as in (19) can be applied to the inequality (27c), following a reasoning as in the proof of item 2 in Lemma 3.5. Similarly the conditions $\tilde{Q}_1 \succeq 0$ and $\tilde{Q}_2 \succeq 0$ in Lemma 3.3, can be studied more deeply, still using the decompositions of $Q_1, Q_2$, and of $-P_{12}L_2H_1 - (P_{12}L_2H_1)^\top$ and of $-P_{12}^\top L_1H_2 - (P_{12}^\top L_1H_2)^\top$.

* In case $\tilde{D} + \tilde{D}^\top = 0$, then (13) reduces to conditions (13a) and (13c). Thus it is sufficient to look at Lemmata 3.1 and 3.3.

* The conditions in Theorem 3.9 guarantee that the inequality in (8) is strict, by imposing small enough perturbation to the matrix

$$
\begin{pmatrix}
Q_1 & 0 & P_1 B_1 - C_1^\top & 0 \\
0 & Q_2 & 0 & P_2 B_2 - C_2^\top \\
(P_1 B_1 - C_1^\top)^\top & 0 & D_1 + D_1^\top & 0 \\
0 & (P_2 B_2 - C_2^\top)^\top & 0 & D_2 + D_2^\top
\end{pmatrix}.
\quad (40)
$$

* Most of the foregoing results and conditions imply restrictions on the couplings between both subsystems, i.e., conditions on matrices $L_i$. $E_i$, $H_i$ and $G_i$, $i = 1, 2$. Some of these are of geometric nature and impose restrictions on linear spaces (see for instance (14) (15), items 1 and 2 in Lemma 3.2), others characterize the smallness of the couplings (see for instance the singular values conditions in items 1 and 2 of Lemma 3.2, equality in item 3 of Lemma 3.2, equalities (22a)–(22d) in Corollary 3.7, equality in item 2 in Proposition 3.8, inequality (28) in Theorem 3.8, conditions (i)–(iv) in Corollary 3.10, see also section 6.1). It is interesting to note that the matrix $E_1G_2 + (E_2G_1)^\top$, which concerns the multipliers feedback inside complementarity constraints, plays an important role and appears in several of the foregoing results. Specifically, several of the above smallness conditions are automatically satisfied when this coupling matrix vanishes.

4. Maximal Monotonicity and Incremental Passivity

In this section, it is assumed that both LCS in (4) are autonomous, i.e.:

**Assumption 4.1.** In (4), we have $f_1 \equiv 0$, $f_2 \equiv 0$, $g_1 \equiv 0$ and $g_2 \equiv 0$.

Under some basic conditions (see Appendix C and Theorem C.1), the interconnection (6) defines a differential inclusion (DI):

$$
\dot{x} \in -\mathcal{F}(x) \stackrel{\Delta}{=} \tilde{A}x - \tilde{B}(\tilde{D} + \tilde{N}_{\tilde{K}^*}^{-1})^{-1}(\tilde{C}x) \quad (41)
$$

with $\mathcal{F}(\cdot)$ a maximal monotone operator, $\text{Dom}(\mathcal{F}) = \{x \in \mathbb{R}^n \mid \tilde{C}x \in \text{Dom}(\tilde{D} + \tilde{N}_{\tilde{K}^*}^{-1})^{-1} = \text{Im}(\tilde{D} + \tilde{N}_{\tilde{K}^*}^{-1})\} = \tilde{C}^{-1}(\text{Im}(\tilde{D} + \tilde{N}_{\tilde{K}^*}^{-1}))$. This motivates the next result.
(d'1): rank($\tilde{D} + \tilde{D}^\top$) = d, $\tilde{D} + \tilde{D}^\top \neq 0$ and:

$$
\begin{align}
& \sigma_{\max} \left\{ \begin{array}{@{}l@{}} \operatorname{diag}(P_1 B_1 - C_1^\top, P_2 B_2 - C_2^\top) \\ \qquad + \begin{pmatrix} P_{12} L_2 G_1 & P_1 L_1 G_2 + P_{12} B_2 - H_1^\top E_2^\top \\ P_{12}^\top B_1 + P_2 L_2 G_1 - H_2^\top E_1^\top & P_{12}^\top L_1 G_2 \end{pmatrix} \end{array} \right\} \nonumber \\
& < \lambda_d^{\frac{1}{2}} \left( \operatorname{diag}(D_1 + D_1^\top, D_2 + D_2^\top) + \begin{pmatrix} 0 & E_1 G_2 + G_1^\top E_2^\top \\ E_2 G_1 + G_2^\top E_1^\top & 0 \end{pmatrix} \right) \nonumber \\
& \quad \times (\sigma_{\max}(\operatorname{diag}(Q_1, Q_2) + \bar{Q}) - \mu \lambda_{\max}(P))^{\frac{1}{2}} \tag{36}
\end{align}
$$

with $\bar{Q}$ in (10).

(d'2): $\tilde{D}+\tilde{D}^\top = 0$, equivalently: $D_1+D_1^\top = 0$ and $D_2+D_2^\top = 0$ and $E_2G_1+G_2^\top E_1^\top = 0$. Then the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is strictly state passive with storage function matrix $P$ and dissipation constant $\mu > 0$.

Proof. The LMI for strict state passivity is $P = P^\top \succ 0$, (13) (a) (b) and

$$ (c') - \tilde{A}^\top P - P \tilde{A} - \mu P \succ 0 \quad (37) $$

$$ (d') - \tilde{A}^\top P - P \tilde{A} - \mu P \succ (P \tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger (\tilde{B}^\top P - \tilde{C}) \quad (38) $$

for some $\mu > 0$.

(a) Equality (13) (a) holds under (14), hence under (1).

(b) The necessary and sufficient conditions such that condition (13) (b) holds are the same as in Lemma 2.

(c') Condition (37) is analysed as follows. Notice that $P \succ 0$ since item (i) in Proposition 1 holds. We have: $-\tilde{A}^\top P - P \tilde{A} = \operatorname{diag}(Q_1, Q_2) + \bar{Q}$ with $\bar{Q}$ in (10), with $Q_1 - \mu_1 P_1 \succ 0$ and $Q_2 - \mu_2 P_2 \succ 0$. Thus $-\tilde{A}^\top P - P \tilde{A} \succ \operatorname{diag}(\mu_1 P_1, \mu_2 P_2) + \bar{Q}$. Let us study the conditions such that $\operatorname{diag}(\mu_1 P_1, \mu_2 P_2) + \bar{Q} \succ \mu P \Leftrightarrow \operatorname{diag}(\mu_1 P_1, \mu_2 P_2) + \bar{Q} - \mu P \succ 0$ for some $\mu > 0$. To this aim let us analyse the inequality:

$$ \left( \begin{array}{@{}l@{}} (\mu_1 - \mu)P_1 + P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top \\ \qquad \bar{Q}_{12}^\top - \mu P_{12}^\top \end{array} \right) \quad \left( \begin{array}{@{}l@{}} \bar{Q}_{12} - \mu P_{12} \\ (\mu_2 - \mu)P_2 + P_{12}^\top L_1 H_2 + H_2^\top L_1^\top P_{12} \end{array} \right) \succ 0. \quad (39) $$

Recall that $P_{12}L_2H_1 \in \mathbb{R}^{n_1 \times n_1}$, and $P_{12}^\top L_1H_2 \in \mathbb{R}^{n_2 \times n_2}$. The necessary and sufficient conditions for (39) to hold are in [10, Proposition 10.2.5, Fact 8.4.3] and are given in (3) i)-iv) (these are the counterparts of (13) for (39)). Let us examine the conditions given in (35) i) to iv) one by one. i) This matrix inequality is satisfied strictly if $\mu_1 > \mu$, $\sigma_{\max}(P_{12}L_2H_1 + H_1^\top L_2^\top P_{12}^\top) \le \sigma_{\max}(P_{12}L_2H_1) + \sigma_{\max}(H_1^\top L_2^\top P_{12}^\top) = 2\sigma_{\max}(P_{12}L_2H_1) < (\mu_1 - \mu)\sigma_{\max}(P_1) = (\mu_1 - \mu)\lambda_{\max}(P_1)$ (where we used [9, Fact 9.14.15] and Corollary 8); or it is satisfied if: $\mu_1 \ge \mu$, $P_{12}L_2H_1 \succ 0$, which holds if $P_{12} = H_1^\top L_2^\top$ (this secures the positive semidefiniteness). ii) Similarly this matrix inequality is satisfied strictly if $\mu_2 > \mu$, $\sigma_{\max}(P_{12}^\top L_1H_2 + H_2^\top L_1^\top P_{12}) \le \sigma_{\max}(P_{12}^\top L_1H_2) + \sigma_{\max}(H_2^\top L_1^\top P_{12}) = 2\sigma_{\max}(P_{12}^\top L_1H_2) < (\mu_2 - \mu)\sigma_{\max}(P_2) = (\mu_2 - \mu)\lambda_{\max}(P_2)$; or is satisfied if $\mu_2 \ge \mu$, $P_{12}^\top L_1H_2 \succ 0$, which
**Corollary 4.2.** Consider subsystems in (4) under Assumptions 2.1 and 4.1, and their interconnection in (6). Let $\mathrm{Im}(\tilde{C}) \cap \mathrm{rint}(\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})) \neq \emptyset$. If the quadruplet $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive (resp. strongly passive, resp. strictly state passive), then the LCS in (6) has a maximal monotone (resp. single valued Lipschitz strongly monotone; resp. strongly monotone) right-hand side mapping $\mathcal{F}(\cdot)$. Therefore, the interconnected system possesses unique AC solutions for all $x(0) = x_0 \in \mathrm{Dom}(\mathcal{F})$.

*Proof.* If $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive, hence under the stated condition, the mapping $\mathcal{F}(\cdot)$ is maximal monotone, see Theorem C.1. Let the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be strongly passive, then there exists $\mu > 0$ such that $(\tilde{A} + \mu I_n, \tilde{B}, \tilde{C}, \tilde{D})$ is passive (this holds for all $0 < \mu < \lambda_{\min}(\tilde{Q})$ with $\tilde{Q}$ in (8), applying for instance Corollary A.1). Therefore the mapping $\tilde{\mathcal{F}}: x \mapsto -(\tilde{A} + \mu I_n)x + \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$ is maximal monotone, and $\langle x_a - x_b, y_a - y_b \rangle \ge \mu \|x_a - x_b\|^2$ for all $x_i \in \mathrm{Dom}(\mathcal{F})$, $y_i \in \mathcal{F}(x_i)$, $i = a, b$. In addition, strong passivity implies $\tilde{D} > 0$, hence applying [15, Proposition 1] the mapping $(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}$ is single-valued, well-defined and Lipschitz continuous with constant $\frac{1}{\lambda_{\min}(\tilde{D}+\tilde{D}^-)}$. Let now $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be strictly state passive, i.e., $\tilde{Q} \succeq 0$ and $-\tilde{A}^\top P + P\tilde{A} - \mu P \succeq 0$ for some $\mu > 0$. Then the quadruple $(\tilde{A} + \frac{\mu}{2}I_n, \tilde{B}, \tilde{C}, \tilde{D})$ is passive. Thus the mapping $x \mapsto -(\tilde{A} + \mu I_n)x + \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$ is maximal monotone. Hence $\mathcal{F}(\cdot)$ is $\mu$-strongly monotone (but not necessarily single-valued since $\tilde{D}$ may be low-rank). The well-posedness is a consequence of Theorem C.1. $\square$

Let us analyze the condition in Theorem C.1 and in Corollary 4.2.

**Lemma 4.3.** Let the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be passive. Assume that: $\tilde{D} = \tilde{D}^\top$ ($\Leftrightarrow D_1 = D_1^\top, D_2 = D_2^\top, E_2 G_1 = (E_1 G_2)^\top$), $\{\mathrm{Im}(\begin{pmatrix} C_1 \\ E_2 H_1 \end{pmatrix}) + \mathrm{Im}(\begin{pmatrix} E_1 H_2 \\ C_2 \end{pmatrix})\} \cap \{\mathrm{rint}(K^*) + \mathrm{rint}(\mathrm{Im}(\tilde{D})) \neq \emptyset\}$. Then $\mathrm{Im}(\tilde{C}) \cap \mathrm{rint}(\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})) \neq \emptyset$.

