Monotonicity and incremental passivity are closely related one to each other [16]. 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 $\dot{x}_1 \in -\mathcal{F}(x_1) + u_1$ and $\dot{x}_2 \in -\mathcal{F}(x_2) + u_2$, with $u_i$ such that the system is well-posed with AC solutions. Then $s(x_1, x_2, u_1, u_2) \stackrel{\Delta}{=} \langle x_1 - x_2, u_1 - u_2 \rangle = \langle x_1 - x_2, \dot{x}_1 - \dot{x}_2 \rangle + \langle x_1 - x_2, \xi_1 - \xi_2 \rangle$, where $\xi_i \in \mathcal{F}(x_i)$. Thus defining the storage function $V(x_1, x_2) = \frac{1}{2}\langle x_1 - x_2, x_1 - x_2 \rangle$, it follows that $\dot{V} = s(x_1, x_2, u_1, u_2) - \langle x_1 - x_2, \xi_1 - \xi_2 \rangle \le s(x_1, x_2, u_1, u_2)$. This motivates us to enlarge the interconnection (6) to

$$ \begin{aligned} \dot{x} &= \tilde{A}x + \tilde{B}\lambda + Lv \\ K \ni \lambda \perp w &= \tilde{C}x + \tilde{D}\lambda \in K^*, \end{aligned} \qquad (58) $$

for some admissible control input $u$. The system (58) is rewritten equivalently as:

$$ \dot{x} \in \tilde{A}x - \tilde{B}(\tilde{D} + \mathcal{N}_{K^*}^{-1})^{-1}(\tilde{C}x) + Lv = -\mathcal{F}(x) + Lv. \qquad (59) $$

The system in (59) 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 6.** Consider subsystems in (4) under Assumption 1, and their interconnection in (6), and the input/output mapping $\Sigma_{vz} : v \mapsto z$, with $z = Hx + Gv$, in (58). Consider the storage function $V(x_1, x_2) = \frac{1}{2}\langle x_1 - x_2, P(x_1 - x_2) \rangle$, for some $P = P^\top \succcurlyeq 0$.

1. Let the mapping $P\mathcal{F}(\cdot)$ be maximal monotone. Then $\Sigma_{vz}$ is incrementally passive if $G + G^\top \succcurlyeq 0$ and $PL = H^\top$.

2. Let the mapping $P\mathcal{F}(\cdot)$ be maximal $\alpha$-strongly monotone, $\alpha > 0$. Then $\Sigma_{vz}$ is incrementally passive if $\begin{pmatrix} 2\alpha I_n & PL - H^\top \\ (PL - H^\top)^\top & G + G^\top \end{pmatrix} \succcurlyeq 0$. If this inequality is strict then $\Sigma_{vz}$ is incrementally strictly state and input passive. If $G + G^\top = 0$ then incremental strict state passivity holds if $PL = H^\top$.

*Proof.* From (59) we have $\dot{x} \in -\mathcal{F}(x) + Lv$, $z = Hx + Gv$. For two admissible inputs $v_1$ and $v_2$ it follows that $\dot{x}_i \in -\mathcal{F}(x_i) + Lv_i$, $z_i = Hx_i + Gv_i$, $i = 1, 2$. Let $\xi_i \in \mathcal{F}(x_i)$, and $\omega = \langle z_1 - z_2, v_1 - v_2 \rangle$.

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

$$ \begin{aligned} \dot{V} &= -(x_1 - x_2)^\top P(\xi_1 - \xi_2) + (x_1 - x_2)^\top PL(v_1 - v_2) \\ &\quad \pm (z_1 - z_2)^\top (v_1 - v_2) \\ &= -(x_1 - x_2)^\top P(\xi_1 - \xi_2) + (x_1 - x_2)^\top (PL - H^\top)(v_1 - v_2) \\ &\quad + \frac{1}{2}(v_1 - v_2)^\top (G + G^\top)(v_1 - v_2) + \omega. \end{aligned} \qquad (60) $$
*Proof.* The proposition's conditions imply $\tilde{D} + \tilde{D}^{\top} = 0$, hence from (13a) we have $P\tilde{B} = \tilde{C}^{\top}$, and the LCS (6) can be transformed into a FOSwP [19, 18]. □

Both subsystems and their interconnection may have AC or RCLBV solutions.

## 7. Conclusions

The generic interconnection of passive linear cone complementarity systems (LCCS), which are nonsmooth non-linear systems, is analysed. The matrix inequality (MI) which guarantees the passivity of the interconnection is studied in detail. Interconnections of passive, strictly passive and strongly passive systems are analysed. Conditions stemming from the passivity MI and involving the complementarity system matrices and control gains, are given in each case. The relationships with differential inclusions with a maximal monotone right-hand side, and with incrementally passive systems, are analysed. The existence of state jumps in these interconnections (that is, can interconnections of passive LCCS create or suppress state jumps?) is studied. Examples from circuits, switching DAEs, interconnected Moreau's sweeping processes and observer design are presented. Future research could concern the extension towards general maximal monotone mappings, discrete-time systems, stability and invariance analysis.

## Appendix A. Positive Definite Matrices under Perturbations

Let us provide a corollary of [29, Theorem 2.11]. For $M \in \mathbb{R}^{n \times n}$, $\|M\|_{2,2}$ is the induced matricial norm such that $\|M\|_{2,2} = \sigma_{\max}(M)$ (the largest singular value) [8, Proposition 9.4.9], it is a submultiplicative norm [8, Corollary 9.4.12].

**Corollary A.1.** Let $D = P + N$, where $D$, $P$ and $N$ are $n \times n$ real matrices, and $P > 0$, not necessarily symmetric. If $\|N\|_{2,2} < \frac{1}{\|((\frac{P+P^{\top}}{2})^{-1})\|_{2,2}}$, then $D > 0$.

## Appendix B. Passive Linear Systems

A quadruple $(A, B, C, D)$ is said passive if there exists $P = P^{\top} \succcurlyeq 0$ such that:

$$ \begin{pmatrix} -A^{\top}P - PA & -PB + C^{\top} \\ (-PB + C^{\top})^{\top} & D + D^{\top} \end{pmatrix} \succcurlyeq 0. \qquad (B.1) $$

This linear matrix inequality (LMI) implies that $\mathrm{Im}(PB - C^{\top}) \subseteq \mathrm{Im}(D + D^{\top})$, and $\mathrm{Im}(PB - C^{\top}) \subseteq \mathrm{Im}(A^{\top}P + PA)$, using [9, Proposition 10.2.5, Fact 8.4.3], or [20, Lemma 1 iv] and [8, Theorem 2.4.3, Fact 2.9.14]. The observability of the pair $(C, A)$ guarantees that $P > 0$ [20] [16, p.116]. The system is said strictly state passive if in addition $P > 0$ and there exists $\mu > 0$ such that (B.1) holds with $-A^{\top}P - PA - \mu P$ (hence (B.1) implies $-A^{\top}P - PA \succcurlyeq \mu P > 0$). It is said strongly passive if $P > 0$ and the inequality in (B.1) is strict. Let us consider the system $\dot{x} = Ax + Bu$, $y = Cx + Du$. The above passivity LMI is equivalent to the dissipation inequality: $V(x(t_2)) - V(x(t_1)) \le \int_{t_1}^{t_2} u(t)^{\top}y(t)dt$, for all $t_1 \le t_2$, and with storage function $V(x) = \frac{1}{2}x^{\top}Px$, supply rate $u^{\top}y$, where $x(\cdot)$ is a solution of the system. The system is said strictly input passive if there exists $\alpha > 0$ such that $V(x(t_2)) - V(x(t_1)) \le \int_{t_1}^{t_2} u(t)^{\top}y(t)dt - \alpha \int_{t_1}^{t_2} u(t)^{\top}u(t)dt$, while strict state passivity is equivalent to $V(x(t_2)) - V(x(t_1)) \le \int_{t_1}^{t_2} u(t)^{\top}y(t)dt - \mu \int_{t_1}^{t_2} x(t)^{\top}Px(t)dt$. Positive real (PR) transfer matrices with minimal state space realization are passive. Strictly positive real (SPR) transfer matrices with minimal state space realization are strictly state passive, and vice-versa [48] [16, Theorem 4.73]. Strongly SPR transfer matrices with minimal state space realization are strongly passive [16, p.162], and output and input strictly passive [46] [16, Theorem 2.81]. Incremental passivity is defined in [16, Definition 4.62] [52, 71]. Extension of passivity to measure differential inclusions as (47), using generalized supply rates, is made in [19] [13, Section 5.4.4.5].
If $P\mathcal{F}(\cdot)$ is monotone, then $-(x_1 - x_2)^\top P(\xi_1 - \xi_2) \le 0$. Then incremental passivity holds if and only if $-(x_1 - x_2)^\top P(\xi_1 - \xi_2) + (x_1 - x_2)^\top (PL - H^\top)(v_1 - v_2) + \frac{1}{2}(v_1 - v_2)^\top (G + G^\top)(v_1 - v_2) \le 0$ for all $v_1, v_2, x_1, x_2$ (see, e.g., [35, Definition 1]). The result is proved.

2. This time we get:

$$ 
\begin{aligned}
\dot{V} & \le -\alpha(x_1 - x_2)^\top P(x_1 - x_2) + (x_1 - x_2)^\top (PL - H^\top)(v_1 - v_2) \\
& \quad + \frac{1}{2}(v_1 - v_2)^\top (G + G^\top)(v_1 - v_2) + w \\
& = (x_1^\top - x_2^\top, v_1^\top - v_2^\top) \begin{pmatrix} 2\alpha I_n & PL - H^\top \\ (PL - H^\top)^\top & G + G^\top \end{pmatrix} \begin{pmatrix} x_1 - x_2 \\ v_1 - v_2 \end{pmatrix} + \omega.
\end{aligned} 
\quad (61) $$

The last result follows using [16, Lemma A.69] or [9, Proposition 8.2.4].

<figure><img src="image_5.png" /></figure>

Proposition 6 shows that if the conditions in Propositions 1,2 and 3 hold, then the interconnection (6) defines incrementally passive systems.

**Corollary 4.** Let the conditions of item 2 in Proposition 6 be satisfied. Then $\Sigma_{vz}$ is incrementally passive if $\sigma_{\max}^2(PL - H^\top) \le 2\alpha\lambda_{\min}(G + G^\top)$.

*Proof.* Directly from [10, Fact 10.12.17].

<figure><img src="image_6.png" /></figure>

**Remark 1.** Assumption 1 plus $\mathrm{Im}(C_i) \cap \mathrm{rint}(\mathrm{Im}(\mathcal{N}_{K_i^*}^{-1} + D_i)) \ne \emptyset$, $i = 1,2$, allow us to interpret each subsystem with $u_i \equiv 0$, as a maximal monotone mapping, using Theorem 3. Under conditions similar to those in Proposition 6, the interconnection is that of two incrementally passive systems (with inputs $\lambda_i$ and outputs $w_i$, $i = 1,2$). This topic has been studied for a long time [44, 35, 43], however with different interconnections.

