# State Jumps Analysis in Differential Algebraic Linear Complementarity Systems

Bernard Brogliato

December 23, 2025

## Abstract

This article is largely concerned with a class of finite dimensional nonlinear nonsmooth systems named Differential Algebraic Linear Complementarity Systems (DALCS). Such dynamical systems can undergo state discontinuities. The objective is to provide characterizations of the jumps in both the differential and the algebraic state variables. In particular it is shown that algebraic states can undergo discontinuities at time instants strictly larger than the initial time, in the absence of exogenous excitation (a phenomenon sharing similarities with impasse points in differential algebraic equations (DAE)). This is shown under passivity properties to be closely related to the analysis of various generalized equations of the form $0 \in A(\eta) + B(\eta)$ with $A(\cdot)$ single-valued, $B(\cdot)$ set-valued, both with maximal monotonicity properties. Circuits with ideal diodes, switching DAEs, descriptor-variable systems with set-valued feedback, illustrate the theory.

## 1 Introduction

State discontinuities are ubiquitous in finite-dimensional systems subjected to equality and/or inequality state constraints. Among those systems we may find linear complementarity systems (LCS)[49, 25, 26, 22, 29, 28, 27, 21, 9, 46, 63], nonlinear complementarity systems [21], nonsmooth, linear, or nonlinear and time-varying differential-algebraic equations (DAE) [75, 78, 77, 76, 66, 68, 67, 69], switching DAE with state-dependent [71, 32] or exogenous switches [55, 52, 83, 31], singular zero-order sweeping process (ZOSwP) [20, section 2.1.1], first-order sweeping processes (FOSwP) [58, 57, 56, 39, 81, 20], second order sweeping process (SOSwP) and mechanical systems with bilateral or unilateral contact and friction [59, 60, 56, 11], where friction induces Painlevé paradoxes and velocity jumps without normal impacts [47, 30], higher-order sweeping process (HOSwP) [1, 12]. As can be found in the above-cited references, applications are in electrical circuits, power systems, hydraulic circuits, mechanical systems, chemistry, thermodynamical systems, DAEs with nonsmooth or switching constraints. Under given conditions, these formalisms can possess (Lipschitz, absolutely) continuous solutions. However they can also undergo state jumps of various kinds (the most well-known being mechanical impacts, electrical spikes). In general state jumps can be triggered by exogenous terms (it is the case in some LCS
[25, 20] and DAE [66, 69]), or be state-dependent (mechanical systems with unilateral contact [11]) or state-triggered (impasse points and surfaces in DAE [51, 66, 69, 75, 33, 70]), or occur at the initial time due to inconsistency (in LCS [26] and in DAE [35, 36, 68, 69, 82]). Differential Algebraic LCS [13, 14, 19] (or descriptor variable LCS) and differential algebraic inclusions [28] have been introduced and analysed recently. In this article it is shown (mostly under passivity-like assumptions, which are closely related to maximal monotonicity), that state jumps in DALCS obey generalized equations (GE) of the form

$$0 \in A(\eta) + B(\eta), \quad (1)$$

where $A(\cdot)$ and $B(\cdot)$ are maximal monotone mappings, $A(\cdot)$ is single-valued (affine) and $B(\cdot)$ may be set-valued. Also in most of the cases the structure of $B(\cdot)$ stems from the Lur'e set-valued systems framework [10, 16, 17, 20], and may be seen as an extended resolvant (see [73, p.539] [5] for classical resolvants analysis). Hence, the analysis of such GE is pivotal. One advantage of working with such GE is that well-posedness results [2, 5, 40, 85] and efficient algorithms exist to solve them [5, 8, 84, 48, 45]. State jumps in DALCS is a totally open issue which needs to be addressed as a step towards well-posedness analysis. The motivation stems from numerical experiments performed in [19]. They show (a fact proved in this article) that state jumps can occur in the absence of exogenous signals at times other than initial time (while LCS undergo only initial jumps if no exogenous signals are present in the complementarity constraints). Notice that the works in [14, 13] are limited to continuous solutions, as well as the related works in [28, 78, 77, 76].