*Proof.* Let us characterize $\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})$. Both mappings $x \mapsto \tilde{D}x$ and $\mathcal{N}_{K^*}^{-1}(\cdot)$ are maximal monotone, and they are both the subdifferential of proper, convex lower semicontinuous functions, because $\tilde{D}$ is symmetric, and $\mathcal{N}_{K^*}^{-1}(\cdot) = \partial\sigma_{K^*}(\cdot)$, with $\sigma_{K^*}$ being the support function of $K^*$. Hence both mappings satisfy the property (*) stated in [11, section 1] (a monotone operator $\mathcal{M}(\cdot)$ satisfies the property (*) if for all $\xi \in \mathrm{Im}(\mathcal{M})$ and all $\zeta \in \mathrm{Dom}(\mathcal{M})$, we have $\sup_{(c,d) \in \mathrm{graph}(\mathcal{M})}(h-\xi)^\top(\zeta-z) < +\infty$). From [7, Corollary 24.4 (ii)], the sum is maximal monotone. Using [11, Theorems 3, 4], we have $\overline{\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})} = \overline{\mathrm{Im}(\mathcal{N}_{K^*}^{-1})} + \overline{\mathrm{Im}(\tilde{D})} = \overline{K^*} + \overline{\mathrm{Im}(\tilde{D})}$, and $\mathrm{int}(\overline{\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})}) = \mathrm{int}(\overline{\mathrm{Im}(\mathcal{N}_{K^*}^{-1})}) + \overline{\mathrm{Im}(\tilde{D})} = \mathrm{int}(K^* + \overline{\mathrm{Im}(\tilde{D})})$. Therefore $\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D}) \approx K^* + \overline{\mathrm{Im}(\tilde{D})}$. Using [57, Corollary 6.6.2], we have $\mathrm{rint}(K^* + \overline{\mathrm{Im}(\tilde{D})}) = \mathrm{rint}(K^*) + \mathrm{rint}(\mathrm{Im}(\tilde{D}))$. The proof finishes by applying [9, Fact 3.14.8] to $\mathrm{Im}(\tilde{C})$. $\square$

If $K$ is polyhedral then it holds that $\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D}) = K^* + \mathrm{Im}(\tilde{D})$ [53]. Monotonicity and incremental passivity are closely related to each other [4, 16, 71]. Let us consider the system $\dot{x} \in -\mathcal{F}(x)+u$, with $\mathcal{F}(\cdot)$ maximal monotone. Then the mapping $u \mapsto x$ is incrementally passive. Indeed, let $\dot{x}_a \in -\mathcal{F}(x_a) + u_a$ and $\dot{x}_b \in -\mathcal{F}(x_b) + u_b$, with $u_a, u_b$ such that the system is well-posed with AC solutions. Let $s(x_a, x_b, u_a, u_b) \stackrel{\Delta}{=} \langle x_a - x_b, u_a - u_b \rangle = \langle x_a - x_b, \dot{x}_a - \dot{x}_b \rangle + \langle x_a - x_b, \xi_a - \xi_b \rangle$, where $\xi_i \in \mathcal{F}(x_a)$, for $i = a, b$. Thus defining the storage function $V(x_a, x_b) = \frac{1}{2}\langle x_a - x_b, x_a - x_b \rangle$, it follows that $\dot{V} = s(x_a, x_b, u_a, u_b) - \langle x_a - x_b, \xi_a - \xi_b \rangle \le s(x_a, x_b, u_a, u_b)$. This motivates us to enlarge the interconnection (6) to

$$ \begin{gathered} \dot{x} = \tilde{A}x + \tilde{B}\lambda + Lv \\ K \ni \lambda \perp w = \tilde{C}x + \tilde{D}\lambda \in K^*, \end{gathered} \quad (42) $$

for some admissible control input $v$. The system (42) is rewritten equivalently as:

$$ \dot{x} \in \tilde{A}x - \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x) + Lv = -\mathcal{F}(x) + Lv. \quad (43) $$

The system in (43) has exactly the same form as the subsystems in (4), indicating that further interconnections of interconnections can be made. This is an interesting feature, similar to nested negative feedback interconnections of passive systems. In the next proposition an output $z$ is considered for the interconnected system.

**Proposition 4.4.** Consider subsystems in (4) under Assumptions 2.1 and 4.1, their interconnection in (6), and the input/output mapping $\Sigma_{vz}: v \mapsto z$, with $z = Hx + Gv$, in (42). Consider the storage function $V(x_a, x_b) = \frac{1}{2}\langle x_a - x_b, P(x_a - x_b) \rangle$, for some $P = P^\top \succeq 0$.
holds if $P_{12} = L_1 H_2$ (this secures the positive semidefiniteness). iii) this condition uses [10, Fact 8.4.3]; the first set of sufficient conditions ii1 in (35) ii) guarantees strict inequalities, consequently iii) holds since the right-hand side matrix in the inclusion is full-rank. iv) stems from [10, Proposition 10.2.5, iii)]; let conditions i1 and ii1 be true (this secures that both matrices in i) and ii) are > 0, hence both matrices in the left-hand and the right-hand sides in inequality (3) iv) are > 0); we have:

$$
\begin{align*}
& \sigma_{\max}((\bar{Q}_{12} - \mu P_{12})((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2)^{-1}(\bar{Q}_{12}^\top - \mu P_{12}^\top)) \\
& \leq \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\sigma_{\max}((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2)^{-1}) \\
& = \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\lambda_{\max}((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2)^{-1}) \\
& = \sigma_{\max}^2(\bar{Q}_{12} - \mu P_{12})\lambda_{\min}^{-1}((\mu_2 - \mu)P_2 + H_2^\top L_1^\top P_{12} + P_{12}^\top L_1 H_2)
\end{align*}
$$

where [9, Corollary 9.6.5] is used. Then we can apply Corollary 8 to obtain the sufficient condition stated in (3) iv).

(d') If (38) holds for $\mu > 0$, then it holds strictly for some $\mu' < \mu$ since $P \succ 0$. Hence without loss of generality, we consider a strict inequality. Applying Corollary 8, (38) holds if $\sigma_{\max}((P\tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger(\tilde{B}^\top P - \tilde{C}) + \mu P) < \sigma_{\max}(-\tilde{A}^\top P - P\tilde{A})$. Using [9, Facts 9.14.15, Corollary 9.6.5], $\sigma_{\max}(P\tilde{B} - \tilde{C}^\top)(\tilde{D} + \tilde{D}^\top)^\dagger(\tilde{B}^\top P - \tilde{C}) + \mu P) \le \sigma_{\max}^2((P\tilde{B} - \tilde{C}^\top)\sigma_{\max}((\tilde{D} + \tilde{D}^\top)^\dagger) + \sigma_{\max}(\mu P)).$ (d'1) If $\mathrm{rank}(\tilde{D} + \tilde{D}^\top) = d$ and $\tilde{D} + \tilde{D}^\top \ne 0$, then $\sigma_{\max}((\tilde{D} + \tilde{D}^\top)^\dagger) = \sigma_d^{-1}(\tilde{D} + \tilde{D}^\top)$ [9, Fact 6.3.28]. Consequently (38) holds if $\sigma_{\max}^2((P\tilde{B} - \tilde{C}^\top)\sigma_d^{-1}(\tilde{D} + \tilde{D}^\top) + \mu\lambda_{\max}(P)) < \sigma_{\max}(\mathrm{diag}(Q_1, Q_2) + \bar{Q})$. Since $\tilde{D} + \tilde{D}^\top \succeq 0$ by item (b), $\sigma_d(\tilde{D} + \tilde{D}^\top) = \lambda_d(\tilde{D} + \tilde{D}^\top)$. Thus we get equivalently $\sigma_{\max}^2(P\tilde{B} - \tilde{C}^\top) < \lambda_d(\tilde{D} + \tilde{D}^\top)(\sigma_{\max}(\mathrm{diag}(Q_1, Q_2) + \bar{Q}) - \mu\lambda_{\max}(P))$. Using (11) and (12) yields (36). (d'2) We have $\tilde{D} + \tilde{D}^\top = 0 \iff D_1 + D_1^\top = 0, D_2 + D_2^\top = 0, E_1 G_2 + G_1^\top E_2^\top = 0$. From item (a) and (14), it follows that $P\tilde{B} - \tilde{C}^\top = 0$. $\square$

## 2.4 Comments

Some comments on the foregoing results arise:

* Many conditions in the above three propositions imply some "smallness" of the interconnections, in the following sense. Consider Lemma 2: all the conditions are satisfied if $E_1 G_2 = -(E_2 G_1)^\top$, which is a kind of "skew-symmetry" condition. The conditions of Lemma 2 state how much this skew-symmetry property can be deteriorated. Similar conclusions are drawn for Lemma 1 in (15).

* LMIs can be solved numerically efficiently. However it is unclear how to compute the interconnections terms from scratch so that the interconnected system's matrix inequality is solvable. Notice in particular that the matrix inequalities are nonlinear if the interconnection matrices $H_1, H_2, G_1$ and $G_2$ (and possibly in some design problems $L_1, L_2, E_1$ and $E_2$) are considered as unknowns in addition to $P_{12}$. The usefulness of the above results is that they provide analytical conditions which pave the way towards a criterion
1. Let the mapping $\mathcal{PF}(\cdot)$ be maximal monotone. Then $\Sigma_{vz}$ is incrementally passive if $G + G^\top \succcurlyeq 0$ and $PL = H^\top$.

2. Let the mapping $\mathcal{PF}(\cdot)$ be maximal $\alpha$-strongly monotone, $\alpha > 0$. Then,

* $\Sigma_{vz}$ is incrementally passive if

$$ \left( \begin{array}{cc} 2\alpha I_n & H^\top - PL \\ (H^\top - PL)^\top & G + G^\top \end{array} \right) \succcurlyeq 0. \qquad (44) $$

* $\Sigma_{vz}$ is incrementally strictly state and input passive, if inequality (44) is strict.

* $\Sigma_{vz}$ is incrementally strictly state passive if $G + G^\top = 0$ and $PL = H^\top$.

*Proof.* From (43), we have $\dot{x} \in -\mathcal{F}(x) + Lv$, with output $z = Hx + Gv$. For two admissible inputs $v_a$ and $v_b$, it follows that $\dot{x}_i \in -\mathcal{F}(x_i) + Lv_i$, $z_i = Hx_i + Gv_i$, $i = a, b$. Let $\xi_i \in \mathcal{F}(x_i)$, and $\omega = \langle z_a - z_b, v_a - v_b \rangle$.

1. From $P\dot{x}_i \in -\mathcal{PF}(x_i) + PLv_i$:

$$ \begin{aligned} \dot{V} &= -(x_a - x_b)^\top P(\xi_a - \xi_b) + (x_a - x_b)^\top PL(v_a - v_b) \pm (z_a - z_b)^\top (v_a - v_b) \\ &= -(x_a - x_b)^\top P(\xi_a - \xi_b) + (x_a - x_b)^\top (PL - H^\top)(v_a - v_b) - \frac{1}{2}(v_a - v_b)^\top (G + G^\top)(v_a - v_b) + \omega. \end{aligned} \qquad (45) $$

If $\mathcal{PF}(\cdot)$ is monotone, then $-(x_a - x_b)^\top P(\xi_a - \xi_b) \le 0$. Then incremental passivity holds if and only if $-(x_a - x_b)^\top P(\xi_a - \xi_b) + (x_a - x_b)^\top (PL - H^\top)(v_a - v_b) - \frac{1}{2}(v_a - v_b)^\top (G + G^\top)(v_a - v_b) \le 0$ for all $v_a, v_b, x_a, x_b$ (see, e.g., [52, Definition 1]). Since $PL - H^\top = 0$ the result is proved.

2. From $\alpha$-strong monotonicity and doing same computations as in (45) we get:

$$ \begin{aligned} \dot{V} &\le -\alpha(x_a - x_b)^\top(x_a - x_b) + (x_a - x_b)^\top (PL - H^\top)(v_a - v_b) - \frac{1}{2}(v_a - v_b)^\top (G + G^\top)(v_a - v_b) + \omega \\ &= -\frac{1}{2}(x_a^\top - x_b^\top, v_a^\top - v_b^\top) \left( \begin{array}{cc} 2\alpha I_n & H^\top - PL \\ (H^\top - PL)^\top & G + G^\top \end{array} \right) \left( \begin{array}{c} x_a - x_b \\ v_a - v_b \end{array} \right) + \omega. \end{aligned} \qquad (46) $$

The first result is proved. The second result follows using [16, Theorem A.65]. The last result follows using [16, Lemma A.69] or [9, Proposition 10.2.5].

**Corollary 4.5.** Suppose that Assumptions 2.1 and 4.1 hold and the mapping $\mathcal{PF}(\cdot)$ is maximal $\alpha$-strongly monotone, for some $\alpha > 0$. Then, $\Sigma_{vz}$ is incrementally passive if $\sigma_{\max}^2(PL - H^\top) \le \alpha\lambda_{\min}(G + G^\top)$.

*Proof.* Directly from [9, Fact 10.12.17].

## 5. Analysis of Interconnections State Jumps

It is well-known that LCS can undergo state jumps under some conditions. The objective of this section is to analyze how the interconnection (6) influences the state jumps behaviours. In this section it is assumed that $K_i = \mathbb{R}_+^{m_i}$, $i = 1, 2$. When state jumps are present, the LCS in (4) and (6) have to be embedded in the framework of Measure Differential Inclusions (MDI) [50, 68, 18]. For systems of the form (6), when there are possible state jumps due to discontinuous input $g$, it is of interest to take the following mathematical model:

$$ dx = (Ax + f)dt + B\lambda \qquad (47a) $$

$$ 0 \le \lambda \perp w = Cx + D\lambda + g \ge 0. \qquad (47b) $$

Here, we assume that the exogenous signal $g$ is RCLBV. The derivative of such functions is considered in the sense of distributions and is a Radon measure, which we denote by $dg$. Similarly, the measure associated with the total
guaranteeing that (8) possesses at least a solution for $H_1, H_2, G_1, G_2$ and $P_{12}$. In an even more general setting $P_1$ and $P_2$ could be considered unknowns of the problem also.