# 5 Asymptotic Stability

In this section some asymptotic stability results are given for various cases of interconnections of LCS, i.e., $K_1 = \mathbb{R}_+^{m_1}$ and $K_2 = \mathbb{R}_+^{m_2}$. Since the systems we deal with are usually set-valued, stability and invariance principles have to rely on specific results for DIs [17, 7]. As a prerequisite it has to be assumed that the system (6), equivalently (9), is well-posed with AC solutions and uniqueness. The maximal monotonicity (deduced from Corollary 3) and the fact that $x_1^* = 0$, $x_2^* = 0$, $\lambda_1^* = 0$, $\lambda_2^* = 0$ is an equilibrium point of (6), guarantee that it satisfies the conditions 1, 2, 3 in [17, Section 6.2.1]. In this section it is assumed that $x(0) = x_0 \in \mathrm{Dom}(\mathcal{F})$, where the set-valued right-hand side of (9) is denoted as above as $-\mathcal{F}(x) = \tilde{A}x - \tilde{B}(\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(\tilde{C}x)$. If conditions in Corollary 3 hold, then the interconnection in (6) has bounded AC solutions from Theorems 3 and 2.
## Appendix C. LCCS, Maximal Monotone Mappings and DIs

Let $\mathcal{M}: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ be a set-valued mapping, $\alpha > 0$. It is monotone (resp. $\alpha$-strongly monotone) if for any $x_1, x_2 \in \text{Dom}(\mathcal{M})$, for any $y_1 \in \mathcal{M}(x_1)$, $y_2 \in \mathcal{M}(x_2)$, it holds $\langle x_1 - x_2, y_1 - y_2 \rangle \ge 0$ (resp. $\alpha \|x_1 - x_2\|^2$). It is maximal if its graph cannot be enlarged without destroying monotonicity. Let us first recall how LCCS as in (1) can be equivalently reformulated as DIs. Firstly $K$ being a convex cone the following equivalences hold:

$$ 
\begin{aligned}
K \ni \lambda \perp w &= Cx + D\lambda + Eu + g \in K^* &\Leftrightarrow &\lambda \in -\mathcal{N}_{K^*}(Cx + D\lambda + Eu + g) \\
&\Leftrightarrow (D + \mathcal{N}_{K^*}^{-1})(-\lambda) \ni Cx + Eu + g &\Leftrightarrow &\lambda \in -(D + \mathcal{N}_{K^*}^{-1})^{-1}(Cx + Eu + g)
\end{aligned} 
\quad (C.1) 
$$

see [37, 58] for the first equivalence, and mapping inversion is used. Inserting the last expression into (1a) we obtain the right-hand side

$$ -\mathcal{F}(x, u, f, g) \stackrel{\Delta}{=} Ax + Lu + f - B(D + \mathcal{N}_{K^*}^{-1})^{-1}(Cx + Eu + g) \quad (C.2) $$

To study the well-posedness of system (1) using the maximal monotonicity of the mapping on the right-hand side, we consider the mapping, $\mathcal{F}: \mathbb{R}^n \times \mathbb{R}^m \rightrightarrows \mathbb{R}^n$, defined in (C.2). By introducing the functions $v_1: [0, T] \to \mathbb{R}^n$ and $v_2: [0, T] \to \mathbb{R}^m$, the differential inclusion

$$ \dot{x}(t) \in -\mathcal{F}(x(t), v_1(t)) + v_2(t) \quad (C.3) $$

provides an equivalent representation of system (1) with $v_1 = Eu + g$ and $v_2 = Lu + f$. We arrive at the following statement using the main results in [25, 23], see also [15, 68], for the existence and uniqueness of AC solutions.

**Theorem C.1.** Suppose that $(A, B, C, D)$ in (C.2) is passive with storage function $x \mapsto \frac{1}{2}x^\top P x$, $P = P^\top > 0$, $K \subseteq \mathbb{R}^m$ a constant closed convex nonempty cone. For each $v \in \mathbb{R}^m$, if $\text{Im}(C) \cap \text{rint}(\text{Im}(\mathcal{N}_{K^*}^{-1} + \tilde{D}) - v) \neq \emptyset$, then the mapping $\mathcal{F}(\cdot, v): \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ is maximal monotone. Moreover, if $v_1: [0, T] \to \mathbb{R}^n$ and $v_2: [0, T] \to \mathbb{R}^m$ are AC, and $\text{Im}(C) \cap \text{rint}(\text{Im}(\mathcal{N}_{K^*}^{-1} + D) - v_1(t)) \neq \emptyset$ for each $t \in [0, T]$, then there exists a unique AC solution to the DI (C.3) for all initial data $x(0) = x_0 \in \text{Dom}(\mathcal{F}(\cdot, v))$.

## Appendix D. Well-posedness of Linear Complementarity Systems with State Jumps

In order to complete section 5, let us present results on the existence and uniqueness of solutions to LCS (i.e., LCCS in (1) with $K = \mathbb{R}_+^m$ in (1b)) including state discontinuities. This issue is tackled in [22, 19, 20, 68]. As recalled in the introduction of section 5, LCS with state jumps can be embedded into MDIs, which is the approach taken in [19, 68]. Let us summarize the results in [68]. Performing similar manipulations as in (C.1) the LCS in (1) with $u \equiv 0$ is rewritten equivalently as:

$$ \dot{x} \in Ax + f - B(D + \mathcal{N}_{\tilde{\Phi}(t)}^{-1})^{-1}(Cx), \quad (D.1) $$

where $\tilde{\Phi}(t) \stackrel{\Delta}{=} \{z \in \mathbb{R}^m | z + g \in \mathbb{R}_+^m\}$ is closed, convex, polyhedral, nonempty. Let us assume that (the rationale behind these various assumptions is explained in [68]):

1. $D \ge 0$ and there exists $P = P^\top > 0$ such that $\text{ker}(D + D^\top) \subseteq \text{ker}(PB - C^\top)$.

2. The time-function $f$ is Lebesgue integrable on $[0, +\infty)$.

3. For each $t \ge 0$, $\text{Im}(C) \cap \text{rint}(\text{Im}(D + \mathcal{N}_{\tilde{\Phi}(t)}^{-1})) \neq \emptyset$.

4. For each $t \ge 0$ and each $v \in \text{Im}(C) \cap \text{Im}(D + \mathcal{N}_{\tilde{\Phi}(t)}^{-1})$, it holds that $\text{Im}(D + D^\top) \cap (D + \mathcal{N}_{\tilde{\Phi}(t)}^{-1})^{-1}(v) \neq \emptyset$.

5. There exists a RCLBV function $\mu_f: [0, +\infty) \to [0, +\infty)$ whose variation over $(0, t]$ is denoted $\mu(t) \stackrel{\Delta}{=} \text{var}(\mu_{f(0,t)})$, which satisfies:

$$ d_{\text{Haus}}(\tilde{\Phi}(t_1) \cap \text{Im}(C), \tilde{\Phi}(t_2) \cap \text{Im}(C)) \le d\mu((t_1, t_2]) = \mu(t_1) - \mu(t_2), \quad (D.2) $$

for each $t_1 \le t_2$, where $d\mu$ denotes the Lebesgue-Stieltjes measure associated with $\mu$, and $d_{\text{Haus}}(S, S')$ is the Hausdorff distance between two set $S$ and $S'$.
## 5.1 Passive Interconnection

Let the conditions in Proposition 4 be satisfied. The dissipation inequality in (42) is important because it paves the way to the application of Krasovskii-LaSalle's invariance principle. Notice using (9) that $\lambda_1 = \lambda_1(x_1, x_2)$, hence (42) can be rewritten as $\dot{V}(x) \le -W(x_1, x_2)$, $W: \mathbb{R}^n \to \mathbb{R}_+$. The application of the Krasovskii-LaSalle invariance principle as in [17, sections 6.2, 6.3] [7], requires that $W(\cdot)$ be continuous. We may therefore split the analysis into two parts: (i) $W(x) \stackrel{\Delta}{=} \alpha_1||x_1||^2 + \alpha_2||\lambda_1(x_1, x_2)||^2$ if $\lambda_1(x_1, x_2)$ is continuous, and (ii) $W(x) \stackrel{\Delta}{=} \alpha_1||x_1||^2$ if it is not. Since $\lambda_1 = -(D_1 + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(C_1x_1 + E_1H_2x_2 + E_1G_2\lambda_2)$ and $D_1 \succ 0$ by assumption, then if $E_1G_2 = 0$, continuity of $\lambda_1(x_1, x_2)$ is guaranteed. One basic assumption for the invariance principle is that $\text{Dom}(-\mathcal{F})$ contains a positively invariant set $\Psi$. Here we have $\text{Dom}(-\mathcal{F}) = \text{dom}(\tilde{A}) \cap \text{Dom}(-\tilde{B}(\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(\tilde{C})) = \text{Dom}(-\tilde{B}(\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(\tilde{C})) = \text{Dom}((\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(\tilde{C})) = \{x \in \mathbb{R}^n \mid \tilde{C}x \in \text{Dom}((\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}) = \text{Im}(\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})\} = \tilde{C}^{-1}(\text{Im}(\tilde{D} + \mathcal{N}_{\mathbb{R}_+^m}^{-1})) = \tilde{C}^{-1}(\text{Im}(\tilde{D}) + \mathbb{R}_+^m) \neq \emptyset$ from Lemma 8 and following the reasoning in its proof. This guarantees the existence and uniqueness of solutions, and from (42) the existence of $\Psi$.

### 5.1.1 Case (i)

So we assume that $E_1G_2 = 0$. Our goal is to characterize the largest invariant set $S_{lis}$ of the interconnection inside $\mathcal{Z} = \{x \in \text{Dom}(\mathcal{F}) \mid x_1 = 0 \text{ and } \lambda_1(x_1, x_2) = 0\}$. In $\mathcal{Z}$ it follows that $L_1(H_2x_2 + G_2\lambda_2) = 0$, and $0 \le 0 \perp E_1H_2x_2 \ge 0$. We therefore end up with the nonsmooth constrained dynamics on $\mathcal{Z}$:

$$
\begin{align}
(a) \quad \dot{x}_2 &= A_2x_2 + B_2\lambda_2 \nonumber \\
(b) \quad 0 &\le \lambda_2 \perp w_2 = C_2x_2 + D_2\lambda_2 \ge 0 \tag{62} \\
(c) \quad L_1(H_2x_2 + G_2\lambda_2) &= 0 \nonumber \\
(d) \quad E_1H_2x_2 &\ge 0. \nonumber
\end{align}
$$

This system is an LCS with equality and inequality constraints. The constraint in (62) (b) is equivalent to: $\lambda_2 \in -(D_2 + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(C_2x_2)$, hence (62) is equivalent to:

$$
\begin{align}
(a) \quad \dot{x}_2 \in -\mathcal{F}_2(x_2) &\stackrel{\Delta}{=} A_2x_2 - B_2(D_2 + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(C_2x_2) \nonumber \\
(b) \quad 0 \in L_1\mathcal{G}(x_2) & \nonumber \\
(c) \quad E_1H_2x_2 &\ge 0. \tag{63}
\end{align}
$$

where $\mathcal{G}(x_2) \stackrel{\Delta}{=} H_2x_2 - G_2(D_2 + \mathcal{N}_{\mathbb{R}_+^m}^{-1})^{-1}(C_2x_2)$. Notice that $x_2 = 0$ and $\lambda_2 = 0$ is always a solution of (62) or of (63), because $0 \in \mathcal{G}(0)$ since $\text{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^m}^{-1}) = \text{Im}(D_2) + \mathbb{R}_+^{m_2} \ni 0$ (proceeding as in the proof of Lemma 8). This equilibrium is not an asymptotically stable equilibrium of (62) (a) (b) in view of the assumption in Proposition 4, which implies that $\dot{V}_2(t) = \frac{d}{dt}(x_2^\top P_2x_2) = 0$ along the trajectories of (62)(a) (b) (or of (63) (a)), $V_2(x_2) = \frac{1}{2}x_2^\top P_2x_2$.
Let us define the notion of solutions to (D.1). Basically they are RCLBV functions of time (hence they may possess countable sets of discontinuities with possible finite accumulations of jump times), obtained as outlined in the introduction of section 5 (see (48)). Let us define strong and weak solutions to (D.1), as done in [68], inspired from [36] [10, Definition 3.1].

**Definition 2.** Consider the DI in (D.1) and assume items 1 through 5 are satisfied. A RCLBV function $x: [0, +\infty) \to \mathbb{R}^n$ is called a strong solution to (D.1) if there exists a Radon measure $dv$ which is absolutely continuously equivalent to $dt + d\mu$, such that the differential measure $dx$ is absolutely continuous with respect to $dv$, $\frac{dx}{dv} \in L_1^{loc}([0, +\infty), \mathbb{R}^n; v)$, and

$$ \frac{dx}{dv}(t) \in (Ax(t) + f(t)) \frac{dt}{dv}(t) - B(D + N_{\tilde{\Phi}(t)}^{-1})^{-1}(Cx(t^{+})) \quad (D.3) $$

holds $v$-almost everywhere, $t \in [0, +\infty)$. The RCLBV function $x: [0, +\infty) \to \mathbb{R}^n$ is called a weak solution to (D.1) if there exists a sequence of strong solutions $\{x_n\}_{n=1}^{+\infty}$ to the DIs: $\dot{x}_n \in Ax_n + f_n - B(D + N_{\tilde{\Phi}(t)}^{-1})^{-1}(Cx_n)$, with $f_n \to f$ in $L_1^{loc}([0, +\infty), \mathbb{R}^n; v)$, such that $\{x_n\}_{n=1}^{+\infty}$ converges uniformly to $x$.

The DI in (D.3) is an inclusion of functions of time, which are densities. Roughly speaking, it enables us to separate times which are atoms of $dv$ (the state jump times) at which $\frac{dt}{dv}(t) = 0$ and $\frac{dx}{dv}(t) = x(t^{+}) - x(t^{-})$, and times outside jumps at which densities can be assimilated to the usual derivatives. Then the following holds true:

**Theorem D.1.** [68, Theorem 1, Corollary 1] Assume that the differential inclusion (D.1) satisfies assumptions 1 through 5, with $x(0) = x_0$. Then there exists a unique weak solution for any $x_0$.

Solutions can undergo discontinuities which are calculated as in section 5. Let us note that (49) is equivalently written as

$$ x(t^{+}) - x(t^{-}) \in -B(D + N_{\tilde{\Phi}(t^{+})}^{-1})^{-1}(Cx(t^{+})). \quad (D.4) $$

The relation between the variation of the set $\tilde{\Phi}(t)$ and the time-function $g$, is explained by Robinson's inequality [56] [68, Lemma 6].

## Appendix E. Continuity of Sets

Continuity of the sets $\Phi: t \mapsto \Phi(t)$ can be studied using results from [56, p.760], [58, Example 9.35], recalling that $S_D^*$ is a polyhedral closed convex cone. In the next lemma Haus denotes the Hausdorff distance between sets.

**Lemma E.1.** Let $S = \{x \in \mathbb{R}^n | Ax - b \in K\}$, $S' = \{x \in \mathbb{R}^n | Ax - b' \in K\}$, where A is an $m \times n$ matrix and K is a closed convex polyhedral cone of $\mathbb{R}^m$ of the form $K = \{y \in \mathbb{R}^m: My \ge 0\}$ for some matrix M. Then there exists a constant $\delta$ depending only on A and M such that $Haus(S, S') \le \delta\|M\| \|b - b'\|$ whenever S and S' are non empty, that is $b, b' \in Im A - K$.

Therefore $Haus(\Phi(t_1), \Phi(t_2)) \le \delta\|g(t_1) - g(t_2)\|$ for some $\delta \ge 0$ and all $t_1, t_2$. If $g$ is continuous at $t$ then $\Phi$ is Hausdorff-continuous at $t$ [49, section 2.2].

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**The case $G_2 = 0$ (passivity and Nagumo's Theorem approach)** (63) is a DI with constraints, and the problem is a viability problem. However the right-hand side of (63) (a) may not be compact, hence the material in [5, Chapter 4] does not apply. If $\{0\} \subset S_c \stackrel{\Delta}{=} -E_1H_2\text{Ker}(L_1H_2) \cap \mathbb{R}_+^{m_1}$, the set of solutions to (63) (b) (c) is nontrivial (i.e., not reduced to $x_2 = 0$ and $\lambda_2 = 0$). Both sets of the intersection are nonempty convex pointed cones (since the image of a subspace by a linear map is a subspace [9, Fact 2.9.26]). Thus $S_c$ is a convex nonempty polyhedral cone [9, Fact 2.9.9], pointed since it contains the origin. Hence the following is true:

**Lemma 9.** Let $G_2 = 0$, the conditions of Proposition 4 hold, and $\text{Im}(C_2) \cap \text{rint}(\text{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}})) \neq \emptyset$. Then (63) is a maximal monotone DI subjected to the convex conic polyhedral constraint $x_2 \in S_c$.

The maximal monotonicity of $\mathcal{F}_2(\cdot)$ in (62) (a) follows from Theorem 3 and the assumptions of Proposition 4, and we infer from Theorem 2 that $x_2(\cdot)$ as a solution of (63) (a) is AC and bounded, with uniqueness.

**Corollary 5.** Let the conditions of Lemma 9 hold. If $\text{rank}(L_1H_2) = n_2$ (i.e., $L_1H_2$ is full-column rank), the unique solution of (63) is $x_2^* = 0$ with multiplier $\lambda_2^* \in \text{SOLLCP}(D_2, 0)) \cap \text{Ker}(B_2)$. Hence the origin of the interconnection is asymptotically stable.

Notice that (63) (b) implies $x_2 = 0$ under the rank condition. Then the largest invariant set $S_{lis} = \{0\}$ and the proof follows from [17, Theorem 6.5]. Matrix $D_2$ is not nondegenerate [27, Definition 3.6.1], so it is not an $\mathbb{R}_0$ matrix [27, Definition 3.8.7] and the LCP in (62) (b) with $x_2 = 0$ may have other solutions than $\lambda_2^* = 0$. The results in [3] extend Nagumo's Theorem to maximal monotone DI, and apply to (63) with $G_2 = 0$. Let Lemma 9 apply, the fact that $x_2 \mapsto \mathcal{F}_2(x_2)$ is maximal monotone secures that [3, Corollaries 6, 8] can be applied. A basic condition in [3] is that an invariant set $S$ satisfies $S \subseteq \overline{\text{Dom}(\mathcal{F}_2)}$. Assume that $D_2 = 0$, then $\text{Dom}(\mathcal{F}_2) = \mathbb{R}^{n_2} \cap \text{Dom}(B_2\mathcal{N}_{\mathbb{R}_+^{m_2}}(C_2\cdot)) = C_2^{-1}(\mathbb{R}_+^{m_2})$, thus the requirement is that $S \subseteq C_2^{-1}(\mathbb{R}_+^{m_2})$. For instance, for $S_c$ to be an invariant set of (62) (a), it is necessary and sufficient (because solutions are strong solutions in finite dimension [3, Section 3]) that, when $D_2 = 0$:

$$S_c = -E_1H_2\text{Ker}(L_1H_2) \cap \mathbb{R}_+^{m_1} \subseteq C_2^{-1}(\mathbb{R}_+^{m_2}). \quad (64)$$

Let us define as in [3, Lemma 3]: $S_m = \{x_2 \in S \cap \text{Dom}(\mathcal{F}_2) \mid \|\mathcal{F}_2^0(x_2)\| \le m\}$, for some $m \ge 0$, where $\mathcal{F}_2^0(x_2) = \{y \in \mathcal{F}_2(x_2) \mid \|y\| = \min_{z \in \mathcal{F}_2(x_2)} \|z\|\} = \text{proj}[0; \mathcal{F}_2(x_2)]$ is the closed set of least-norm elements of $\mathcal{F}_2(x_2)$. Thus $\mathcal{F}_2^0(x_2) = \{y \in \mathbb{R}^{n_2} \mid y = -A_2x_2 + B_2z, z = \text{argmin}_{z \in \mathcal{F}_2(x_2)} \|y\|\}$. Assume that $C_2x_2 > 0$, then the set $B_2\mathcal{N}_{\mathbb{R}_+^{m_2}}(C_2x_2) = \{0\}$, and $\mathcal{F}_2^0(x_2) = \{-A_2x_2\}$. If $C_2x_2 = 0$, it is inferred that $\mathcal{N}_{\mathbb{R}_+^{m_2}}(0) = \mathbb{R}_-^{m_2}$, $\mathcal{F}_2(x_2) = -A_2x_2 + B_2\mathbb{R}_-^{m_2}$, and $\mathcal{F}_2^0(x_2) = \{y \in \mathbb{R}^{n_2} \mid y = -A_2x_2 + B_2z, z = \text{argmin}_{z \le 0} \|y\|\}$. Using [27, Theorem 3.8.6] (or [27,
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**Lemma 10.** Let $D_2 = 0$ and the conditions of Lemma 9 hold. Then the nonempty set $S_c$ of solutions to (63) (b)(c) is an invariant set of (63) (a) if and only if (64) holds and

$$ -\mathcal{F}_2^0(x_2) \in \mathcal{T}_{S_c,m}(x_2), \quad \text{for all } m \ge \|\mathcal{F}_2^0(x_2)\| \\ \Downarrow \\ \text{for each } m \ge \|\mathcal{F}_2^0(x_2)\|, \sup_{\xi \in \mathcal{N}_{S_c,m}(x_2)} \langle \xi, -\mathcal{F}_2^0(x_2) \rangle \le 0, \qquad (65) $$

where $S_{c,m} = \{x_2 \in S_c \cap \text{Dom}(\mathcal{F}_2) \mid \|\mathcal{F}_2^0(x_2)\| \le m\}$, $\mathcal{T}_{S_c,m}(x_2)$ is the Bouligand tangent cone, $\mathcal{N}_{S_c,m}(x_2)$ is the Frèchet or the proximal normal cone to $S_{c,m}$ at $x_2$.