• The decomposition in item 4.1 in Lemma 4 can be done with $N = (P\tilde{B} - \tilde{C}^T)((\tilde{D} + \tilde{D}^T)^\dagger)^{\frac{1}{2}}$. The Cholesky decomposition [9, Fact 8.9.38], or Gram matrix [9, Fact 8.9.37] are possible choices, also in items 4.2 and 4.4. Similar condition as in (25) can be applied to the inequality (27) (iii), following a reasoning as in the proof of item 4.4 in Lemma 4. Similarly the conditions $\tilde{Q}_1 \ge 0$ and $\tilde{Q}_2 \ge 0$ in Lemma 3, can be studied more deeply, still using the decompositions of $Q_1, Q_2$, and of $-P_{12}L_2H_1 - (P_{12}L_2H_1)^\top$ and of $-P_{12}^\top L_1H_2 - (P_{12}^\top L_1H_2)^\top$.

• In case $\tilde{D} + \tilde{D}^\top = 0$, then (13) reduces to conditions (a) and (c). Thus it is sufficient to look at Lemmata 1 and 3.

• Item 4.4 in Lemma 4 yields an algorithm for checking that (13) (d) is satisfied.

• The storage matrices $P_1$ and $P_2$ may be nonunique. Therefore conditions involving these matrices (like (16) and (19)) have to be true for at least one pair of storage functions. Thus the interconnection matrices may vary while keeping the same property of the interconnection.

• The sufficient conditions proposed in the case $P_{12} = 0$ imply that the interconnections terms cancel the coupling terms between both subsystems.

• The condition in (30) can be replaced by the diagonal dominance theorem.

• The conditions in Proposition 2 guarantee that the inequality in (8) is strict, by imposing small enough perturbation to the matrix

$$ \begin{pmatrix} Q_1 & 0 & P_1 B_1 - C_1^\top & 0 \\ 0 & Q_2 & 0 & P_2 B_2 - C_2^\top \\ (P_1 B_1 - C_1^\top)^\top & 0 & D_1 + D_1^\top & 0 \\ 0 & (P_2 B_2 - C_2^\top)^\top & 0 & D_2 + D_2^\top \end{pmatrix}. \quad (40) $$

## 2.5 Lack/excess of passivity

It is often of interest to investigate cases with lack/excess of passivity, i.e., can the excess of passivity in one subsystem, compensate for the lack of passivity in the other one.

**Proposition 4.** Assume that $(A_1, B_1, C_1, D_1)$ is strongly passive, while system 2 is marginally passive with $D_2 + D_2^\top = 0$, and with $A_2^\top P_2 + P_2 A_2 = -Q_2 = 0$. Then the interconnection (4) (5) is strictly passive with respect to $x_1$ and $\lambda_1$, with $P = \text{blockdiag}(P_1, P_2)$, if:

$$ \begin{aligned} P_1 L_1 H_2 + (P_2 L_2 H_1)^\top &= 0, & P_1 L_1 G_2 - H_1^\top E_2^\top &= 0 \\ P_2 L_2 G_1 - H_2^\top E_1^\top &= 0, & E_1 G_2 + G_1^\top E_2^\top &= 0. \end{aligned} \quad (41) $$

In this case (assuming AC solutions exist for all $x(0) = x_0 \in \text{Dom}(\mathcal{F})$ with $\mathcal{F}(x) = -\tilde{A}x + \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$) the dissipation inequality:

$$ \dot{V}(x(t)) \le -\alpha_1 ||x_1||^2 - \alpha_2 ||\lambda_1||^2, \quad \alpha_1 > 0, \alpha_2 > 0, \quad (42) $$

holds along the interconnected system's trajectories, with $V(x) = x_1^\top P_1 x_1 + x_2^\top P_2 x_2$.
variation of g is denoted by $|dg|$. Because of the discontinuities in g, it becomes necessary to allow for state trajectories to be discontinuous for satisfying (47b). These discontinuities in state-trajectories are modeled by interpreting $\lambda$ as a measure and this allows us to consider the solution x of (47) in the space of RCLBV functions. The density of $dx$ with respect to a positive Radon measure $v$ over an interval $I$ is defined as $\frac{dx}{dv}(t) = \lim_{\epsilon \to 0} \frac{dx(I(t, \epsilon))}{v(I(t, \epsilon))}$, where $I(t, \epsilon) := I \cap [t - \epsilon, t + \epsilon]$. We say that $x: [0, T] \to \mathbb{R}^n$ is a solution to MDI (47) if there exists a Radon measure $v$, absolutely continuous with respect to $dt + |dg|$, such that

$$ \frac{dx}{dv}(t) = (Ax + f) \frac{dt}{dv}(t) + B \frac{d\lambda}{dv}(t) \qquad (48) $$

and (47b) holds when evaluating the measures and functions at each time instant from the right. For the article's completeness results on LCS' well-posedness with RCLBV solutions are recalled in Appendix D. Consider the MDI in (47), with RCLBV solutions. To start with, let us look at (47) at state jump times, then the system equations are equivalently written as:

$$ x(t^+) - x(t^-) \in -B(D + N_{\mathbb{R}^m}^{-1})^{-1}(Cx(t^+) + g(t^+)). \qquad (49) $$

Using the passivity of (A, B, C, D), plus assumptions (A3) (A4) (A5) in [68, Theorem 3.1], the inclusion in (49) is rewritten equivalently as

$$ z(t^+) - z(t^-) \in -\Psi(t, z(t^+)) \Leftrightarrow z(t^+) = (\mathrm{Id} + \Psi(t, \cdot))^{-1}(z(t^-)) \qquad (50) $$

where $z = Rx$, $R = R^\top > 0$, $R^2 = P$, $P$ is a solution of the LMI in (B.1), and $\Psi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ is a maximal monotone mapping for each $t$ [68, Lemma 4]. Thus its resolvent $(\mathrm{Id} + \Psi(t, \cdot))^{-1}$ is single-valued Lipschitz continuous in $z$, and $x(t^+) = R^{-1}z(t^+)$ is uniquely defined from (50). Let us introduce the following, which provides a more tractable expression for the jump [22, 41, 18]:

**Definition 1.** Consider the MDI in (47) with RCLBV solutions. Let us assume that (A, B, C, D) is passive, with storage function $V(x) = \frac{1}{2}x^\top P x$, $P = P^\top > 0$. For any $x(t^-)$, the state $x(t^+)$ is given as the solution to:

$$ P(x(t^+) - x(t^-)) \in -N_{\Phi(t^+)}(x(t^+)) \Leftrightarrow x(t^+) = (\mathrm{Id} + P^{-1}N_{\Phi(t^+)})^{-1}(x(t^-)), \qquad (51) $$

where $\Phi(t^+) \stackrel{\Delta}{=} \{z \in \mathbb{R}^n | Cz + g(t^+) \in S_D^*\}$, $S_D = \{z \in \mathbb{R}^m | 0 \le z \perp Dz \ge 0\}$, and the constraint qualification (CQ): $g(t^+) \in S_D^* + \mathrm{Im}(C)$ holds.

Given $x(t^-)$ and $g(t^-)$, the existence or absence of state jumps depend on $D$ and $g(t^+)$. Assume that $g(\cdot)$ is continuous at $t$, so that $\Phi(\cdot)$ is also continuous at $t$ (see Appendix E). Then clearly $x(t^-) \in \Phi(t^-) = \Phi(t^+)$ implies that $x(t^+) = x(t^-)$ which is the unique solution to (51). It is inferred that state jumps can occur only (but not necessarily) at jumps of $g(\cdot)$. Notice that $\mathrm{Im}(C) = \mathbb{R}^m \not\Leftrightarrow \Phi(t^+) = \mathbb{R}^n$ so that even in this case state jumps may occur because it may be that $g(t^+)$ is such that $x(t^-) \notin \Phi(t^+)$. In other words the set $\Phi(t^+) = C^{-1}(S_D^* - g(t^+)) \subset \mathbb{R}^n$ excepted when $D > 0$. Along a trajectory it may happen that although $x(t^-) \in \Phi(t^-)$, a discontinuity in $g$ implies that $x(t^-) \notin \Phi(t^+)$ and a state jump has to occur. The operator $(\mathrm{Id} + P^{-1}N_{\Phi(t^+)})^{-1}(\cdot)$ is the projection onto $\Phi(t^+)$ in the metric defined by $P = P^\top > 0$. Hence the conditions stated in Definition 1 secure that the post-jump state is admissible. The two extreme cases are $D > 0$ ($S_D = \{0\} \Leftrightarrow S_D^* = \mathbb{R}^m \Rightarrow x(t^+) = x(t^-)$), and $D = 0$ ($\Rightarrow S_D = \mathbb{R}_+^m \Leftrightarrow S_D^* = \mathbb{R}_+^m$ and there may exist state jumps for suitable $g(t^+)$ and $x(t^-)$. Notice that since at $t$ the set $\Phi(t^+)$ is nonempty (since the (CQ) holds) closed convex, $x(t^+)$ is uniquely defined from (51) as a projection onto a closed nonempty convex set. The state jump law (51) secures that at any time $x(t^+) \in \Phi(t^+)$, hence the LCP: $0 \le \lambda(t) \perp Cx(t^+) + D\lambda(t) + g(t^+) \ge 0$ has a unique solution $\lambda(t) \in \mathbb{R}^m$, relying on [34, Theorem 3.8.6].

**Lemma 5.1.** A state jump occurs at $t$ along (51), if and only if there is no multiplier $\lambda(t) \in \mathbb{R}^m$ such that

$$ 0 \le \lambda(t) \perp Cx(t^{-}) + D\lambda(t) + g(t^{+}) \ge 0. \qquad (52) $$

*Proof.* ($\Leftarrow$) From [34, Theorem 3.8.6, Corollary 3.8.10] and $D \succ 0$ it follows that the LCP in (52) has a solution $\lambda(t)$ if and only if $Cx(t^-) + g(t^+) \in S_D^* \Leftrightarrow x(t^-) \in \Phi(t^+) \Leftrightarrow x(t^+) = x(t^-)$ is a solution of (51). Hence, $\lambda(t)$ is the unique solution of (52). Thus $x(t^+) = x(t^-)$ implies solvability of the LCP (equivalently, unsolvability implies a state jump). ($\Rightarrow$) Assume now that the lemma's LCP is solvable. Then $x(t^+) = x(t^-)$ is a solution and it is the unique one. $\square$
Proof. We have $\begin{pmatrix} -A_1^\top P_1 - P_1 A_1 & P_1 B_1 - C_1^\top \\ (P_1 B_1 - C_1^\top)^\top & D_1 + D_1^\top \end{pmatrix} \succ 0$, $P_1 = P_1^\top \succ 0$, and also $-A_2^\top P_2 - P_2 A_2 = 0$, $P_2 = P_2^\top \succ 0$, $D_2 + D_2^\top = 0 \Rightarrow P_2 B_2 = C_2^\top$. The inequality in (8) is composed of the matrix in (40) which is equal to:

$$ \begin{pmatrix} Q_1 & 0 & P_1 B_1 - C_1^\top & 0 \\ 0 & 0 & 0 & 0 \\ (P_1 B_1 - C_1^\top)^\top & 0 & D_1 + D_1^\top & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad (43) $$

where $Q_1 = -A_1^\top P_1 - P_1 A_1$, with the perturbation:

$$ \begin{pmatrix} 0 & P_1 L_1 H_2 + (P_2 L_2 H_1)^\top & 0 & P_1 L_1 G_2 - H_1^\top E_2^\top \\ (P_1 L_1 H_2)^\top + P_2 L_2 H_1 & 0 & P_2 L_2 G_1 - H_2^\top E_1^\top & 0 \\ 0 & (P_2 L_2 G_1 - H_2^\top E_1^\top)^\top & 0 & E_1 G_2 + G_1^\top E_2^\top \\ (P_1 L_1 G_2 - H_1^\top E_2^\top)^\top & 0 & (E_1 G_2 + G_1^\top E_2^\top)^\top & 0 \end{pmatrix} \qquad (44) $$

Permuting the third and second columns, then the third and second rows, these matrices are transformed into:

$$ \begin{pmatrix} Q_1 & P_1 B_1 - C_1^\top & 0 & 0 \\ (P_1 B_1 - C_1^\top)^\top & D_1 + D_1^\top & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}, \qquad (45) $$

and

$$ \begin{pmatrix} 0 & 0 & P_1 L_1 H_2 + (P_2 L_2 H_1)^\top & P_1 L_1 G_2 - H_1^\top E_2^\top \\ 0 & 0 & (P_2 L_2 G_1 - H_2^\top E_1^\top)^\top & E_1 G_2 + G_1^\top E_2^\top \\ (P_1 L_1 H_2)^\top + P_2 L_2 H_1 & P_2 L_2 G_1 - H_2^\top E_1^\top & 0 & 0 \\ (P_1 L_1 G_2 - H_1^\top E_2^\top)^\top & (E_1 G_2 + G_1^\top E_2^\top)^\top & 0 & 0 \end{pmatrix} \qquad (46) $$

From [9, Proposition 8.2.4] [16, Lemmata A.69, A.70], the sum of the matrices in (45) and (46) is positive semidefinite, if and only if:

$$ \begin{pmatrix} P_1 L_1 H_2 + (P_2 L_2 H_1)^\top & P_1 L_1 G_2 - H_1^\top E_2^\top \\ (P_2 L_2 G_1 - H_2^\top E_1^\top)^\top & E_1 G_2 + G_1^\top E_2^\top \end{pmatrix} = 0. \qquad (47) $$

The proof of the passivity is complete. To end the proof, let us take into account (41), calculations yield $\dot{V}(t) = (x_1^\top \lambda_1^\top) \begin{pmatrix} A_1^\top P_1 + P_1 A_1 & P_1 B_1 - C_1^\top \\ (P_1 B_1 - C_1^\top)^\top & -D_1 - D_1^\top \end{pmatrix} \begin{pmatrix} x_1 \\ \lambda_1 \end{pmatrix}$ and the result follows from strong passivity of $(A_1, B_1, C_1, D_1)$ and [9, Corollary 8.4.2]. □

The conditions for asymptotic stability of this interconnection will be studied in section 5.1.
Let us present an example which shows the complexity of interconnected passive LCS from the point of view of state jumps.