In both above cases ($C_2x_2 > 0$ and $C_2x_2 = 0$), $S_{c,m}$ is convex polyhedral so the tangent and normal cones are the polyhedral linearization cones [5, 17]. Notice that $S_c \cap \text{Dom}(\mathcal{F}_2) = S_c$ from (64). If $S_c = \{0\}$ ($\Rightarrow S_{c,m} = \{0\}$) then (64) and (65) hold.

**Corollary 6.** Let $D_2 = 0$ and the conditions of Lemma 9 hold. (i) If (64) (65) hold and $S_c$ is nontrivial, trajectories of the interconnection converge to $S_c$. (ii) If either (64) or (65) do not hold and $\rho\mathbb{B}_{n_2} \subset S_c$ for some $\rho > 0$, then the origin of the interconnection is not asymptotically stable. (iii) If such a $\rho$ does not exist, the origin $(x_1, x_2) = (0, 0)$ is asymptotically stable. (iv) If $S_c = \{0\}$ the origin $(x_1, x_2) = (0, 0)$ is asymptotically stable.

*Proof.* (i) The first result is a consequence of the fact that $S_{lis} \subseteq S_c$, and using [17, Theorem 6.5] trajectories converge asymptotically to $S_{lis}$. However $S_c$ is an invariant set of (63) (a) from Lemma 10, and it must be the largest invariant set of (63) (a) (b) (c). (ii) The level sets of $V_2(\cdot)$ satisfy $V_2^{-1}(l) \subseteq S_c$ for all $l \in [0, l^*]$ and some $l^* > 0$. As seen above these level sets are invariant sets of (63) (a). (iii) In this case $l^* = 0$ and the only invariant set is $S_{lis} = \{0\}$. (iv) If $S_c = \{0\}$ then the only solution of (63) (a)(b)(c) is the origin, and the conclusion follows from [17, Theorem 6.5]. □

Since $S_c$ is a convex pointed polyhedral cone, the condition in item (ii) is restrictive. Conditions such that it holds are not pursued here for the sake of briefness.

**The case $G_2 \ne 0$ (complementarity approach)** In a more general setting, the constrained system (62) can be studied as follows, noticing that (62) (a)(b)(c) is an LCS with extended equality constraint. If $\text{Im}(G_2) \subseteq \text{Im}(H_2)$, the equality (62) (c) yields $x_2 = -(L_1H_2)^{\dagger}L_1G_2\lambda_2 + (I_{n_2} - (L_1H_2)^{\dagger}L_1H_2)x'_2$, for some $x'_2 \in \mathbb{R}^{n_2}$ [10, Proposition 8.1.9], where $(I_{n_2} - (L_1H_2)^{\dagger}L_1H_2)$ is the projector onto $\text{Ker}(L_1H_2)$ [10, Proposition 8.1.7 xiv), equ. (4.8.1)]. Inserting this into (62) (b) yields the LCP: $0 \le \lambda_2 \perp \bar{D}_2\lambda_2 + q_2(x'_2) \ge 0$, with $q_2(x'_2) \triangleq C_2(I_{n_2} - (L_1H_2)^{\dagger}L_1H_2)x'_2$ and $\bar{D}_2 \triangleq -C_2(L_1H_2)^{\dagger}L_1G_2 + D_2 \in \mathbb{R}^{m_2 \times m_2}$. The inequality (62) (d) yields $-E_1H_2(L_1H_2)^{\dagger}L_1G_2\lambda_2 + E_1H_2(I_{n_2} - (L_1H_2)^{\dagger}L_1H_2)x'_2 \ge 0$. Therefore the following is proved:
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**Lemma 11.** Let Im($G_2$) ⊆ Im($H_2$). System (62) is rewritten equivalently as:

$$
\begin{align}
(a) \quad \dot{x}_2 &= (-A_2(L_1 H_2)^\dagger L_1 G_2 + B_2)\lambda_2 + A_2(I_{n_2} - (L_1 H_2)^\dagger L_1 H_2)x'_2 \nonumber \\
(b) \quad x_2 &= -(L_1 H_2)^\dagger L_1 G_2 \lambda_2 + (I_{n_2} - (L_1 H_2)^\dagger L_1 H_2)x'_2 \nonumber \\
(c) \quad &- E_1 H_2(L_1 H_2)^\dagger L_1 G_2 \lambda_2 + E_1 H_2(I_{n_2} - (L_1 H_2)^\dagger L_1 H_2)x'_2 \ge 0 \tag{66} \\
(d) \quad 0 \le \lambda_2 \perp \bar{D}_2 \lambda_2 + q_2(x'_2) \ge 0 \nonumber
\end{align}
$$

for some $x'_2$.

Thus (66) (d) is a distorted LCP that incorporates (62) (c) (equivalently (66) (b)), and (66) (c) is (62) (d) incorporating (62) (c). From (66) a necessary condition for the existence of solutions of (62) is SOLLCP($\bar{D}_2$, $q_2(x'_2)$) ≠ $\emptyset$. From [27, Theorem 3.8.6], a sufficient condition for the solvability of (66) (d) when $\bar{D}_2$ is copositive [27, Definition 3.8.1] is: $q_2(x'_2) \in \text{SOLLCP}(\bar{D}_2, 0)^*$. This result needs the computation of SOLLCP($\bar{D}_2, 0$), with $D_2 = -D_2^\top$ and (41), to be constructive, as well as criteria which guarantee the copositiveness [30]. Assume that $\bar{D}_2$ is a P-matrix. The LCP($\bar{D}_2$, $q_2(x'_2)$) in (66) (d) has a solution $\lambda_2(x'_2)$, which is continuous piecewise linear [27], and $\lambda_2(0) = 0$. A necessary and sufficient condition for (66) (a) (b) (d) to be well-posed, is that (a) and (b) be compatible. This yields that $x'_2$, possibly a time-function, has to be a solution of:

$$
\begin{equation}
\begin{split}
& \left[-(L_1 H_2)^\dagger L_1 G_2 \frac{\partial \lambda_2}{\partial x'_2} + (I_{n_2} - (L_1 H_2)^\dagger L_1 H_2)\right] \dot{x}'_2 + (A_2 (L_1 H_2)^\dagger L_1 G_2 - B_2) \lambda_2(x'_2) \\
& \qquad - A_2 (I_{n_2} - (L_1 H_2)^\dagger L_1 H_2) x'_2(t) = 0
\end{split}
\tag{67}
\end{equation}
$$

when $\frac{\partial \lambda_2}{\partial x'_2}$ exists, in which case it is a constant matrix (i.e., (67) holds almost everywhere). Therefore, we can rewrite (67) as the switching system:

$$
M_1^{\sigma(x'_2(t))}(x'_2(t))\dot{x}'_2 + M_2^{\sigma(x'_2(t))}(x'_2(t))x'_2 = 0, \quad (68)
$$

with the index step function $\sigma \circ x'_2 : \mathbb{R}_+ \to \{0, \dots, 2^{m_2} - 1\} \subseteq \mathbb{N}$, $M_i^{\sigma \circ x'_2(t)}(x'_2(t))$ are constant matrices for each $t$. If $x'_2$ is constant, then $\sigma \circ x'_2(t) = 0$ for all $t \ge 0$, otherwise it may switch depending on the LCP modes. Each $\sigma(t)$ corresponds to a partitioning of the index set $\{1, \dots, m_2\}$ into two subsets $\alpha$ and $\bar{\alpha} = \{1, \dots, m_2\} \setminus \alpha$, such that the solution of LCP($\bar{D}_2$, $q_2(x'_2)$) is equal to: $\lambda_2^\alpha(x'_2) = -(\bar{D}_2^{\alpha\alpha})^{-1}q_2^\alpha(x'_2)$, $\lambda_2^{\bar{\alpha}}(x'_2) = 0$, where $M^{\alpha\beta}$ denotes the submatrix constructed from $M \in \mathbb{R}^{m \times m}$ by taking rows indexed in $\alpha$ and columns indexed in $\beta$, $\alpha$ and $\beta$ two index subsets of $\bar{m} = \{1, \dots, m\}$ [22]. Therefore $q_2^\alpha = (I_{n_2} - (L_1H_2)^\dagger L_1H_2)_{\alpha\bullet} x'_2 = (I_{n_2} - (L_1H_2)^\dagger L_1H_2)_{\alpha\bar{m}} x'_2$. Assuming that indices are ordered such that $\lambda_2 = \begin{pmatrix} \lambda_2^\alpha \\ \lambda_2^{\bar{\alpha}} \end{pmatrix}$, we have $\frac{\partial \lambda_2}{\partial x'_2} = \begin{pmatrix} -(\bar{D}_2^{\alpha\alpha})^{-1}(I_{n_2} - (L_1H_2)^\dagger L_1H_2)_{\alpha\bar{m}} \\ 0 \end{pmatrix}$, which shows the structure of $M_1^{\sigma(x'_2(t))}$ and $M_2^{\sigma(x'_2(t))}$, and $M_3^{\sigma(x'_2(t))}$ shown below. Let $x'_2(\cdot)$ be a solution of (68), $\lambda_2(x'_2(\cdot))$ the solution of LCP($\bar{D}_2$, $q_2(x'_2)$). This means that $x_2(\cdot)$ as above is a solution of (62) (a)(b)(c). Thus it is a solution of (62) if and only if in addition (66) (c) holds for $x'_2$ and $\lambda_2(x'_2)$. Notice that (66) (c) is, for each $\sigma(t)$, a linear inequality
of the form $M_3^{\sigma(x'_2(t))}x'_2 \ge 0$, which defines a convex nonempty cone. We are therefore led to characterize the set of solutions to (68) subject to $M_3^{\sigma(x'_2(t))}x'_2 \ge 0$, keeping in mind that solutions $x_2(\cdot)$ to (62) are AC and bounded since (42) holds. Assume that $\text{rank}(M_1^{\sigma(x'_2(t))}(x'_2(t))) = n_2$ for all $t \ge 0$, and let us analyse possibly time-varying $x'_2(\cdot)$, solutions to the constrained piecewise linear ODE:

$$ \begin{aligned} \dot{x}'_2 &= -(M_1^{\sigma(x'_2(t))}(x'_2(t)))^{-1} M_2^{\sigma(x'_2(t))}(x'_2(t))x'_2, \\ M_3^{\sigma(x'_2(t))}(x'_2(t))x'_2(t) &\ge 0. \end{aligned} \qquad (69) $$

System (69) is a switching system with switching inequality constraints, where the inequality represents a switching polyhedral set (that has to be positively invariant for the dynamical part). Solutions to (69) are AC. All bounded solutions to (69) yield a solution $x_2(\cdot)$ to (66), hence to (62). Notice that the LCP($\bar{D}_2$, $q_2(x'_2)$) has $2^{m_2}$ modes. Thus all possible matrices $M_i^{\sigma(x'_2(t))}(x'_2(t))$ can, in principle, be calculated. In the following $M_i^{\sigma}$ denotes a matrix in the set $\{M_i^0, \dots, M_i^{2^{n_2}}\}$. Basically there are two situations for the solutions to (69): (a) either $x'_2(\cdot)$ is such that there exists $T < +\infty$ such that for all $t \ge T$, $\sigma \circ x'_2(t) = \sigma_f$ is constant; or (b) $x'_2(\cdot)$ is such that $\sigma \circ x'_2(\cdot)$ never attains a constant integer inside $\{0, \dots, 2^{m_2} - 1\}$, and the matrices $M_i^{\sigma(t)}$ keep on switching indefinitely. Let us analyze (a).

**Lemma 12.** Let $\text{Im}(G_2) \subseteq \text{Im}(H_2)$ and $\bar{D}_2$ be a P-matrix. Situation (a) occurs if and only if there exists an essentially nonnegative matrix $M$ such that $MM_3^{\sigma_f} = -M_3^{\sigma_f}(M_1^{\sigma_f})^{-1}M_2^{\sigma_f}$, which is true only if $\text{Ker}(M_3^{\sigma_f})$ is a $-(M_1^{\sigma_f})^{-1}M_2^{\sigma_f}$-invariant subspace. Then $x_2(\cdot)$ is given by $x_2(t) = (I_{n_2} - (L_1H_2)^{\dagger}L_1H_2)x'_2(t)$ for all $t \ge T$. If the above conditions are not satisfied, then the only global solution to (69) is $x'_2 = 0$, so that the only solution to (66) is $x_2(t) = x'_2 = 0$.

*Proof.* It follows from [24, Proposition 1, Lemma 1]. The given condition is necessary for positive invariance of the set $\{z \in \mathbb{R}^{n_2} | M_3^{\sigma_f}z \ge 0\}$ with respect to the dynamics $\dot{x}'_2 = -(M_1^{\sigma_f})^{-1}M_2^{\sigma_f}x'_2$. In this case, the projection of $x'_2(t)$ onto $\text{Ker}(L_1H_2)$ is bounded for all $t \ge T$, $x'_2(t) = \exp(-(M_1^{\sigma_f})^{-1}M_2^{\sigma_f})(t-T)x'_2(T)$, for all $t \ge T$, and $x_2(\cdot)$ is a (bounded) solution to (66) (a): $x_2(t) = x_2(T)+(-A_2(L_1H_2)^{\dagger}L_1G_2+B_2)\lambda_2^f(t-T)+A_2(I_{n_2} - (L_1H_2)^{\dagger}L_1H_2) \int_t^T x'_2(s)ds$. Boundedness of the state implies $\lambda_2^f \in \text{Ker}(-A_2(L_1H_2)^{\dagger}L_1G_2+B_2))$ and from (63) (b) $\lambda_2^f \in \text{Ker}((L_1H_2)^{\dagger}L_1G_2)$, which are compatible since (66)(a) and (b) are compatible from (67). Thus the result for $x_2(t)$ follows. The second result follows from the fact that if the condition on positive invariance is not satisfied, the only solution is the null solution. $\square$

The following holds:

**Corollary 7.** Assume that there is no index $\sigma \in \{0, \dots, 2^{m_2} - 1\}$ such that $\text{Ker}(M_3^{\sigma})$ is a $-(M_1^{\sigma})^{-1}M_2^{\sigma}$-invariant subspace. Then $S_{\text{lis}}$ reduces to $x'_2 = 0$.
Proof. The unique global solution to (69) is $x'_2 = 0 \Rightarrow q_2(x'_2) = 0 \Rightarrow \lambda_2 = 0$ since $\bar{D}_2$ is a P-matrix. From the boundedness of $x_2(\cdot)$, and from (66)(b), $x_2 = 0$. $\square$

**The case $G_2 \neq 0$ (passivity and VI approach)** Checking the above conditions may not be easy in general if $m_2$ is too large. Let us provide an alternative. First let us study the set of solutions to (63) (b), relying on maximal monotonicity properties of the mappings. In the following we shall assume that $D_2 = D_2^\top$, which in view of $D_2 + D_2^\top = 0$ in Proposition 4 implies $D_2 = 0$.

**Lemma 13.** Let $D_2 = 0$. Assume that the quadruple $(L_1H_2, L_1G_2, C_2, 0)$ is passive ($\Rightarrow n_1 = n_2$), and that $\mathrm{Im}(C_2) \cap \mathbb{R}_+^{m_2}$ is not reduced to $\{0\}$. Then the set $\mathcal{S}$ of solutions to (63) (b) (which is (62) (b) (c)), is closed and convex. If in addition this quadruple is strictly state passive, then $\mathcal{S} = \{0\} = \mathcal{S}_{\mathrm{lis}}$.

Proof. Let us first check that $\mathrm{Im}(C_2) \cap \mathrm{rint}(\mathrm{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})) \neq \emptyset$, for any $D_2 = D_2^\top \succ 0$. Both mappings $z \mapsto D_2z$ and $\mathcal{N}_{\mathbb{R}_+^{m_2}}(\cdot)$ are maximal monotone, and they verify property (*) in [12, section 1], since they are subdifferentials of convex proper lower semi-continuous functions [12, p.167]. The mapping $(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}})(\cdot)$ is maximal monotone from [8, Corollary 24 (ii)]. Thus using [12, Theorems 3, 4], one finds $\overline{\mathrm{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})} = \overline{\mathrm{Im}(D_2) + \mathrm{Im}(\mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})} = \overline{\mathrm{Im}(D_2) + \mathbb{R}_+^{m_2}} = \mathbb{R}_+^{m_2}$ in our case since $D_2 = 0$. Also $\mathrm{int}(\mathrm{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})) = \mathrm{int}(\mathrm{Im}(D_2) + \mathrm{Im}(\mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})) = \mathrm{int}(\mathrm{Im}(D_2) + \mathbb{R}_+^{m_2})$. Thus $\mathrm{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1}) = \mathrm{Im}(D_2) + \mathbb{R}_+^{m_2} \supset \mathrm{Im}(D_2) \cup \mathbb{R}_+^{m_2}$ [9, Fact 2.9.11]. Since $C_2 \in \mathbb{R}^{m_2 \times n_2}$, $\mathrm{Im}(C_2) \subseteq \mathbb{R}^{m_2}$, and $\mathrm{Im}(C_2) \cap \mathrm{rint}(\mathrm{Im}(D_2 + \mathcal{N}_{\mathbb{R}_+^{m_2}}^{-1})) = \mathrm{Im}(C_2) \cap \mathrm{rint}(\mathrm{Im}(D_2) + \mathbb{R}_+^{m_2})$, which in our case ($D_2 = 0$) reduces to $\mathrm{Im}(C_2) \cap \mathrm{int}(\mathbb{R}_+^{m_2}) \neq \emptyset$ from the assumption. Then passivity guarantees that the map $L_1\mathcal{G}(\cdot)$ is maximal monotone using Theorem 3. From [8, Proposition 23.39] the first result is proved. Using Corollary 3, strong monotonicity follows from strict state passivity. From [8, Proposition 23.37] it is inferred that $\mathcal{S} = \{0\}$, the unique solution to (63) is $x'_2 = 0$. $\square$

### 5.1.2 Case (ii)

The goal is to characterize the largest invariant set inside $\mathcal{Z} = \{x \in \mathbb{R}^n | x_1 = 0\}$. On the set $\mathcal{Z}$, the interconnection (6) has the dynamics:

$$ 
\begin{align} 
(a) \quad & \dot{x}_2 = A_2x_2 + (L_2G_1 \ B_2)\lambda \nonumber \\ 
(b) \quad & L_1H_2x_2 + (B_1 \ L_1G_2)\lambda = 0 \tag{70} \\ 
(c) \quad & 0 \le \lambda \perp \tilde{D}\lambda + \begin{pmatrix} E_1H_2 \\ C_2 \end{pmatrix} x_2 \ge 0, \nonumber 
\end{align} 
$$

which is a mixed LCS (MLCS), and $x_2 = 0$, $\lambda_2 = 0$ is an equilibrium pair. This MLCS can be studied along the same lines as (62).
## 5.2 Strictly State and Strongly Passive Interconnection

Let us place ourselves in the framework of Proposition 3. The strict state passivity of the interconnection guarantees that along its trajectories: $\dot{V}(x_1, x_2) = x^\top P \dot{x} \leq -\epsilon x^\top P x$. On the subspace $x_1 = 0$ and $x_2 = 0$, the subsystems in (4) reduce to:

$$ 
\begin{array}{l}
0 = B_1 \lambda_1^* + L_1 G_2 \lambda_2^* \\
0 = B_2 \lambda_2^* + L_2 G_1 \lambda_1^* \\
(a) \quad 0 \le \lambda_1^* \perp D_1 \lambda_1^* + E_1 G_2 \lambda_2^* \ge 0 \\
(b) \quad 0 \le \lambda_2^* \perp D_2 \lambda_2^* + E_2 G_1 \lambda_1^* \ge 0,
\end{array}
\qquad (71) $$

equivalently: $\lambda^* \in \text{Ker}(\tilde{B}) \cap \text{SOLLCP}(\tilde{D}, 0)$, which is nonempty since $\lambda^* = 0$ is a solution. Since $\tilde{D} \ge 0$, it follows that the set of solutions to the LCP (71) (a) (b), is convex and contains a least norm solution, which is $\lambda^* = 0$. We have SOLLCP($\tilde{D}, 0$) = $\{0\}$ if and only if $\tilde{D}$ is an $R_0$-matrix [27, Definition 3.8.7], equivalently (since $\tilde{D}$ is a $P_0$-matrix) if and only if it is an $R$- or a $Q$-matrix [27, Theorem 3.9.22]. There exists finite algorithm to check the $Q$ness of a matrix [34, Exercise 3.87].