**Example 1** (From MDIs to ODE). Let us consider an interconnection which has a continuous solution at some time t while both LCS would undergo a state jump at the same time t, if these LCS were disconnected (i.e., with $u_1 \equiv 0$ and $u_2 \equiv 0$). To that aim, let us consider two scalar LCS, with $(A_i, B_i, C_i, D_i) = (A_i, 1, 1, 0)$, $P_i = 1$, $A_i < 0$, so that both quadruples are passive. Here $n_i = 1$, $m_i = 1$, $n_{u_i} = 1$, $i = 1, 2$. Then $S_{D_i} = \mathbb{R}_+ = S_{D_i}^*, \Phi_i(t^+) = \{z \in \mathbb{R} \mid z + g_i(t^+) \ge 0\}$. For any value of $x_i(t^-)$, there exists a right-continuous exogenous function $g_i(\cdot)$ such that $x_i(t^-) + g_i(t^+) < 0$. In this case, according to Lemma 5.1, a jump has to occur in $x_i$ at t, in order to guarantee that the state's right-limit belongs to the admissible domain $\Phi(t^+)$ (see Definition 1). The CQ in Definition 1 is satisfied for any $C_i \neq 0$ in this example, since $g_i(t^+) \in \mathbb{R}_+ + \text{Im}(C_i) = \mathbb{R}$. We have $\tilde{D} + \tilde{D}^\top = \begin{pmatrix} 0 & E_1G_2 + E_2G_1 \\ E_1G_2 + E_2G_1 & 0 \end{pmatrix} \succ 0$ (this is (13b)) only if $\tilde{D} + \tilde{D}^\top = 0$, which holds if and only if $E_1G_2 = -E_2G_1$, where $E_1G_2 \in \mathbb{R}$, $E_2G_1 \in \mathbb{R}$, $\tilde{D} \in \mathbb{R}^{2 \times 2}$. Hence, assuming $E_1G_2 = -E_2G_1$, we obtain from the expression of $\tilde{D}$ in (6): $S_{\tilde{D}} = \{z \in \mathbb{R}^2 \mid z_1 \ge 0, z_2 \ge 0, E_1G_2z_2 \ge 0, E_1G_2z_1 \le 0\}$. Consequently: $S_{\tilde{D}} = \{z_1 = 0, z_2 \ge 0\}$ if $E_1G_2 > 0$, and $S_{\tilde{D}} = \{z_1 \ge 0, z_2 = 0\}$ if $E_1G_2 < 0$, so that: $S_{\tilde{D}}^* = \mathbb{R} \times \mathbb{R}_+$ if $E_1G_2 > 0$, and $S_{\tilde{D}}^* = \mathbb{R}_+ \times \mathbb{R}$ if $E_1G_2 < 0$. Let $E_1G_2 > 0$ (a similar analysis can be made if $E_1G_2 < 0$). Therefore $\Phi(t^+) = \{z \in \mathbb{R}^2 \mid \tilde{C}z + g(t^+) \in S_{\tilde{D}}^*\} = \{z \in \mathbb{R}^2 \mid E_2H_1z_1 + z_2 + g_2(t^+) \ge 0\}$, where $\tilde{C}$ is defined in (6): it is a closed convex polyhedral set for each t, nonempty under the CQ in Definition 1. Let $x_i(t^-) + g_i(t^+) < 0$, $i = 1, 2$, i.e., $x_i(t^-) \notin \Phi_i(t^+)$: from Lemma 5.1 and since $D_i = 0$, each subsystem (4) when considered independently, i.e., $u_1 = u_2 = 0$, undergoes a state jump at t, with $x_i(t^+) \in \Phi_i(t^+)$. In fact from (51): $x_i(t^+) = (\text{Id} + N_{\Phi_i(t^+)})^{-1}(x_i(t^-)) = \text{proj}[\Phi_i(t^+); x_i(t^-)] = -g_i(t^+)$, the orthogonal projection of $x_i(t^-)$ onto $\Phi_i(t^+)$. Assume now that $H_1 \in \mathbb{R}$ is chosen such that $E_2H_1x_1(t^-) + x_2(t^-) + g_2(t^+) > 0 \Leftrightarrow x(t^-) \in \Phi(t^+)$. The generalized equation in (51) reduces to $x(t^+) = (\text{Id} + N_{\Phi_i(t^+)})^{-1}(x(t^-))$, where the resolvent $(\text{Id} + N_{\Phi_i(t^+)})^{-1}(x) = \text{proj}[\Phi(t^+); x]$. Since the set $\Phi(t^+)$ is closed convex nonempty and $x(t^-) \in \Phi(t^+)$, the solution is unique and it satisfies $x(t^+) = x(t^-)$. Thus, let us choose $E_1G_2 + E_2G_1 = 0$, so that $\tilde{D} + \tilde{D}^\top = 0$ (hence (13b) holds), then passivity of the interconnection holds if $\alpha_1 + P_{12}L_2G_1 = 1$, $\alpha_1L_1G_2 + P_{12} = E_2H_1$, $P_{12} + \alpha_2L_2G_1 = E_1H_2$, $P_{12}L_1G_2 + \alpha_2 = 1$ (this guarantees $P\tilde{B} = \tilde{C}^\top$ from (13a) and $(\tilde{D} + \tilde{D}^\top)^\dagger = 0$), $A_1 < 0$, $A_2 < 0$ (this is (13c)), which implies that (13d) holds since $(\tilde{D} + \tilde{D}^\top)^\dagger = 0$. These conditions are satisfied with $E_1G_2 > 0$, $P_{12} = 0$, $\alpha_1 = \alpha_2 = 1$, $H_1 = \frac{-L_1}{L_2}H_2$, $G_1 = \frac{E_1}{L_2}H_2$, $G_2 = \frac{E_2}{L_2}H_2$, $-\frac{E_2L_1}{L_2}H_2x_1(t^-) > -x_2(t^-) - g_2(t^+)$. For instance, let $E_1G_2 = 1$, $E_2G_1 = -1$, $H_2 = 1$, the interconnection (6) is equal to:

$$ \begin{aligned} \dot{x} &= \begin{pmatrix} A_1 & L_1 \\ -L_1 & A_2 \end{pmatrix} x + \begin{pmatrix} 1 & \frac{-L_1E_2}{L_2} \\ E_1 & 1 \end{pmatrix} \lambda + f(t) \\ 0 \le \lambda \perp \begin{pmatrix} \frac{1}{-L_1E_2} & E_1 \\ \frac{1}{L_2} & 1 \end{pmatrix} x + \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \lambda + g(t) \ge 0. \end{aligned} \tag{53} $$

It is inferred that there exist $H_2$, $L_1$, $L_2$, $E_1$, $E_2$, $g_1(t^+)$ and $g_2(t^+)$ such that the subsystems' states would jump at times t if they were disconnected, while the interconnected system's state remains continuous at such times, while all three quadruples $(A_i, B_i, C_i, D_i)$, $i = 1, 2$, and $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ are passive. Hence, the interconnection of two MDIs becomes an ordinary differential inclusion with continuous solutions.

Let us express various sets:

$$ \begin{aligned} S_D &= \mathbb{R}_+^m \cap \tilde{D}^{-1}(\mathbb{R}_+^m) \cap \text{Ker}(\tilde{D} + \tilde{D}^\top) \Leftrightarrow S_{\tilde{D}}^* &= \mathbb{R}_+^m + \text{Im}(\tilde{D} + \tilde{D}^\top) + \tilde{D}^\top \mathbb{R}_+^m \\ \tilde{\Phi}(t^+) &= \{z = (z_1^\top, z_2^\top)^\top \in \mathbb{R}^n \mid \begin{pmatrix} C_1z_1 + E_1H_2z_2 + g_1 \\ E_2H_1z_1 + C_2z_2 + g_2 \end{pmatrix} \in \mathbb{R}_+^m + \text{Im}(\tilde{D} + \tilde{D}^\top) + \tilde{D}^\top \mathbb{R}_+^m \}, \end{aligned} \tag{54} $$

where [59, Corollary 16.4.2] is used. Let $D_1 = 0$, $D_2 = 0$, so that both subsystems in (4) (with $u_1 \equiv 0$ and $u_2 \equiv 0$) are equivalent to first-order sweeping processes (FOSwP) after the state-space transformation $z_i = R_i x_i$, $R_i = R_i^\top > 0$, $R_i^2 = P_i$ [18]:

$$ dz_i \in R_i A_i R_i^{-1} z_i(t) dt - N_{S_i(t^+)}(z_i(t^+)), \tag{55} $$

with $S_i(t^+) = R_i U_i(t^+)$, $U_i(t^+) = \{z \in \mathbb{R}^{m_i} \mid C_i z + g_i(t^+) \ge 0\}$ are closed convex polyhedral nonempty sets under suitable
## 3 Analysis of State Jumps

The objective is to analyze how the interconnection (6), equivalently (9), influences the state jumps behaviours. In this section it is assumed that $K_i = \mathbb{R}_+^{m_i}$, $i = 1, 2$. When state jumps are present, the LCS in (4) and (6) have to be embedded in the framework of Measure Differential Inclusions (MDI) [33, 42, 17]. For systems of the form (6), when there are possible state jumps due to discontinuous inputs $f, g$, it is of interest to take the following mathematical models:

$$dx = (Ax + f)dt + B\lambda \quad (48a)$$

$$0 \le \lambda \perp w = Cx + D\lambda + g \ge 0. \quad (48b)$$

Here, we assume that the exogenous signal $g$ is right-continuous of locally bounded variation (RCLBV). The derivative of such functions is considered in the sense of distributions and is a Radon measure, which we denote by $dg$. Similarly, the measure associated with the total variation of $g$ is denoted by $|dg|$. Because of the discontinuities in $g$, it becomes necessary to allow for state trajectories to be discontinuous for satisfying (48b). These discontinuities in state-trajectories are modeled by interpreting $\lambda$ as a measure and this allows us to consider the solution $x$ of (48) in the space of RCLBV functions. The density of $dx$ with respect to a positive Radon measure $\nu$ over an interval $I$ is defined as $\frac{dx}{d\nu}(t) = \lim_{\epsilon \to 0} \frac{dx(I(t, \epsilon))}{\nu(I(t, \epsilon))}$, where $I(t, \epsilon) := I \cap [t - \epsilon, t + \epsilon]$. We say that $x: [0, T] \to \mathbb{R}^m$ is a solution to MDI (48) if there exists a Radon measure $\nu$ absolutely continuous with respect to $dt + |dg|$ such that

$$\frac{dx}{d\nu}(t) = (Ax + f)\frac{dt}{d\nu}(t) + B\frac{d\lambda}{d\nu}(t) \quad (49)$$

and (48b) holds when evaluating the measures and functions at each time instant from the right. Consider the MDI in (48), with RCLBV solutions. To start with, let us look at (48) at state jump times, then the system equations are equivalently written as:

$$x(t^+) - x(t^-) \in -B(D + \mathcal{N}_{\mathbb{R}_+^m}^{-1})(Cx(t^+) + g(t^+)). \quad (50)$$

Using the passivity of $(A, B, C, D)$, plus assumptions (A3) (A4) (A5) in [42, Theorem 3.1], the inclusion in (50) is rewritten equivalently as

$$z(t^+) - z(t^-) \in -\Psi(t, z(t^+)) \Leftrightarrow z(t^+) = (I_d + \Psi(t, \cdot))^{-1}(z(t^-)) \quad (51)$$

where $z = Rx$, $R = R^\top > 0$, $R^2 = P$, $P$ is a solution of the LMI in (77), and $\Psi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ is a maximal monotone mapping for each $t$ [42, Lemma 4]. Thus its resolvent $(I_d + \Psi(t, \cdot))^{-1}$ is single-valued Lipschitz continuous in $z$, and $x(t^+) = R^{-1}z(t^+)$ is uniquely defined from (51). Let us introduce the following, which provides a more tractable expression for the jump [20, 29, 17]:
constraint qualification [19]. Then:

$$ S_{\tilde{D}}^{\star} = \left( \frac{\operatorname{Im}(E_1 G_2 + G_1^{\top} E_2^{\top}) + G_1^{\top} E_2^{\top} \mathbb{R}_+^{m_2} + \mathbb{R}_+^{m_1}}{\operatorname{Ker}^{\perp}(E_1 G_2 + G_1^{\top} E_2^{\top}) + G_2^{\top} E_1^{\top} \mathbb{R}_+^{m_1} + \mathbb{R}_+^{m_2}} \right) \quad (56) $$

where [9, Theorem 3.5.3] is used. The next result extends Example 1 to the interconnection of two FOSwP. Let $\lambda = (\lambda_1^\top, \lambda_2^\top)^\top$, $x = (x_1^\top, x_2^\top)^\top$.

**Proposition 5.2** (State-jump suppression via the interconnection of FOSwPs). Consider (6) with $D_1 = 0$, $D_2 = 0$ and Assumption 2.1 holds, so both subsystems in (4) are controlled FOSwP as in (55). Let $x_i(t^-)$, $i = 1, 2$, and $g_i(t^+)$ be such that the problem: $0 \le \lambda_i(t) \perp C_i x_i(t^-) + g_i(t^+) \ge 0$, $i = 1, 2$, has no solution. Then the LCP: $0 \le \lambda(t) \perp \tilde{C}x(t^-) + \tilde{D}\lambda(t) + g(t^+) \ge 0$ has a solution and (13) holds, if the conditions in (22) are satisfied and the following set of constraints is solvable:

$$ E_1 H_2 x_2(t^-) \in \{C_1 x_1(t^-) + g_1(t^+)\} + G_1^\top E_2^\top \mathbb{R}_+^{m_2} + \mathbb{R}_+^{m_1} \quad (57a) $$

$$ E_2 H_1 x_1(t^-) \in \{C_2 x_2(t^-) + g_2(t^+)\} - E_2 G_1 \mathbb{R}_+^{m_1} + \mathbb{R}_+^{m_2}, \quad (57b) $$

with $\alpha_1 > 0, \alpha_2 > 0$.

*Proof.* From Corollary 3.7, passivity of the interconnection holds if the conditions listed in (22) hold. Let us now use Lemma 5.1, the generalized equation stemming from (55) (which is equivalent to (51)): $z_i(t^+) - z_i(t^-) \in -\mathcal{N}_{S_i(t^+)}(z_i(t^+))$, has a unique solution $z_i(t^+) \neq z_i(t^-)$: both FOSwPs undergo a state jump at $t$. Conditions in (22) and in the Proposition imply that $\tilde{D} + \tilde{D}^\top = 0$. Then from (56):

$$ \tilde{\Phi}(t^+) = \left\{ z \in \mathbb{R}^m \mid \begin{pmatrix} C_1 z_1 + E_1 H_2 z_2 + g_1(t^+) \\ E_2 H_1 z_1 + C_2 z_2 + g_2(t^+) \end{pmatrix} \in \begin{pmatrix} G_1^\top E_2^\top \mathbb{R}_+^{m_2} + \mathbb{R}_+^{m_1} \\ -E_2 G_1 \mathbb{R}_+^{m_1} + \mathbb{R}_+^{m_2} \end{pmatrix} \right\}. \quad (58) $$

Let $x_1(t^-)$ and $x_2(t^-)$ be given which satisfy the Proposition's assumption, i.e., $C_i x_i(t^-) < -g_i(t^+)$ (equivalently $z_i(t^-) \notin S_i(t^+, 0)$). Under the conditions (57a), (57b), it follows that $x(t^-) \in \tilde{\Phi}(t^+)$, equivalently the LCP $0 \le \lambda(t) \perp \tilde{C}x(t^-) + \tilde{D}\lambda(t) + g(t^+) \ge 0$ is solvable and the interconnection undergoes no state jump. $\square$

The set of conditions in (57) could also include nonzero $P_{12}$ for greater generality. The reverse case (from LCS-ODEs with AC solutions to a MDI interconnection with RCLBV solutions) is tackled now.

**Example 2** (From ODEs to MDI). Consider a scalar LCS ($n_1 = n_2 = m_1 = m_2 = n_{u_1} = n_{u_2} = 1$), with $(A_i, B_i, C_i, D_i) = (A_i, 1, 1, 1)$. Let $\alpha_1 = \alpha_2 = 1$, $E_1 H_2 = E_2 H_1 = 1$, $E_1 G_2 = E_2 G_1 = -1$, $L_2 G_1 = L_1 G_2 = 1$, $P_1 = P_2 = 1$, $P_{12} = 0$. Then both matrices in (7) are equal to $\begin{pmatrix} -2A_1 & 0 \\ 0 & 2 \end{pmatrix} > 0$ for any $A_i < 0$: both subsystems are strongly passive, with $S_{D_i} = \{0\}$, $S_{D_i}^* = \mathbb{R}$, $\Phi_i(t^+) = \mathbb{R}^2$, $N_{\Phi_i(t^+)} = \{0\}$. It follows that both subsystems are ordinary differential equations (ODE), with Lipschitz continuous (in $x_i$) right-hand side [18]. Measurability of $f_i(\cdot)$ and $g_i(\cdot)$ allows to assure global existence of locally AC solutions of each independent subsystem [62, Proposition C.3.8]. Now, $\tilde{D} + \tilde{D}^\top = \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix} \ge 0$ (conditions $(\alpha)$, $(\beta)$ and $(\alpha')$, $(\beta')$ in the proof of Lemma 3.2) hold), $\tilde{B} = \tilde{C} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, $P\tilde{B} - \tilde{C}^\top = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$, $-\tilde{A}^\top P - P\tilde{A} = \begin{pmatrix} -2A_1 & -L_1 H_2 - L_2 H_1 \\ -L_1 H_2 - L_2 H_1 & -2A_2 \end{pmatrix}$. It can be checked that all conditions in (13) are satisfied, provided that $A_1 < 0$ and $A_2 < 0$ with big enough magnitudes. The set $S_{\tilde{D}} = \{z \in \mathbb{R}^2 \mid z_1 \ge 0, z_2 \ge 0, z_1 = z_2\}$, and $S_{\tilde{D}}^* = \{w \in \mathbb{R}^2 \mid w_2 \ge -w_1\}$, $\tilde{\Phi}(t^+) = \{z \in \mathbb{R}^2 \mid z_1 + z_2 \ge -\frac{1}{2}(g_2(t^+) + g_1(t^+))\}$. The CQ in Definition 1 holds if $g_1(t) = g_2(t) \ge 0$ for all $t$, and (51) can possess solutions $x(t^+) \neq x(t^-)$ for suitable choices of $g(t^+)$ for which there is no bounded $\lambda(t)$ such that $\tilde{C}x(t^-) + \tilde{D}\lambda(t) + g(t^+) \ge 0$. Notice that (51) is equivalent to $x(t^+) = \operatorname{proj}[\tilde{\Phi}(t^+); x(t^-)]$, so it is sufficient that $x(t^-) \notin \tilde{\Phi}(t^+)$ to get a state jump at $t$ in the interconnection (this is always possible with suitable choice of $g(t^+)$ and $x(t^-)$, at least at $t = 0$ with specific initial state $x(0^-)$). This is due to the interconnection terms which do not allow to preserve strong passivity as in section 3.2. It is inferred that there exists passive interconnections of strongly passive LCS, which possess state jump for some exogenous excitations $g(\cdot)$ with discontinuities. Thus, two passive LCS-ODEs are transformed into a passive MDI after interconnection.
**Definition 1.** Consider the MDI in (48) with RCLBV solutions. Let us assume that $(A, B, C, D)$ is passive, with storage function $V(x) = \frac{1}{2}x^\top P x$, $P = P^\top > 0$. For any $x(t^-)$, the state $x(t^+)$ is given as the solution to:

$$P(x(t^+) - x(t^-)) \in -\mathcal{N}_{\Phi(t^+)}(x(t^+)) \Leftrightarrow x(t^+) = (I_d + P^{-1}\mathcal{N}_{\Phi(t^+)})^{-1}(x(t^-)), \quad (52)$$

where $\Phi(t^+) \triangleq \{z \in \mathbb{R}^m \mid Cz + g(t^+) \in S_D^*\}$, $S_D = \{z \in \mathbb{R}^m \mid 0 \le z \perp Dz \ge 0\}$, and the constraint qualification (CQ): $g(t) \in S_D^* + \text{Im}(C)$ holds.

Notice that since at $t$ the set $\Phi(t^+)$ is nonempty closed convex, $x(t^+)$ is uniquely defined from (52) as a projection onto a closed nonempty convex set. The state jump law (52) secures that at any time $x(t^+) \in \Phi(t^+)$, hence the LCP: $0 \le \lambda(t) \perp Cx(t^+) + D\lambda(t) + g(t^+) \ge 0$ has a unique bounded solution $\lambda(t)$, relying on [27, Theorem 3.8.6].

**Lemma 7.** A state jump occurs at $t$ along (52), if and only if there is no bounded multiplier $\lambda(t) \in \mathbb{R}^m$ such that $0 \le \lambda(t) \perp Cx(t^-) + D\lambda(t) + g(t^+) \ge 0$.

*Proof.* (=$\Leftarrow$) From [27, Theorem 3.8.6, Corollary 3.8.10] and $D \succcurlyeq 0$ it follows that the lemma's LCP has a solution $\lambda(t)$ if and only if $Cx(t^-) + g(t^+) \in S_D^* \Leftrightarrow x(t^-) \in \Phi(t^+) \Leftrightarrow x(t^+) = x(t^-)$ is a solution of (52). Hence it is the unique solution. Thus $x(t^+) = x(t^-)$ implies solvability of the LCP (equivalently, unsolvability implies a state jump). ($\Rightarrow$) Assume now that the lemma's LCP is solvable. Then $x(t^+) = x(t^-)$ is a solution and it is the unique one. $\square$

We have $g(t^+) = g(t^-) \Rightarrow x(t^+) = x(t^-)$. Let us present an example which shows the complexity of interconnected passive LCS from the point of view of state jumps.

**Example 1.** (From ODEs to MDI) Consider a scalar LCS $(n_1 = n_2 = m_1 = m_2 = n_{u_1} = n_{u_2} = 1)$, with $(A_i, B_i, C_i, D_i) = (A_i, 1, 1, 1)$. Let $\alpha_1 = \alpha_2 = 1$, $E_1H_2 = E_2H_1 = 1$, $E_1G_2 = E_2G_1 = -1$, $L_2G_1 = L_1G_2 = 1$, $P_1 = P_2 = 1$, $P_{12} = 0$. Then both matrices in (7) are equal to $\begin{pmatrix} -2A_i & 0 \\ 0 & 2 \end{pmatrix} \succcurlyeq 0$ for any $A_i < 0$: both subsystems are strongly passive, with $S_{D_i} = \{0\}$, $S_{D_i}^* = \mathbb{R}$, $\Phi_i(t^+) = \mathbb{R}^2$, $\mathcal{N}_{\Phi_i(t^+)} = \{0\}$. It follows that both subsystems are ordinary differential equations (ODE), with Lipschitz continuous (in $x_i$) right-hand side [17]. Measurability of $f_i(\cdot)$ and $g_i(\cdot)$ allows to assure global existence of AC solutions of each independent subsystem. Now, $\tilde{D} + \tilde{D}^\top = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix} \succcurlyeq 0$, $\tilde{B} = \tilde{C} = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$, $P\tilde{B} - \tilde{C}^\top = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$, $-\tilde{A}^\top P - P\tilde{A} = \begin{pmatrix} -2A_1 & -L_1H_2 - L_2H_1 \\ -L_1H_2 - L_2H_1 & -2A_2 \end{pmatrix}$. It can be checked that all conditions in (13) are satisfied, provided that $A_1 < 0$ and $A_2 < 0$ with big enough magnitudes. The set $S_{\tilde{D}} = \{z \in \mathbb{R}^2 \mid z_1 \ge 0, z_2 \ge 0, z_1 = z_2\}$, and $S_{\tilde{D}}^* = \{w \in \mathbb{R}^2 \mid w_2 \ge -w_1\}$, $\Phi(t^+) = \{z \in \mathbb{R}^2 \mid z_1 + z_2 \ge \frac{1}{2}(g_2(t^+) - g_1(t^+))\}$. The CQ in Definition 1 holds, and (52) can possess solutions $x(t^+) \neq x(t^-)$ for suitable choices of
**Lemma 5.3.** Assume that (13b) holds, $D_1 > 0$, $D_2 > 0$. Then $\mathrm{rank}(\tilde{D}) < m$ if and only if

$$ \mathrm{rank}(D_2 + D_2^\top - (E_2 G_1 + G_2^\top E_1^\top)(D_1 + D_1^\top)^{-1}(E_1 G_2 + G_1^\top E_2^\top)) = r_2 < m_2, \text{ or} \quad (59a) $$

$$ \mathrm{rank}(D_1 + D_1^\top - (E_1 G_2 + G_1^\top E_2^\top)(D_2 + D_2^\top)^{-1}(E_2 G_1 + G_2^\top E_1^\top)) = r_1 < m_1. \quad (59b) $$

*Proof.* Given the lemma's assumptions, and using the Schur complement [16, Theorem A.65], both rank conditions are necessary for $\tilde{D}$ not to be full-rank, hence not to be positive definite (but just semidefinite). $\square$

**Proposition 5.4** (State-jump creation via the interconnection of two LCS-ODEs). Consider the interconnection system (6). Assume that (i) (13) is satisfied, (ii) $D_i > 0$, $i = 1, 2$, and Assumption 2.1 holds with $P_i > 0$, (iii) The rank condition in (59a) or (59b) holds. Then both LCS with $u_i = 0$ undergo no state jumps, while there exists $x(t^-)$ and $g(t^+)$ such that a state jump occurs at $t$ in (6).

*Proof.* Both LCS when considered separately (i.e., $u_i = 0$ in (4)), possess AC solutions (without state jumps), since $S_{D_i}^* = \mathbb{R}^{m_i} \Rightarrow \Phi_i(t^+) = \mathbb{R}^{n_i} \Rightarrow x_i(t^-) \in \Phi_i(t^+)$ always (i.e., for any $g_i(t^+)$). Another way to conclude this is that the operators $(D_i + N_{K_i^*}^{-1})(\cdot)$ are Lipschitz continuous, single-valued with domain equal to $\mathbb{R}^{m_i}$ [15]. Either condition in item (iii) implies that $\tilde{\Phi}(t^+) \subset \mathbb{R}^n$. Hence, there always exist pairs $(x(t^-), g(t^+))$ such that $x(t^-) \notin \tilde{\Phi}(t^+)$, equivalently the LCP: $0 \le \lambda(t) \perp \tilde{C}x(t^-) + \tilde{D}\lambda(t) + g(t^+) \ge 0$ has no bounded solution $\lambda(t)$. $\square$

# 6. Examples

Let us illustrate the above developments with several examples. Due to space limitations complete calculations are led only for the circuits in section 6.1.