**Proposition 7.** Let the conditions of Proposition 3 (resp. Proposition 2) be satisfied. Then the interconnection (6) possesses the equilibrium point $x^* = 0$ and multipliers $\lambda^* \in \text{Ker}(\tilde{B}) \cap \text{SOLLCP}(\tilde{D}, 0)$, and $x^*$ is asymptotically stable (resp. has a unique equilibrium point $x^* = 0$ and multiplier $\lambda^* = 0$, $x^*$ is globally asymptotically stable and $\lambda(t) \to 0$ as $t \to +\infty$).

*Proof.* Follows using [17, Theorem 6.3] (resp. Theorem 2 and Corollary 3). $\square$

# 6 Examples

Let us analyse the interconnection of two circuits as in Fig. 2 (b). The dynamics is given by the LCS:

$$ 
\left\{
\begin{array}{l}
\dot{x}_{11} = \frac{-1}{\mathbf{R}_1 \mathbf{C}_1} x_{11} + x_{12} + \frac{1}{\mathbf{R}_1} \lambda_1 + (\frac{1}{\mathbf{R}_1} \ 0) u_1 \\
\dot{x}_{12} = \frac{-1}{\mathbf{L}_1 \mathbf{C}_1} x_{11} + \frac{1}{\mathbf{L}_1} \lambda_1 + (\frac{1}{\mathbf{L}_1} \ \frac{1}{\mathbf{L}_1}) u_1 \\
0 \le \lambda_1 \perp w_1 = \frac{-1}{\mathbf{R}_1 \mathbf{C}_1} x_{11} + x_{12} + \frac{1}{\mathbf{R}_1} \lambda_1 + (\frac{1}{\mathbf{R}_1} \ 0) u_1 \ge 0,
\end{array}
\right.
\qquad (72) $$

where $x_{11}$ is the capacitor's voltage ($C_1 \dot{x}_{11} = i_{c1}$), $x_{12} = i_{l1}$ (the current through the inductance), $u_1 = (u_{11} \ u_{12})^\top$ the voltage sources, $\lambda_1$ the voltage across the diode, $w_1$ the current through the diode. We have $x_1 = (x_{11} \ x_{12})^\top$, $A_1 = \begin{pmatrix} \frac{-1}{\mathbf{R}_1 \mathbf{C}_1} & 1 \\ \frac{-1}{\mathbf{L}_1 \mathbf{C}_1} & 0 \end{pmatrix}$, $B_1 = \begin{pmatrix} \frac{1}{\mathbf{R}_1} \\ \frac{1}{\mathbf{L}_1} \end{pmatrix}$, $C_1 = (\frac{-1}{\mathbf{R}_1 \mathbf{C}_1} \ 1)$, $D_1 = \frac{1}{\mathbf{R}_1}$, $L_1 = \begin{pmatrix} \frac{1}{\mathbf{R}_1} & 0 \\ \frac{1}{\mathbf{L}_1} & \frac{1}{\mathbf{L}_1} \end{pmatrix}$, $E_1 = (\frac{1}{\mathbf{R}_1} \ 0)$. We shall consider the interconnection of two identical circuits, the interconnection matrices in (5) are: $H_i = \begin{pmatrix} H_i^{11} & H_i^{12} \\ H_i^{21} & H_i^{22} \end{pmatrix}$,
$G_i = \begin{pmatrix} G_{i1} \\ G_{i2} \end{pmatrix}, i = 1, 2$. The transfer function $H_i(s) = \frac{w_i(s)}{\lambda_i(s)} = \frac{sC_i(sL_i+R_i)}{s^2L_iR_iC_i+sL_i+R_i}$, $i = 1, 2$, is PR but not SPR. The conditions in Lemma 1, (14) are given by ($P_{12} \in \mathbb{R}^{2 \times 2}$):

$$ \mathrm{Im} \left( \begin{pmatrix} \frac{1}{\mathbf{R}_1} & \frac{1}{\mathbf{L}_1} \\ \frac{1}{\mathbf{R}_1} & \frac{1}{\mathbf{C}_1} \end{pmatrix} P_1 + \left( \frac{1}{\mathbf{R}_1} \frac{1}{\mathbf{C}_1} - 1 \right) + \left( \frac{\mathbf{G}_{11}}{\mathbf{R}_2} \frac{\mathbf{G}_{11}+\mathbf{G}_{12}}{\mathbf{L}_2} \right) P_{12}^{\top} \right. \\ \left. \begin{pmatrix} \frac{\mathbf{G}_{21}}{\mathbf{R}_1} & \frac{\mathbf{G}_{22}+\mathbf{G}_{21}}{\mathbf{L}_1} \\ \frac{\mathbf{G}_{21}}{\mathbf{R}_1} & \frac{\mathbf{G}_{22}+\mathbf{G}_{21}}{\mathbf{R}_2} \end{pmatrix} P_1 + \left( \frac{1}{\mathbf{R}_2} \frac{1}{\mathbf{L}_2} \right) P_{12}^{\top} - \left( \frac{\mathbf{H}_1^{21}}{\mathbf{R}_2} \frac{\mathbf{H}_1^{22}}{\mathbf{R}_2} \right) \right) \\ + \mathrm{Im} \left( \begin{pmatrix} \frac{1}{\mathbf{R}_1} & \frac{1}{\mathbf{L}_1} \\ \frac{1}{\mathbf{R}_1} & \frac{1}{\mathbf{C}_1} \end{pmatrix} P_{12} + \left( \frac{\mathbf{G}_{11}}{\mathbf{R}_2} \frac{\mathbf{G}_{11}+\mathbf{G}_{12}}{\mathbf{L}_2} \right) P_2 - \left( \frac{\mathbf{H}_2^{21}}{\mathbf{R}_1} \frac{\mathbf{H}_2^{22}}{\mathbf{R}_1} \right) \right) \right. \\ \left. \begin{pmatrix} \frac{1}{\mathbf{R}_2} & \frac{1}{\mathbf{L}_2} \\ \frac{1}{\mathbf{R}_2} & \frac{1}{\mathbf{C}_2} \end{pmatrix} P_2 + \left( \frac{1}{\mathbf{R}_2} \frac{1}{\mathbf{C}_2} - 1 \right) + \left( \frac{\mathbf{G}_{21}}{\mathbf{R}_1} \frac{\mathbf{G}_{21}+\mathbf{G}_{22}}{\mathbf{L}_1} \right) P_{12} \right) \right) \quad (73) $$

$$ \subseteq \mathrm{Im} \left( \frac{2}{\mathbf{R}_1} \frac{\mathbf{R}_1}{\mathbf{R}_2} + \frac{\mathbf{G}_{21}}{\mathbf{R}_1} \right) + \mathrm{Im} \left( \frac{\mathbf{G}_{11}}{\mathbf{R}_2} + \frac{\mathbf{G}_{21}}{\mathbf{R}_1} \right) $$

which holds always if $|\frac{\mathbf{G}_{21}}{\mathbf{R}_1} + \frac{\mathbf{G}_{11}}{\mathbf{R}_2}| \neq \frac{2}{\sqrt{\mathbf{R}_1 \mathbf{R}_2}}$. Corollary 1 gives $(\frac{\mathbf{G}_{21}}{\mathbf{R}_1} \frac{\mathbf{G}_{22}+\mathbf{G}_{21}}{\mathbf{L}_1})P_1 - (\frac{\mathbf{H}_1^{11}}{\mathbf{R}_2} \frac{\mathbf{H}_1^{12}}{\mathbf{R}_2}) = 0$ and $(\frac{\mathbf{G}_{11}}{\mathbf{R}_2} \frac{\mathbf{G}_{11}+\mathbf{G}_{12}}{\mathbf{L}_2})P_2 - (\frac{\mathbf{H}_2^{11}}{\mathbf{R}_1} \frac{\mathbf{H}_2^{12}}{\mathbf{R}_1}) = 0$. The conditions in Lemma 2 boil down to: $\mathrm{Im}(\frac{\mathbf{G}_{21}}{\mathbf{R}_1} + \frac{\mathbf{G}_{11}}{\mathbf{R}_2}) \subseteq \mathrm{Im}(\frac{2}{\mathbf{R}_1}) = \mathbb{R}$, $|\frac{\mathbf{G}_{21}}{\mathbf{R}_1} + \frac{\mathbf{G}_{11}}{\mathbf{R}_2}| < \frac{4}{\mathbf{R}_1 \mathbf{R}_2}$. Let us now study the conditions in Lemma 3, which involve the matrices (see section 2 for definitions):

$$ \bar{Q}_1 = Q_1 + P_{12} \overbrace{\left( \begin{pmatrix} \frac{H_1^{11}}{\mathbf{R}_1} & \frac{H_1^{12}}{\mathbf{R}_2} \\ \frac{H_1^{11}+H_1^{21}}{\mathbf{L}_2} & \frac{H_1^{12}+H_1^{22}}{\mathbf{L}_2} \end{pmatrix} \right)}^{L_2 H_1} + \left( \begin{pmatrix} \frac{H_1^{11}}{\mathbf{R}_1} & \frac{H_1^{12}}{\mathbf{R}_2} \\ \frac{H_1^{11}+H_1^{21}}{\mathbf{L}_2} & \frac{H_1^{12}+H_1^{22}}{\mathbf{L}_2} \end{pmatrix} \right)^{\top} P_{12}^{\top} \quad (74) $$

and similarly for $\bar{Q}_2$ and $\bar{Q}_{12}$. We have $P_i = \begin{pmatrix} \frac{1}{C_i} & 0 \\ 0 & \mathbf{L}_i \end{pmatrix}$, so $Q_i = \begin{pmatrix} \frac{2}{\mathbf{R}_i C_i} & 0 \\ 0 & 0 \end{pmatrix}$, $i = 1, 2$.