## 6.1. Circuits with Ideal Diodes

<figure><img src="image_4.png" /><figcaption>Figure 2: Circuit with ideal diode.</figcaption></figure>

Let us consider the circuit in Fig. 2, made of a capacitor C, an ideal diode D, a resistance R, a current source $u_1(t) = i_s(t)$, and a voltage source $u_2(t) = v(t)$. Let us denote $x = u_c$, where $u_c$ is the voltage across the capacitor,
$g(t^+)$ for which there is no bounded $\lambda(t)$ such that $\tilde{C}x(t^-)+\tilde{D}\lambda(t)+g(t^+) \ge 0$, equivalently $\mathrm{Im}(\tilde{D})\cap(\mathbb{R}_+^m-g(t^+)-\tilde{C}x(t^-)) = \emptyset$. Another way to see it is that for any $x_1(t^-)$ and $x_2(t^-)$ there always exists $g_1(t^+)$ and $g_2(t^+)$ such that $x(t^-) \in \mathrm{Int}(\Phi(t^+))$ so that $x(t^+) = x(t^-)$ is the solution of (52). It is inferred that there exists passive interconnections of strongly passive LCS, which possess state jump. Thus, two passive LCS-ODEs are transformed into a passive MDI after interconnection.

**Example 2.** Let us consider an interconnection which has a continuous solution at some times $t$ while the interconnected LCS would undergo a state jump at the same times $t$, if they were disconnected (i.e., with $u_1 \equiv 0$ and $u_2 \equiv 0$). Consider two scalar LCS, with $(A_i, B_i, C_i, D_i) = (A_i, 1, 1, 0)$, $P_i = 1$, $A_i < 0$, so that both quadruples are passive. Then $S_{D_i} = \mathbb{R}_+ = S_{D_i}^*$, $\Phi_i(t^+) = \{z \in \mathbb{R} \mid z + g_i(t^+) \ge 0\}$. For any value of $x_i(t^-)$ it is possible to find $g_i(t^+)$ such that $x_i(t^-) + g_i(t^+) < 0$. In this case a jump has to occur in $x_i$ at $t$. The CQ in Definition 1 is satisfied for any $C_i \ne 0$. We have
$$ \tilde{D} + \tilde{D}^\top = \begin{pmatrix} 0 & E_1G_2 + E_2G_1 \\ E_1G_2 + E_2G_1 & 0 \end{pmatrix} \succ 0 \Rightarrow \tilde{D} + \tilde{D}^\top = 0 \Leftrightarrow E_1G_2 = -E_2G_1. $$
Hence $S_{\tilde{D}} = \{z \in \mathbb{R}^2 \mid z_1 \ge 0, z_2 \ge 0, E_1G_2z_2 \ge 0, E_1G_2z_1 \le 0\}$. Thus $S_{\tilde{D}} = \{z_1 = 0, z_2 \ge 0\}$ if $E_1G_2 \ge 0$, $S_{\tilde{D}} = \{z_1 \ge 0, z_2 = 0\}$ if $E_1G_2 \le 0$, and $S_{\tilde{D}}^* = \mathbb{R} \times \mathbb{R}_+$ if $E_1G_2 \ge 0$, $S_{\tilde{D}}^* = \mathbb{R}_+ \times \mathbb{R}$ if $E_1G_2 \le 0$. Let $E_1G_2 \ge 0$. Therefore $\Phi(t^+) = \{z \in \mathbb{R}^2 \mid E_2H_1z_1 + C_2z_2 + g_2(t^+) \ge 0\}$ is a convex polyhedral set for each $t$, nonempty under the CQ. Let $x_i(t^-) + g_i(t^+) < 0$, $i = 1, 2$, so that $x_i(t^+) = -g_i(t^+) \ne x_i(t^-)$. Assume now that $H_1$ and $H_2$ are chosen such that $x_1(t^-) + E_1H_2x_2(t^-) + g_1(t^+) > 0$ and $E_2H_1x_1(t^-) + x_2(t^-) + g_2(t^+) > 0$. Then the unique solution of (52) satisfies $x(t^+) = x(t^-)$. Let $\tilde{D} = 0$, then passivity of the interconnection holds if $\alpha_1 + P_{12}L_2G_1 = 1$, $\alpha_1L_1G_2 + P_{12} = E_2H_1$, $P_{12} + \alpha_2L_2G_1 = E_1H_2$, $P_{12}L_1G_2 + \alpha_2 = 1$ (this is $P\tilde{B} = \tilde{C}^\top$), $A_1 < 0$, $A_2 < 0$ with sufficiently large magnitudes. This holds with $P_{12} = 0$, $\alpha_1 = \alpha_2 = 1$, $H_1 = \frac{L_1}{L_2}H_2$, $G_1 = \frac{E_1}{E_2}H_2$, $G_2 = \frac{E_2}{L_2}H_2$, $\frac{E_2L_1}{L_2}H_2x_1(t^-) > -x_2(t^-) - g_2(t^+) > 0$, $E_1H_2x_2(t^-) > -x_1(t^-) - g_1(t^+) > 0$. It is inferred that there exist $H_2$, $\frac{E_2L_1}{L_2}$, $g_1(t^+)$ and $g_2(t^+)$ such that the subsystems's states would jump at times $t$ if they were disconnected, while the interconnection's state does not. The interconnection of two MDIs becomes a DI with continuous solutions.

Let us express various sets:

$$ \begin{aligned} S_{\tilde{D}} &= \mathbb{R}_+^m \cap \tilde{D}^{-1}(\mathbb{R}_+^m) \cap \mathrm{Ker}(\tilde{D} + \tilde{D}^\top) \Leftrightarrow S_{\tilde{D}}^* = \mathbb{R}_+^m + \mathrm{Im}(\tilde{D} + \tilde{D}^\top) + \tilde{D}^\top \mathbb{R}_+^m \\ \Phi(t^+) &= \{z \in \mathbb{R}^m \mid \begin{pmatrix} C_1z_1 + E_1H_2z_2 + g_1 \\ E_2H_1z_1 + C_2z_2 + g_2 \end{pmatrix} \in \mathbb{R}_+^m + \mathrm{Im}(\tilde{D} + \tilde{D}^\top) + \tilde{D}^\top \mathbb{R}_+^m \} \end{aligned} \quad (53) $$

where [36, Corollary 16.4.2] is used. Let $D_1 = 0$, $D_2 = 0$, so that both subsystems in (4) (with $u_1 \equiv 0$ and $u_2 \equiv 0$) are equivalent to first-order sweeping processes (FOSwP) after the state-space transformation $z_i = R_i x_i$, $R_i = R_i^\top > 0$, $R_i^2 = P_i$ [17]:

$$ dz_i \in R_i A_i R_i^{-1} z_i(t) dt - \mathcal{N}_{S_i(t^+)}(z_i(t^+)), \quad (54) $$
$\lambda = i$, where $i$ is the current through the diode, $u_d = u_c + u_2 + Ri$ is the voltage across the diode. Using $C\dot{u}_c = i - i_s$ and Kirchhoff's laws, the circuit's dynamics is:

$$ \begin{cases} \dot{x} = \frac{\lambda}{C} - \frac{u_1}{C} = \frac{\lambda}{C} + (-\frac{1}{C}) 0u \\ 0 \le \lambda \perp x + u_2 + R\lambda = R\lambda + x + (0 \ 1)u \ge 0. \end{cases} \qquad (60) $$

Referring to (4) we have for two circuits to be interconnected ($i = 1, 2$): $u_i = (u_{i,1} \ u_{i,2})^T$, $K_i = \mathbb{R}_+$, $A_i = 0$, $B_i = \frac{1}{C}$, $L_i = (l_i \ 0)$ with $l_i = -\frac{1}{C}$, $f_i = 0$, $g_i = 0$, $C_i = 1$, $D_i = R_i$, $E_i = (0 \ 1)$. To simplify notations we denote $\dot{x}_i = b_i\lambda + (l_i \ 0)u_i$, $i = 1, 2$ for the two interconnected circuits. We have $n_{u1} = n_{u2} = 2$, $m_1 = m_2 = 1$, $n_1 = n_2 = 1$, $H_i = \begin{pmatrix} h_{i1} \\ h_{i2} \end{pmatrix} \in \mathbb{R}^{2 \times 1}$, $G_i = \begin{pmatrix} g_{i1} \\ g_{i2} \end{pmatrix} \in \mathbb{R}^{2 \times 1}$, $\tilde{A} = \begin{pmatrix} 0 & l_1 h_{21} \\ l_2 h_{11} & 0 \end{pmatrix}$, $\tilde{B} = \begin{pmatrix} b_1 & l_1 g_{21} \\ l_2 g_{11} & b_2 \end{pmatrix}$, $\tilde{C} = \begin{pmatrix} 1 & h_{22} \\ h_{12} & 1 \end{pmatrix}$, $\tilde{D} = \begin{pmatrix} R_1 & g_{22} \\ g_{12} & R_2 \end{pmatrix}$. We have $P_i = C_i$ (from the energy contained in the capacitor) so that the passivity LMI in (7) is equal to $\begin{pmatrix} 0 & 0 \\ 0 & 2R_i \end{pmatrix} \ge 0$ (hence Assumption 2.1 is satisfied). To simplify calculations let us assume that $C_1 = C_2 = C$ and $R_1 = R_2 = R$. Then $P = \begin{pmatrix} \alpha_1 C & P_{12} \\ P_{12} & \alpha_2 C \end{pmatrix} \in \mathbb{R}^{2 \times 2}$, $Q_1 = Q_2 = 0$, $\bar{Q} = \begin{pmatrix} 2P_{12}l_1h_{11} & \alpha_1 C l_1 h_{21} + \alpha_2 C l_2 h_{11} \\ \alpha_1 C l_1 h_{21} + \alpha_2 C l_2 h_{11} & 2P_{12}l_1h_{21} \end{pmatrix}$, $\bar{Q}_1 = -2P_{12}l_1h_{11}$, $\bar{Q}_2 = -2P_{12}l_1h_{21}$, $\bar{Q}_{12} = \alpha_1 C l_1 h_{21} + \alpha_2 h_{11} l_2 C$.

1. Let us check conditions in (15): (i) $\mathrm{Im}(g_{11}l_2P_{12}) \subseteq \mathrm{Im}(\alpha_1 - 1)$, (ii) $\mathrm{Im}(\frac{P_{12}}{C} + \alpha_2g_{11}l_2C - h_{22}) \subseteq \mathrm{Im}(\alpha_1 - 1)$, (iii) $\mathrm{Im}(g_{21}l_1P_{12}) \subseteq \mathrm{Im}(\alpha_2 - 1)$, (iv) $\mathrm{Im}(\alpha_1g_{21}l_1C + \frac{P_{12}}{C} - h_{12}) \subseteq \mathrm{Im}(\alpha_2 - 1)$, (v) through (viii) are satisfied if $R > 0$. Conditions (i) through (iv) hold if $\alpha_i > 0$ and $\alpha_i \ne 1$, $i = 1, 2$.

2. Lemma 3.2 items 1 and 2 give $(g_{22} + g_{12})^2 < 4R^2$. If $R = 0$ then Lemma 3.2 item 3 applies with $g_{22} = -g_{21}$.

3. Lemma 3.3 item 1 holds in this scalar case only if (i) $\bar{Q}_1 = -2P_{12}l_2h_{11} > 0$, (ii) $\bar{Q}_2 = -2P_{12}l_1h_{21} > 0$, (iii) $\mathrm{Im}(h_{21}l_1C + C(l_2h_{11}) \subseteq \mathrm{Im}(P_{12}l_2h_{11})$, and (iv) $\bar{Q}_{12}^2 < \bar{Q}_1\bar{Q}_2$, i.e., $(\alpha_1 C l_1 h_{21} + \alpha_2 C h_{11} l_2)^2 < 4P_{12}^2 l_1 l_2 h_{11} h_{21}$.

4. Let us now examine Lemma 3.5's conditions. Let $d = 2$ ($\Leftrightarrow 4R^2 > (g_{22} + g_{12})^2$ which is in item 2), $P_{12} = -l_2h_{11} = -l_1h_{21}$ (since $P_{12} \ne 0$ this guarantees (i) (ii) (iii) in item 3, while (iv) is $C^2(\alpha_1 + \alpha_2)^2 < 4(l_1h_{11})^2$ which can always be satisfied by taking $\alpha_i$ small enough or $h_{11} = \frac{l_1}{l_2}h_{21}$ large enough). Inequality (18) holds if $\alpha_1$ and $\alpha_2$ are small enough, $l_2h_{11}$ and $l_1h_{21}$ are large enough, or if $R$ is large enough (indeed the left-hand side of inequality (18) does not involve $R$).

5. Theorem 3.6 yields: (a) $P_{12}^2 < \alpha_1\alpha_2 C^2$, (b) $\alpha_1\alpha_2 C^2 \ge P_{12}^2$, that is: $P_{12}^2 < \alpha_1\alpha_2 C^2$. In view of item 4 this gives $(l_2h_{11})^2 = (l_1h_{21})^2 < \alpha_1\alpha_2 C^2$, which indicates that couplings $h_{11}$ and $h_{21}$ should be small.

Therefore it is always possible to choose coupling $H_i$ and $G_i$, as well as parameters $\alpha_i$ so that the interconnection of these two circuits is passive. It happens that checking the above conditions one by one is simpler than considering (13) as a whole.

Corollary 3.7 and (22) yield the following: $g_{22} = -g_{12}$, $C l_1 g_{21} = h_{12}$, $C l_2 g_{11} = h_{22}$, $l_1 h_{21} = -l_2 h_{11}$. This represents a simple set of interconnections. Notice that taking $\alpha_i = 1$ in item 1 (i)-(iv) implies that the left-hand sides of the inclusions are zero, which is satisfied with Corollary 3.7's conditions. Corollary 3.7 shows the sufficiency of Lemma 3.5's conditions studied in item 4 above. However considering nonzero $P_{12}$ allows the designer to enlarge the set of possible connections.

**Remark 6.1.** This example highlights three facts: 1) neither Theorem 3.9 nor Theorem 3.11 apply to these circuits interconnection since they are neither strongly nor strictly passive, 2) the development of numerical tools is necessary to apply the proposed tests to more complex circuits, 3) considering general interconnections as we did yields complex criteria for passivity preservation, as expected. Point 3 should not come as a surprise, since passivity (neither strong nor strict) can be easily destroyed and is thus logically, using a symmetric argument, hard to preserve.

Let us reconsider this example without current source (i.e., $i_s = u_1 = 0$) and with $R = 0$. Thus the only input is inside complementarity conditions:

$$ \begin{cases} \dot{x} = \frac{\lambda}{C} \\ 0 \le \lambda \perp x + u \ge 0. \end{cases} \qquad (61) $$
with $S_i(t^+) = R_i U_i(t^+)$, $U_i(t^+) = \{z \in \mathbb{R}^{m_i} | C_i z + g_i(t^+) \ge 0\}$ are closed convex polyhedral nonempty sets under suitable constraint qualification [18]. Then:

$$S_{\tilde{D}}^{*} = \left( \begin{array}{l} \mathrm{Im}(E_1 G_2 + G_1^{\top} E_2^{\top}) + G_1^{\top} E_2^{\top} \mathbb{R}_{+}^{m_2} + \mathbb{R}_{+}^{m_1} \\ \mathrm{Ker}^{\perp}(E_1 G_2 + G_1^{\top} E_2^{\top}) + G_2^{\top} E_1^{\top} \mathbb{R}_{+}^{m_1} + \mathbb{R}_{+}^{m_2} \end{array} \right) \quad (55)$$

where [10, Theorem 3.5.3] is used. The next result extends Example 2 to the interconnection of two FOSwP. Let $\lambda = (\lambda_1^\top, \lambda_2^\top)^\top$, $x = (x_1^\top, x_2^\top)^\top$.

**Proposition 5** (State-jump suppression via the interconnection of FOSwPs). Consider (6) with $D_1 = 0$, $D_2 = 0$ and Assumption 1 hold, so both subsystems are controlled FOSwP as in (54). Let $x_i(t^-)$, $i = 1,2$, and $g_i(t^+)$ be such that $0 \le \lambda_i(t) \perp C_i x_i(t^-) + g_i(t^+) \ge 0$, $i = 1,2$, has no solution. Then there exist interconnection matrices such that the LCP: $0 \le \lambda(t) \perp \tilde{C}x(t^-) + \tilde{D}\lambda(t) + g(t^+) \ge 0$ has a solution and (13) holds, if the following set of constraints is solvable:

$$ \begin{align} (a) \quad & E_1 G_2 = -(E_2 G_1)^{\top} \nonumber \\ (b) \quad & P_1 L_1 G_2 = (E_2 H_1)^{\top} \nonumber \\ (c) \quad & P_2 L_2 G_1 = (E_1 H_2)^{\top} \nonumber \\ (d) \quad & P_1 L_1 H_2 = -(P_2 L_2 H_1)^{\top} \tag{56} \\ (e) \quad & E_1 H_2 x_2(t^-) \in -\{C_1 x_1(t^-) + g_1(t^+)\} + G_1^{\top} E_2^{\top} \mathbb{R}_{+}^{m_2} + \mathbb{R}_{+}^{m_1} \nonumber \\ (f) \quad & E_2 H_1 x_1(t^-) \in -\{C_2 x_2(t^-) + g_2(t^+)\} - E_2 G_1 \mathbb{R}_{+}^{m_1} + \mathbb{R}_{+}^{m_2}. \nonumber \end{align} $$

*Proof.* Let us use Lemma 7, the generalized equation stemming from (54) (which is equivalent to (52)): $z_i(t^+) - z_i(t^-) \in -\mathcal{N}_{S_i(t^+)}(z_i(t^+))$, has a unique solution $z_i(t^+) \ne z_i(t^-)$. Assume that (56) (a) holds, equivalently $\tilde{D} + \tilde{D}^\top = 0$. Then from (55):

$$\Phi(t^+) = \left\{ z \in \mathbb{R}^m \mid \left( \begin{array}{l} C_1 z_1 + E_1 H_2 z_2 + g_1(t^+) \\ E_2 H_1 z_1 + C_2 z_2 + g_2(t^+) \end{array} \right) \in \left( \begin{array}{l} G_1^\top E_2^\top \mathbb{R}_+^{m_2} + \mathbb{R}_+^{m_1} \\ -E_2 G_1 \mathbb{R}_+^{m_1} + \mathbb{R}_+^{m_2} \end{array} \right) \right\}. \quad (57)$$

Let $x_1(t^-)$ and $x_2(t^-)$ be given which satisfy the Proposition's assumption, i.e., $C_i x_i(t^-) < -g_i(t^+)$ (equivalently $z_i(t^-) \notin S_i(t^+, 0)$). The goal is to prove that under these conditions, $x(t^-) \in \Phi(t^+)$, equivalently (56) (e) (f) hold. From Lemma 2, Corollary 1, Corollary 2 and Lemma 5, passivity of the interconnection holds if (56) (a) (b) (c) (d) hold. Thus we obtain the six constraints (56). $\square$

By interconnection matrices it is meant $L_i$, $E_i$, $H_i$, $G_i$, $i = 1,2$. These eight matrices plus $\alpha_1 > 0$ and $\alpha_2 > 0$ have to solve (56) for given $C_i x_i(t^-) + g_i(y^+) < 0$, $i = 1,2$. The set of conditions in (56) could also include nonzero $P_{12}$ for greater generality. The reverse case (from decoupled LCS-ODEs to a MDI interconnection) is not presented for the sake of briefness.
Hence $H_i = h_i$, $G_i = \bar{g}_i$, $L_i = 0$, $E_i = 1$, $D_i = 0$, $C_i = 1$, $B_i = \frac{1}{C}$. Then:

1. Conditions in (15): (i) (iii) hold, (v) and (vi) yield $\bar{g}_1 + \bar{g}_2 = 0$, (ii) yields $\mathrm{Im}(P_{12}b_1 - h_2) \subseteq \mathrm{Im}(\alpha_1 - 1)$, (iv) yields $\mathrm{Im}(P_{12}b_2 - h_1) \subseteq \mathrm{Im}(\alpha_2 - 1)$, (vii) yields $\mathrm{Im}(\alpha_1 - 1) \subseteq \{0\}$, (viii) yields $\mathrm{Im}(\alpha_2 - 1) \subseteq \{0\}$. Hence $\alpha_1 = \alpha_2 = 1$ and $P_{12} = \frac{h_1}{b_2} = \frac{h_2}{b_1} = \frac{1}{C}$.

2. Lemma 3.2 item 3 yields $\bar{g}_1 + \bar{g}_2 = 0$.

3. We have $\bar{Q}_1 = 0$, $\bar{Q}_2 = 0$, $Q_1 = Q_2 = 0$, $\bar{Q}_{12} = 0$, so that $\bar{Q} = 0$. Thus the conditions stated in the first part of Lemma 3.3, that is, (i), (ii), and (iii) hold. Consequently, (13c) holds trivially (as well as item 3 Lemma 3.2).

4. Conditions stated in Lemma 3.5 item 1 do not hold since here $P_{12} = 0$, while in item 2 the right-hand side of inequality (19) is null, hence $N_1 = 0$ and $N = 0$. Consequently (19) and then (13d) hold if $P\bar{B} = \bar{C}^\top$, equivalently $\alpha_1 C b_1 = 1$, $P_{12}b_2 = h_1$, $P_{12}b_1 = h_2$, $\alpha_2 C b_2 = 1$. We obtain: $P_{12} = \frac{h_1}{b_2} = \frac{h_2}{b_1}$ (same as in item 1 above) and $C b_1 = C b_2 = 1$ which is always true since $b_i = \frac{1}{C}$. It may also be imposed $\bar{g}_1 + \bar{g}_2 = 0$ which implies $\bar{D} + \bar{D}^\top = 0$ so that (13d) holds.

5. Let us check items (a) and (b) in Theorem 3.6. (a): $P_{12}^2 = \left(\frac{h_1}{b_2}\right)^2 = \left(\frac{h_2}{b_1}\right)^2 < \alpha_1 \alpha_2 C^2 = C^2$. (b) $\alpha_2 C = C \ge \frac{1}{\alpha_1} P_{12}^2 \frac{1}{C} = \left(\frac{h_2}{b_1}\right)^2 \frac{1}{C} = \left(\frac{h_1}{b_2}\right)^2 \frac{1}{C}$, which is implied by (a).

To conclude the interconnection is passive with $\alpha_1 = \alpha_2 = 1$, $P_{12} = \frac{h_1}{b_2} = \frac{h_2}{b_1} < C$, $\bar{g}_1 = -\bar{g}_2$. Corollary 3.7 does not apply since conditions (22b) and (22c) yield $h_1 = h_2 = 0$. A nonzero $P_{12}$ is needed.

Finally let us consider $u_2 = 0$, so that the only input $u_1 = i_s = u$ is in the ODE part. Referring to (4) and considering $C_i = C$, $R_i = R$, $i = 1, 2$, we have $A_i = 0$, $B_i = \frac{1}{C} = b$, $L_i = -\frac{1}{C} = l$, $E_i = 0$, $D_i = R > 0$. The dynamics in (60) becomes:

$$ \begin{cases} \dot{x} = \frac{\lambda}{C} - \frac{u}{C} \\ 0 \le \lambda \perp x + R\lambda \ge 0. \end{cases} \tag{62} $$

We have also $H_i = h_i$, $G_i = \bar{g}_i$, $i = 1, 2$. Proceeding as in the two foregoing cases:

1. Conditions in (15): (i) $\mathrm{Im}(g_1 l P_{12}) \subseteq \mathrm{Im}(\alpha_1 - 1)$, (ii) $\mathrm{Im}(bP_{12} + \alpha_2 g_1 l C) \subseteq \mathrm{Im}(\alpha_1 - 1)$, (iii) $\mathrm{Im}(g_2 l P_{12}) \subseteq \mathrm{Im}(\alpha_2 - 1)$, (iv) $\mathrm{Im}(\alpha_1 g_2 l C + bP_{12}) \subseteq \mathrm{Im}(\alpha_2 - 1)$, (v)–(viii) are satisfied if $R > 0$. Conditions (i)–(iv) hold if $\alpha_i \ne 1$, $\alpha_i > 0$.

2. Lemma 3.2 item 3 yields $\bar{g}_1 + \bar{g}_2 = 0$.

3. We have $\bar{A} = \begin{pmatrix} 0 & lh_2 \\ lh_1 & 0 \end{pmatrix}$, $Q_1 = Q_2 = 0$, $\bar{Q}_1 = -2P_{12}lh_1$, $\bar{Q}_2 = -2P_{12}lh_2$, $\bar{Q}_{12} = lC(\alpha_1h_2 + \alpha_2h_1)$. Lemma 3.3 item 1 gives: $-2P_{12}lh_1 \ge 0$, so that choosing $P_{12} = -l_1h_1$ gives $\mathrm{Im}(lh_2) + \mathrm{Im}(l^2h_1^2) \subseteq \mathrm{Im}(l^2h_1^2)$, which is satisfied, and $C^2l^2(\alpha_1h_2 + \alpha_2h_1)^2 < 4l^4h_1^3h_2$, which holds if $h_1$ is large enough or if $\alpha_i$ are small enough.

4. Lemma 3.5: $d = 2$ since $\tilde{D} + \tilde{D}^\top = \mathrm{diag}(2R) > 0$, so item 1 conditions yield $P_{12} = -lh_1 = -lh_2$, hence $h_1 = h_2 = h$ condition $\sigma_{\max}^2(\bar{Q}_{12}) < \lambda_{\min}(\bar{Q}_1)\lambda_{\min}(\bar{Q}_2)$ is $C^2l^2(\alpha_1h_2 + \alpha_2h_1)^2 < 4l^4h_1^3h_2$ that is $C^2(\alpha_1 + \alpha_2)^2 < 4l^2h^2$, which holds if $h$ is large enough. The right-hand side of inequality (18) involves the maximum eigenvalue of $\begin{pmatrix} 2l^2h^2 & lCh(\alpha_1 + \alpha_2) \\ lCh(\alpha_1 + \alpha_2) & 2l^2h^2 \end{pmatrix}$. So inequality (18) holds provided $\alpha_i$ are small enough or $h$ is large enough.