There are 12 parameters to be tuned in (74) in order to satisfy the conditions of Lemma 3. It is noteworthy that when $Q_1 \succ 0$ or $Q_2 \succ 0$, conditions of Lemmata 3 and 4 may not be easy to check analytically. This is greatly simplified if $Q_1 \succ 0$ and $Q_2 \succ 0$, because it is then always possible to choose a small enough $P_{12}$ so that $\bar{Q}_1 \succ 0$, $\bar{Q}_2 \succ 0$ (using Corollary 8), and $H_1$, $P_{12}$ such that $P_{12}L_2H_1+H_1^{\top}L_2^{\top}P_{12}^{\top}$ has rank $n_1$ so that the inclusion in Lemma 3 holds. This is the case if circuit in Fig. 2 (b) has an additional resistance $\mathbf{R}'_1$ in series with the inductance $\mathbf{L}_1$ (and same for the second circuit). Then $Q_i = \begin{pmatrix} \frac{2}{\mathbf{R}_i C_i^2} & 0 \\ 0 & 2\mathbf{R}'_i \end{pmatrix}$,

$H_i(s) = \frac{s^2 + \frac{\mathbf{R}_i + \mathbf{R}'_i}{\mathbf{L}_i} s}{\mathbf{R}_i s^2 + \left( \frac{1}{\mathbf{R}_i C_i} + \frac{\mathbf{R}'_i}{\mathbf{L}_i} \right) s + \frac{1}{\mathbf{L}_i C_i} + \frac{\mathbf{R}'_i}{\mathbf{R}_i C_i \mathbf{L}_i}}$, $i = 1, 2$, which is not SPR and thus not SSPR, so the systems are neither strongly nor strictly state passive (the material in sections 2.2 and 2.3 does not apply). See comments after Lemma 5 for the application of Lemma 4's

$$ \text{conditions. The equality (19) is } P_1 \begin{pmatrix} \frac{H_2^{11}}{\mathbf{R}_1} & \frac{H_2^{12}}{\mathbf{R}_2} \\ \frac{H_2^{11}+H_2^{21}}{\mathbf{L}_1} & \frac{H_2^{12}+H_2^{22}}{\mathbf{L}_1} \end{pmatrix} = - \begin{pmatrix} \frac{H_1^{11}}{\mathbf{R}_2} & \frac{H_1^{12}}{\mathbf{R}_2} \\ \frac{H_1^{11}+H_1^{21}}{\mathbf{L}_2} & \frac{H_1^{12}+H_1^{22}}{\mathbf{L}_2} \end{pmatrix} P_2, $$
which can be easily satisfied with suitable choice of $H_1$ and $H_2$. The circuit in Fig. 2 (a) can be analysed the same way, setting $\mathbf{R}_1 = \mathbf{R}_2 = +\infty$, and a single voltage source. Proposition 4 and Lemma 13 potentially apply to the interconnection of two circuits as in Fig. 2 (a) and (b).

<figure><img src="image_7.png" /><figcaption>Figure 2: Passive circuits with ideal diode.</figcaption></figure>

Switching DAEs [14, 40, 39, 38] can also be studied within the above framework. Let us consider:

$$ egin{cases} 
\dot{x}_{i1} = a_{i1}x_{i1} + b_{i1}x_{i2} + c_{i1}z_i + u_{i1}(t) \\
\dot{x}_{i2} = a_{i2}x_{i1} + b_{i2}x_{i2} + c_{i2}z_i + u_{i2}(t) \\
0 = a_{i3}x_{i1} + b_{i3}x_{i2} + c_{i3}z_i + u_{i4}(t) & \text{if } x_{i1} + u_{i3}(t) < 0 \\
0 = a_{i4}x_{i1} + b_{i4}x_{i2} + c_{i4}z_i + u_{i4}(t) & \text{if } x_{i1} + u_{i3}(t) > 0,
\end{cases} 
\quad (75) $$

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 $c_{i3} \neq 0$ (index 1 DAEs), (75) is rewritten equivalently as the LCS, for each $i = 1, 2$:

$$ egin{cases} 
\dot{x}_{i1} = a_{i1}x_{i1} + \left(b_{i1} - \frac{c_{i1}}{c_{i3}}b_{i3}\right)x_{i2} + \frac{c_{i1}}{c_{i3}}a_{i3}(\lambda_{i1} - u_{i3}) - \frac{c_{i1}}{c_{i3}}a_{i4}(\lambda_{i2} - u_{i3}) + \bar{u}_{i1} \\
\dot{x}_{i2} = a_{i2}x_{i1} + \left(b_{i2} - \frac{c_{i2}}{c_{i3}}b_{i3}\right)x_{i2} + \frac{c_{i2}}{c_{i3}}a_{i3}(\lambda_{i1} - u_{i3}) - \frac{c_{i2}}{c_{i3}}a_{i4}(\lambda_{i2} - u_{i3}) + \bar{u}_{i2} \\
0 \le \lambda_{i1} \perp \lambda_{i1} + x_{i1} + u_{i3}(t) \ge 0, \quad 0 \le \lambda_{i2} \perp \lambda_{i2} - x_{i1} - u_{i3}(t) \ge 0,
\end{cases} 
\quad (76) $$

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 (75) into (76) allows us to see clearly which operator plays a role for passivity analysis, and which role is played by the various controllers. Given the passivity of $(A_i, B_i, C_i, D_i)$ defined from (76), the ma-

terial in foregoing sections applies to study (75). We have: $A_i = \begin{pmatrix} a_{i1} & b_{i1} - \frac{c_{i1}}{c_{i3}}b_{i3} \\ a_{i2} & b_{i2} - \frac{c_{i2}}{c_{i3}}b_{i3} \end{pmatrix}$, $B_i = \begin{pmatrix} \frac{c_{i1}}{c_{i3}}a_{i3} & -\frac{c_{i1}}{c_{i3}}a_{i4} \\ \frac{c_{i2}}{c_{i3}}a_{i3} & -\frac{c_{i2}}{c_{i3}}a_{i4} \end{pmatrix}$, $C_i = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}$, $D_i = I_2$, $L_i = \begin{pmatrix} 1 & 0 & 0 & -\frac{c_{i1}}{c_{i3}} \\ 0 & 1 & 0 & -\frac{c_{i2}}{c_{i3}} \end{pmatrix}$, $E_i = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 0 \end{pmatrix}$, $H_{i1} \in \mathbb{R}^{4 \times 2}$, $H_{i2} \in \mathbb{R}^{4 \times 2}$, $G_{i1} \in \mathbb{R}^{4 \times 2}$, $G_{i2} \in \mathbb{R}^{4 \times 2}$. Once $P_i \succ 0$
has been calculated, one can use the first four inclusions in (15) to check (14). Conditions in Lemma 2 reduce to assuring a "small enough" $E_1G_2 + G_1^\top E_2^\top$.

**Back to motivating example:** Consider the interconnection in (2) (3). Assume that $D \succ 0$ and that each subsystem is passive with $P_1$ for (2), $P_2$ for (3). The conditions in Proposition 2 can be checked with $A_1 = A + LM$, $L_1H_2 = -LM$, $L_2H_1 = 0$, $A_2 = A - K_1C$, $B_1 = B$, $L_1G_2 = 0$, $L_2G_1 = B$, $B_2 = -B$, $C_1 = E_2H_1 = C + EM$, $E_1H_2 = -EM$, $C_2 = -(C - K_2H + EM)$, $D_1 = D_2 = D$, $E_1G_2 = 0$, $E_2G_1 = 0$. Notice that this proposition can be used even if the subsystems are not strongly passive. Condition (30) holds trivially. Condition (31) 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 (32) 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 $D \succ 0$ can be studied along the lines of section 2.

# 7 Conclusions

In this article the interconnection of passive linear cone complementarity systems is analysed. Conditions which guarantee that the interconnected system defines a maximal monotone mapping are given. The asymptotic stability is studied with an invariance principle. Examples illustrate the theoretical findings. Future research could concern the extension of complementarity towards general maximal monotone mappings.

# A Positive Definite Matrices under Perturbations

Let us provide a corollary of [26, Theorem 2.11]. For $M \in \mathbb{R}^{n \times n}$, $||M||_{2,2}$ is the induced matricial norm such that $||M||_{2,2} = \sigma_{\max}(M)$ (the largest singular value) [9, Proposition 9.4.9], it is a submultiplicative norm [9, Corollary 9.4.12]. The following is an easy consequence of [26].

**Theorem 1.** [26] Let $M \in \mathbb{R}^{n \times n}$ be a positive definite matrix. Then every matrix

$$A \in \{A \in \mathbb{R}^{n \times n} \mid \left\| \left( \frac{M+M^\top}{2} \right)^{-1} \right\|_{2,2} \|M-A\|_{2,2} < 1\}$$

is positive definite.

**Corollary 8.** Let $D = P + N$, where $D$, $P$ and $N$ are $n \times n$ real matrices, and $P \succ 0$, not necessarily symmetric. If $||N||_{2,2} < \frac{1}{\|\left(\frac{P+P^\top}{2}\right)^{-1}\|_{2,2}}$, then $D \succ 0$.
## B Passive Linear Systems

A quadruple $(A, B, C, D)$ is said passive if there exists $P = P^\top \succcurlyeq 0$ such that:

$$ \begin{pmatrix} -A^\top P - PA & -PB + C^\top \\ (-PB + C^\top)^\top & D + D^\top \end{pmatrix} \succcurlyeq 0. \qquad (77) $$

This inequality implies that $\mathrm{Im}(B^\top P - C^\top) \subseteq \mathrm{Im}(D + D^\top)$, and $\mathrm{Im}(PB - C^\top) \subseteq \mathrm{Im}(A^\top P + PA)$, using [9, Proposition 8.2.4] [10, Fact 8.4.3], or [19, Lemma 1 iv] and [9, Theorem 2.4.3, Fact 2.9.14]. The observability of the pair $(C, A)$ guarantees that $P \succcurlyeq 0$ [19] [16, p.116]. The LMI (77) is equivalent to the dissipation equality:

$$ \begin{aligned} V(x(t_1)) - V(x(t_2)) = & \int_{t_1}^{t_2} x^\top(s)(A^\top P + PA)x(s)ds \\ & + \int_{t_1}^{t_2} x^\top(s)(PB - C^\top)u(s)ds \\ & - \int_{t_1}^{t_2} u^\top(s)(D + D^\top)u(s)ds + \int_{t_1}^{t_2} u^\top(s)y(s)ds \end{aligned} \qquad (78) $$

with $V(x) = \frac{1}{2}x^\top P x$, along the trajectories of the system $\dot{x} = Ax + Bu$, $y = Cx + Du$, and for all $t_1 \le t_2$. The system is said strictly state passive if in addition $P \succcurlyeq 0$ and there exists $\mu > 0$ such that (77) holds with $-A^\top P - PA - \mu P$ (hence (77) implies $-A^\top P - PA \succcurlyeq \mu P \succcurlyeq 0$). It is said strongly passive if $P \succcurlyeq 0$ and the inequality in (77) is strict. Positive real (PR) transfer matrices with minimal state space realization are passive. Strictly positive real (SPR) transfer matrices with minimal state space realization are strictly state passive, and vice-versa [32] [16, Theorem 4.73]. Strongly SPR transfer matrices with minimal state space realization are strongly passive [16, p.162], and output and input strictly passive [31]. Incremental passivity is defined in [16, Definition 4.62] [35, 43].