5. Conditions (a) in Theorem 3.6 holds if $l^2h^2 < \alpha_1\alpha_2 C^2$. Condition (b)'s first and second conditions hold trivially since $P_1 = C$, third condition is $\alpha_1\alpha_2 C^2 \ge l^2h^2$.
# 4 Maximal Monotonicity and Incremental Passivity

Under some basic conditions (see Theorem 3), the negative feedback interconnection of a passive system with a maximal monotone static nonlinearity (possibly set-valued) defines a DI: $\dot{x} \in -\mathcal{F}(x) = \tilde{A}x - \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$ with $\mathcal{F}(\cdot)$ a maximal monotone operator with domain $\{x \in \mathbb{R}^n \mid \tilde{C}x \in \text{dom}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1} = \text{Im}(\tilde{D} + \mathcal{N}_{K^*}^{-1})\} = \tilde{C}^{-1}(\text{Im}(\tilde{D} + \mathcal{N}_{K^*}^{-1}))$. This motivates the next result.

**Corollary 3.** Consider subsystems in (4) under Assumption 1, and their interconnection in (6). Let $\text{Im}(\tilde{C}) \cap \text{rint}(\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})) \neq \emptyset$. If the conditions in Proposition 1 (resp. 2, resp. 3) are satisfied, then the LCS in (6) has a maximal monotone (resp. single valued Lipschitz strongly monotone; resp. strongly monotone) right-hand side mapping $\mathcal{F}(\cdot)$.

*Proof.* If the conditions of Proposition 1 hold, then $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ is passive, hence under the stated condition, the mapping $\mathcal{F}(\cdot)$ is maximal monotone, see Theorem 3. Let the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be strongly passive, then there exists $\mu > 0$ such that $(\tilde{A} + \mu I_n, \tilde{B}, \tilde{C}, \tilde{D})$ is passive (this holds for all $0 < \mu < \lambda_{\min}(\tilde{Q})$ with $\tilde{Q}$ in (8), applying for instance Corollary 8). Therefore the mapping $\tilde{\mathcal{F}}: x \mapsto -(\tilde{A} + \mu I_n)x + \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$ is maximal monotone, and $\langle x_1 - x_2, y_1 - y_2 \rangle \ge \mu \|x_1 - x_2\|^2$ for all $x_i \in \text{dom}(\mathcal{F})$, $y_i \in \mathcal{F}(x_i)$, $i = 1, 2$. In addition strong passivity implies $\tilde{D} \succ 0$, hence applying [15, Proposition 1] the mapping $(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}$ is single-valued, well-defined and Lipschitz continuous with constant $\frac{1}{\lambda_{\min}(\tilde{D} + \tilde{D}^\top)}$. Let now $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be strictly state passive, i.e., $\tilde{Q} \succ 0$ and $-\tilde{A}^\top P + P\tilde{A} - \mu P \succ 0$ for some $\mu > 0$. Then the quadruple $(\tilde{A} + \frac{\mu}{2}I_n, \tilde{B}, \tilde{C}, \tilde{D})$ is passive. Thus the mapping $x \mapsto -(\tilde{A} + \mu I_n)x + \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x)$ is maximal monotone. Hence $\mathcal{F}(\cdot)$ is $\mu$-strongly monotone (but not necessarily single valued). $\square$

Let us analyze the condition in Theorem 3 and in Corollary 3.

**Lemma 8.** Let the quadruple $(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D})$ be passive. Assume that $\tilde{D} = \tilde{D}^\top$ ($\Leftrightarrow D_1 = D_1^\top, D_2 = D_2^\top, E_2 G_1 = (E_1 G_2)^\top$). Assume that $\{\text{Im}\left(\begin{pmatrix} C_1 \\ E_2 H_1 \end{pmatrix}\right) + \text{Im}\left(\begin{pmatrix} E_1 H_2 \\ C_2 \end{pmatrix}\right)\} \cap \{\text{rint}(K^*) + \text{rint}(\text{Im}(\tilde{D}))\} \neq \emptyset$. Then $\text{Im}(\tilde{C}) \cap \text{rint}(\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})) \neq \emptyset$.

*Proof.* Let us characterize $\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})$. Both mappings $x \mapsto \tilde{D}x$ and $\mathcal{N}_{K^*}^{-1}(\cdot)$ are maximal monotone, and they are both the subdifferential of proper, convex lower semi-continuous functions, because $\tilde{D}$ is symmetric, and $\mathcal{N}_{K^*}^{-1}(\cdot) = \partial\sigma_{K^*}(\cdot)$. Hence both mappings satisfy the property (*) stated in [12, section 1]. From [8, Corollary 24.4 (ii)], the sum is maximal monotone. Using [12, Theorems 3, 4], we have $\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D}) = \overline{\text{Im}(\mathcal{N}_{K^*}^{-1})} + \text{Im}(\tilde{D}) = \mathbb{R}_+^m + \text{Im}(\tilde{D})$, and $\text{int}(\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D})) = \text{int}(\text{Im}(\mathcal{N}_{K^*}^{-1}) + \text{Im}(\tilde{D})) = \text{int}(K^* + \text{Im}(\tilde{D}))$. Therefore $\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D}) = K^* + \text{Im}(\tilde{D})$. Using [36, Corollary 6.6.2], we have $\text{rint}(K^* + \text{Im}(\tilde{D})) = \text{rint}(K^*) + \text{rint}(\text{Im}(\tilde{D})$. The proof finishes by applying [10, Fact 3.14.8] to $\text{Im}(\tilde{C})$. $\square$
## 6.2. Back to motivating example

Consider the interconnection in (2) (3). Assume that $D > 0$ and that each subsystem is passive with $P_1$ for (2), $P_2$ for (3). The conditions in Theorem 3.9 can be checked with $A_1 = A + LM$, $L_1H_2 = -LM$, $L_2H_1 = 0$, $A_2 = A - K_1H$, $B_1 = B$, $L_1G_2 = 0$, $L_2G_1 = B$, $B_2 = -B$, $C_1 = E_2H_1 = C + EM$, $E_1H_2 = -EM$, $C_2 = -(C - K_2H + EM)$, $D_1 = D_2 = D$, $E_1G_2 = 0$, $E_2G_1 = 0$. Notice that this proposition can be used even if the subsystems are not strongly passive. Condition (28) holds trivially. Condition (29) holds if there exists $M$ and $K_1$ such that the poles of each matrix $A_1$ and $A_2$ can be placed arbitrarily, or if $\alpha_1$ and $\alpha_2$ are large enough. Condition (30) holds if $D$ is large enough and if the poles of each matrix $A_1$ and $A_2$ can be placed arbitrarily. Other cases with $D \ge 0$ can be studied along the lines of section 2.

## 6.3. Switching DAEs

Switching (or nonsmooth) DAEs are an area of interest for several control related problems [14, 65, 64, 70, 69]. Particular classes of such nonsmooth systems can be studied within the above framework. Let us consider:

$$ egin{cases} 
\dot{x}_{i1} = a_{i1}x_{i1} + b_{i1}x_{i2} + c_{i1}z_i + u_{i1}(t) \\ 
\dot{x}_{i2} = a_{i2}x_{i1} + b_{i2}x_{i2} + c_{i2}z_i + u_{i2}(t) \\ 
0 = a_{i3}x_{i1} + b_{i3}x_{i2} + c_{i3}z_i + u_{i4}(t) & \text{if } x_{i1} + u_{i3}(t) < 0 \\ 
0 = a_{i4}x_{i1} + b_{i3}x_{i2} + c_{i3}z_i + u_{i4}(t) & \text{if } x_{i1} + u_{i3}(t) > 0, 
\end{cases} 
\quad (63) $$

for $i = 1, 2$, where $x_{i1}, x_{i2}$ are the differential state variables, $z_i$ is the algebraic state variable, $u_{i1}, u_{i2}, u_{i3}, u_{i4}$ are inputs. Assuming $c_{i3} \ne 0$ (i.e., we deal with index 1 DAEs), (63) is rewritten equivalently as the LCS, for each $i = 1, 2$:

$$ egin{cases} 
\dot{x}_{i1} = a_{i1}x_{i1} + \left(b_{i1} - \frac{c_{i1}}{c_{i3}}b_{i3}\right)x_{i2} + \frac{c_{i1}}{c_{i3}}a_{i3}(\lambda_{i1} - u_{i3}) - \frac{c_{i1}}{c_{i3}}a_{i4}(\lambda_{i2} - u_{i3}) + \bar{u}_{i1} \\ 
\dot{x}_{i2} = a_{i2}x_{i1} + \left(b_{i2} - \frac{c_{i2}}{c_{i3}}b_{i3}\right)x_{i2} + \frac{c_{i2}}{c_{i3}}a_{i3}(\lambda_{i1} - u_{i3}) - \frac{c_{i2}}{c_{i3}}a_{i4}(\lambda_{i2} - u_{i3}) + \bar{u}_{i2} \\ 
0 \le \lambda_{i1} \perp \lambda_{i1} + x_{i1} + u_{i3}(t) \ge 0, \quad 0 \le \lambda_{i2} \perp \lambda_{i2} - x_{i1} - u_{i3}(t) \ge 0, 
\end{cases} 
\quad (64) $$

with $\bar{u}_{i1} = u_{i1} - \frac{c_{i1}}{c_{i3}}u_{i4}$ and $\bar{u}_{i2} = u_{i2} - \frac{c_{i2}}{c_{i3}}u_{i4}$. The transformation of (63) into (64) allows us to see clearly which operator plays a role for passivity analysis, and which role is played by the various controllers. The new control input

for (64) is $\bar{u} = (\bar{u}_{i1}, \bar{u}_{i2}, \bar{u}_{i3})^T$. We have: $A_i = \begin{pmatrix} a_{i1} & b_{i1} - \frac{c_{i1}}{c_{i3}}b_{i3} \\ a_{i2} & b_{i2} - \frac{c_{i2}}{c_{i3}}b_{i3} \end{pmatrix}$, $B_i = \begin{pmatrix} \frac{c_{i1}}{c_{i3}}a_{i3} & -\frac{c_{i1}}{c_{i3}}a_{i4} \\ \frac{c_{i2}}{c_{i3}}a_{i3} & -\frac{c_{i2}}{c_{i3}}a_{i4} \end{pmatrix}$, $C_i = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}$, $D_i = I_2$,

$L_i = \begin{pmatrix} 1 & 0 & -\frac{c_{i1}}{c_{i3}}(a_{i3} - a_{i4}) \\ 0 & 1 & -\frac{c_{i2}}{c_{i3}}(a_{i3} - a_{i4}) \end{pmatrix}$, $E_i = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & -1 \end{pmatrix}$, $H_1, H_2 \in \mathbb{R}^{3 \times 2}$, $G_1, G_2 \in \mathbb{R}^{3 \times 2}$. Given the passivity of $(A_i, B_i, C_i, D_i)$,

the material in foregoing sections applies to study interconnections of the class of switching DAEs as in (64). Since $D_i = I_2$ we may assume that the passivity LMI for strict or strong passivity holds true (see Appendix Appendix B), and it is possible in general to check if conditions in Theorems 3.11, 3.9 and Corollary 3.10 apply.

## 6.4. Sweeping Processes Interconnections

First order sweeping processes (FOSwP) are an important class of differential inclusions [18]. Let $D_i = 0$ in (4). Following [19], the LCCS (4) can be transformed to the FOSwP:

$$ \dot{z}_i \in R_i A_i R_i^{-1} z_i + R_i L_i u_i + R_i f_i - N_{\Phi_i(u_i, g_i)}(z_i), \quad (65) $$

with $z_i = R_i x_i$, $R_i = R_i^T > 0$, $R_i^2 = P_i$, $P_i = P_i^T > 0$ solves (7) with $P_i B_i = C_i^T$, and there exists $z$ such that $C_i R_i^{-1} z \in \text{rint}(\Phi_i(u_i, g_i))$, $\Phi_i(u_i, g_i) = K_i^* - E_i u_i - g_i$, and $\tilde{\Phi}_i(u_i, g_i) = \{z \in \mathbb{R}^{n_i} | C_i R_i^{-1} z \in \Phi_i(u_i, g_i)\}$. Then (65) is obtained following [19]. The FOSwP (65) is a controlled FOSwP, whose optimal control has been analysed [32, 31, 27, 33].

The above analysis shows how to guarantee that the interconnection of two FOSwP is passive. Since passivity is an invariant property with respect to state space change, passivity of the interconnection is guaranteed by (13), where

$$ D = \begin{pmatrix} 0 & E_1 G_2 \\ E_2 G_1 & 0 \end{pmatrix}. \text{ Notice that when } \mathbf{R} = 0, \text{ the circuit in (60) is the FOSwP: } \dot{x} \in \frac{-u_1}{C} - N_{\mathbb{R}_+ - u_2}(x). $$

**Proposition 6.2.** Consider the two FOSwP in (65), $i = 1, 2$. Let $E_1G_2 + (E_2G_1)^T = 0$, and assume (13a) holds with $P = P^T > 0$. Let $R = R^T > 0$, $R^2 = P$. Assume there exists $y$ such that $\tilde{C}R^{-1}y \in \text{rint}(\Phi(g_1, g_2))$, where $\Phi(g_1, g_2) = K^* - (g_1^T g_2^T)^T$. Then the interconnection of these FOSwP with $u_1$ and $u_2$ in (5) is again a FOSwP.