## C Maximal Monotone Mappings and DIs

Let $\mathcal{M}: \mathbb{R}^n \Rightarrow \mathbb{R}^n$ be a set-valued mapping, $\alpha > 0$. Is is monotone (resp. $\alpha$-strongly monotone) if for any $x_1, x_2 \in \mathrm{Dom}(\mathcal{M})$, for any $y_1 \in \mathcal{M}(x_1)$, $y_2 \in \mathcal{M}(x_2)$, it holds $\langle x_1 - x_2, y_1 - y_2 \rangle \ge 0$ (resp. $\ge \alpha ||x_1 - x_2||^2$). It is maximal if its graph cannot be enlarged without destroying monotonicity. The following result appeared in [11, Theorem 3.1].

**Theorem 2.** Consider the system $\dot{x} \in -\mathcal{M}(x)$ with $\mathcal{M}(\cdot)$ maximal monotone. For ever $x_0 \in \mathrm{Dom}(\mathcal{M})$, there exists a unique right-differentiable Lipschitz continuous (hence AC) solution $x: [0, \infty) \to \mathrm{Dom}(\mathcal{M})$, with $x(0) = x_0$, satisfying $||x(t)|| \le ||x(0)||$ and $||\dot{x}(t)|| \le ||\mathcal{M}^0(x(0))||$. Moreover, if $x_1, x_2$ represent two solutions of the system, then

$$ \|x_2(t) - x_1(t)\|^2 \le e^{-\alpha t} \|x_2(0) - x_1(0)\|^2. \qquad (79) $$
Inequality (2) means that the system is incrementally stable. If the system has an equilibrium point then it is Lyapunov stable. In this case, if $M(\cdot)$ is strongly maximal monotone, the equilibrium point is exponentially stable, solutions are bounded and depend continuously on initial data. Using some identities from Convex Analysis, it is possible to rewrite system (1) in the form of a differential inclusion (DI) as:

$$ \dot{x} \in -\mathcal{F}(x, u) \stackrel{\Delta}{=} Ax + Lu - B(D + \mathcal{N}_{K^*}^{-1})^{-1}(Cx + Eu), \quad (80) $$

The next result follows from [23], see also [15, 13, 42] and [37, Theorem 12.43]. It uses passivity of $(A, B, C, D)$ (as defined in Appendix B) to rewrite the LCCS as a DI with a maximal monotone operator, and the fact that the normal cone to a closed convex nonempty set defines a maximal monotone mapping, and so does its inverse mapping [37, Exercise 12.8].

**Theorem 3.** Suppose that $(A, B, C, D)$ in (1) is passive with storage function $x \mapsto x^\top P x$, $P = P^\top > 0$, $K \subseteq \mathbb{R}^m$ a constant closed convex nonempty cone, $u \equiv 0$ and $\mathrm{Im}(C) \cap \mathrm{rint}(\mathrm{Im}(\mathcal{N}_{K^*}^{-1} + D)) \neq \emptyset$. Then, the mapping $\mathcal{F}: \mathbb{R}^n \to \mathbb{R}^n$ is maximal monotone.

# D Useful Matrix Algebra Results

Some of the results from [9, 10] and used at several places in the article are recalled for convenience.

**Proposition 8.** [9, Proposition 8.2.4] Same as Proposition 11 below.

**Corollary 9.** [9, Corollary 9.6.5] Let $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{m \times l}$. Then $\sigma_{\max}(AB) \le \sigma_{\max}(A)\sigma_{\max}(B)$.

**Fact 1.** [10, Fact 8.4.3] Let $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{n \times l}$. Then, $\mathrm{Im}(A) \subseteq \mathrm{Im}(B) \Leftrightarrow BB^\top A = A$.

**Fact 2.** [10, Fact 3.13.15] Let $A, B \in \mathbb{R}^{n \times m}$. Then $\mathrm{Im}(A+B) \subseteq \mathrm{Im}(a) + \mathrm{Im}(B)$. Furthermore $\mathrm{Im}(A) \subseteq \mathrm{Im}(A+B) \Leftrightarrow \mathrm{Im}(B) \subseteq \mathrm{Im}(A+B) \Leftrightarrow \mathrm{Im}(A+B) = \mathrm{Im}(A)+\mathrm{Im}(B)$.

**Fact 3.** [10, Fact 7.17.24] Let $A \in \mathbb{R}^{n \times n}$, assume that $A = A^\top \succ 0$, and $\mathrm{rank}(A) = r$. Then there exists $B \in \mathbb{R}^{n \times r}$ such that $A = BB^\top$.

**Theorem 4.** [9, Theorem 8.6.2] Let $A \in \mathbb{R}^{n \times m}$, $B \in \mathbb{R}^{n \times l}$. Then: There exists $C \in \mathbb{R}^{l \times m}$ such that $A = BC \Leftrightarrow$ there exists $\alpha > 0$ such that $AA^\top \prec \alpha BB^\top \Leftrightarrow \mathrm{Im}(A) \subseteq \mathrm{Im}(B)$.

**Theorem 5.** [9, Theorem 2.4.3] Let $A \in \mathbb{R}^{n \times m}$. Then $\mathrm{Im}^\perp(A) = \mathrm{Ker}(A^\top)$, $\mathrm{Im}(A) = \mathrm{Im}AA^\top$, $\mathrm{Ker}(A) = \mathrm{Ker}(A^\top A)$.
**Theorem 6.** [9, Theorem 8.4.9] Let $A = A^\top \in \mathbb{R}^{n \times n}$, $B = B^\top \in \mathbb{R}^{n \times n}$, with $A \preceq B$. Then for all $i \in \{1, \dots, n\}$: $\lambda_i(A) \le \lambda_i(B)$. If $A \ne B$, there exists $i \in \{1, \dots, n\}$ such that: $\lambda_i(A) < \lambda_i(B)$. If $A \prec B$, then $\lambda_i(A) < \lambda_i(B)$ for all $i \in \{1, \dots, n\}$.

**Corollary 10.** [9, Corollary 8.4.10] Let $A = A^\top \in \mathbb{R}^{n \times n}$, $B = B^\top \in \mathbb{R}^{n \times n}$. If $0 \preceq A \preceq B$ then $0 \le \text{tr}(A) \le \text{tr}(B)$.

(Notice that the trace of a positive semidefinite matrix is nonnegative always).

**Fact 4.** [9, Fact 6.5.21] Let $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{k \times l}$. Then $\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}^\dagger = \begin{pmatrix} A^\dagger & 0 \\ 0 & B^\dagger \end{pmatrix}$.

**Proposition 9.** [10, Proposition 8.1.7 xxx)] Let $A = A^\top \in \mathbb{R}^{n \times m}$, then $A \succ 0 \Leftrightarrow A^\dagger \succ 0$, $A \asymp 0 \Leftrightarrow A^\dagger \asymp 0$.

**Fact 5.** [9, Fact 8.19.2] Let $0 \prec A = A^\top \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $0 \prec C = C^\top \in \mathbb{R}^{m \times m}$. If $\sigma_{\max}^2(B) < \sigma_{\min}(A)\sigma_{\min}(B)$ then $\begin{pmatrix} A & B \\ B^\top & C \end{pmatrix} \succ 0$.

**Theorem 7.** [9, Theorem 5.6.3] Let $A \in \mathbb{R}^{n \times m}$, $A \ne 0$, let $\text{rank}(A) = r$, define $B = \text{diag}(\sigma_1(A), \dots, \sigma_r(A))$. Then, there exist unitary matrices $S_1 \in \mathbb{R}^{n \times n}$ and $S_2 \in \mathbb{R}^{m \times m}$ such that $A = S_1 \begin{pmatrix} B & 0_{r \times (m-r)} \\ 0_{(n-r) \times r} & 0_{(n-r) \times (m-r)} \end{pmatrix} S_2$.

**Corollary 11.** [10, Corollary 11.6.8] Let $A, B \in \mathbb{R}^{n \times n}$, then, for all $i \in \{1, \dots, n\}$: $\sigma_i(A)\sigma_{\min}(B) \le \sigma_i(A)\sigma_{\max}(B) \le \sigma_{\min}(A)\sigma_{\max}(B) \le \sigma_{\max}(A)\sigma_{\min}(B) \le \sigma_{\max}(A)\sigma_{\max}(B)$.

**Proposition 10.** [9, Proposition 6.1.6 vi) and vii)] Let $A \in \mathbb{R}^{n \times m}$. Then $\text{Im}(A) = \text{Im}(A^\dagger)$ and $\text{Im}(A^\dagger) = \text{Im}(A)$.

**Lemma 14.** [9, Lemma 2.4.1] Let $A \in \mathbb{R}^{n \times m}$, $B \in \mathbb{R}^{m \times l}$, $C \in \mathbb{R}^{k \times n}$. Then $\text{Im}(AB) \subseteq \text{Im}(A)$ and $\text{Ker}(A) \subseteq \text{Ker}(CA)$.

**Fact 6.** [9, Fact 9.14.15] Let $A, B \in \mathbb{R}^{n \times n}$, then $\sigma_{\max}(A+B) \le \sigma_{\max}(A) + \sigma_{\max}(B)$.

**Fact 7.** [10, Fact 10.12.17] Let $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $C \in \mathbb{R}^{m \times m}$, assume that $A = A^\top \succ 0$, $C = C^\top \succ 0$, and assume that $\sigma_{\max}^2(B) \le \sigma_{\min}(A)\sigma_{\min}(C)$. Then $\begin{pmatrix} A & B \\ B^\top & C \end{pmatrix} \succ 0$. Assume that in addition $\sigma_{\max}^2(B) < \sigma_{\min}(A)\sigma_{\min}(C)$. Then $\begin{pmatrix} A & B \\ B^\top & C \end{pmatrix} \succ 0$.

**Proposition 11.** [10, Proposition 10.2.5] Let $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^\top & A_{22} \end{pmatrix}$, $A_{11} = A_{11}^\top$, $A_{22} = A_{22}^\top$. Then $A \succ 0 \Leftrightarrow A_{11} \succ 0$ and $A_{12} = A_{11}A_{11}^\dagger A_{12}$ and $A_{12}^\top A_{11}^\dagger A_{12} \preceq A_{22} \Leftrightarrow A_{22} \succ 0$ and $A_{12} = A_{12}A_{22}^\dagger A_{22}$ and $A_{12}A_{22}^\dagger A_{12}^\top \preceq A_{11}$. Also $A \succ 0 \Leftrightarrow A_{11} \succ 0$ and $A_{12}^\top A_{11}^{-1} A_{12} \prec A_{22} \Leftrightarrow A_{22} \succ 0$ and $A_{12}A_{22}^{-1} A_{12}^\top \prec A_{11}$.
(warning: there is a mistake in [10, Fact 10.24.25] and in [9, Fact 8.21.22], copied in [16, Proposition A.68], see https://inria.hal.science/hal-03364924v3/document: so unfortunately these criterion cannot be used to characterize positive semidefiniteness.)

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