This article is organized as follows: dynamical systems are presented in section 2, state jumps are studied in section 3: jumps in the algebraic state when differential states are continuous are characterized in sections 3.2 and 3.4, jumps in differential states with continuous algebraic states are analysed in section 3.3, simultaneous jumps in both state variables are characterized in section 3.5. Section 4 is dedicated to illustrating examples: a circuit with ideal diode in section 4.1, a class of switching DAE in section 4.2, descriptor-variable systems with set-valued feedback in section 4.3. Conclusions and perspectives are in section 5, and useful mathematical tools are in appendices A, B, C.

**Notation and definitions:** [43, 5, 72, 73, 44, 50] Let $K \subseteq \mathbb{R}^n$ be a closed convex set, its normal cone at $x \in K$ is $\mathcal{N}_K(x) = \{w \in \mathbb{R}^n \mid w^\top(z-x) \le 0, \forall z \in K\}$, $\mathcal{N}_K(x) = \emptyset$ if $x \notin K$, its tangent cone at $x$ is $\mathcal{T}_K(x) = \{w \in \mathbb{R}^n \mid w^\top x \le 0, \forall x \in \mathcal{N}_K(x)\}$, its indicator function is $\Psi_K(x) = 0$ if $x \in K$, $\Psi_K(x) = +\infty$ if $x \notin K$, its support function is $\sigma_K(x) = \sup_{y \in K} y^\top x$, its dual cone is $K^* = \{w \in \mathbb{R}^n \mid w^\top x \ge 0, \forall x \in K\}$, its polar cone is $K^\circ = -K^*$ (one has $\mathcal{N}_K(x) = (\mathcal{T}_K(x))^\circ$ and $\mathcal{T}_K(x) = (\mathcal{N}_K(x))^\circ$), its barrier cone is $\text{bar}(K) = \text{dom}(\sigma_K)$ (convex but not necessarily closed). Let $K$ be nonempty convex (not necessarily closed), then for any $x_0 \in K$: $K_\infty = \bigcap_{\lambda>0} \frac{1}{\lambda}(K - x_0) = \{u \in \mathbb{R}^n \mid x + \lambda u \in K, \forall \lambda \ge 0, \forall x \in K\}$ is its recession (nonempty, convex) cone.

Let $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper convex lower semicontinuous (LSC) function, its conjugate function is the convex proper LSC function $f^*(y) = \sup_{x \in \text{dom}(f)}(x^\top y - f(x))$ (one has $\sigma_K = \Psi_K^*$, and $\Psi_K^* = \Psi_{K^\circ}$ if $K$ is a closed convex cone), its subdifferential at $x$ is the set $\partial f(x) = \{w \in \mathbb{R}^n \mid f(y) - f(x) \ge w^\top(y - x), \forall y \in \mathbb{R}^n\}$ (one has $\partial \Psi_K(x) = \mathcal{N}_K(x)$), the inverse mapping of $\partial f: \mathbb{R}^n \to \mathbb{R}^n$ is $(\partial f)^{-1}(\cdot) = \partial f^*(\cdot)$, see also (96) and (97). The recession function of $f(\cdot)$ is defined as follows: let $x_0$ be any element in $\text{dom}(f) = \{x \in \mathbb{R}^n \mid f(x) < +\infty\}$, then for all $x \in \mathbb{R}^n$, $f_\infty(x) = \lim_{\lambda \to +\infty} \frac{1}{\lambda}f(x_0 + \lambda x)$. The function $f_\infty: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is a proper convex LSC function.
The domain of a set-valued function $F: \mathbb{R}^n \to \mathbb{R}^m$ is $\text{dom}(F) = \{x \in \mathbb{R}^n | F(x) \neq \emptyset\}$. The set-valued mapping $F: \mathbb{R}^n \to \mathbb{R}^n$ is monotone if for all $x_1, x_2 \in \text{dom}(F)$, for all $y_1 \in F(x_1)$, $y_2 \in F(x_2): (x_1 - x_2)^\top(y_1 - y_2) \ge 0$. It is maximal monotone if its graph cannot be extended to another monotone mapping. It is $\alpha$-strongly monotone if $(x_1 - x_2)^\top(y_1 - y_2) \ge \alpha |x_1 - x_2|^2$, $\alpha > 0$. The resolvant of index $\alpha > 0$ of a maximal monotone mapping $F(\cdot)$, is defined as the mapping $(\mathbf{I}_d + \alpha F)^{-1}(\cdot)$, $\mathbf{I}_d$ the identity mapping, it is single-valued, Lipschitz continuous with domain $\mathbb{R}^m$.

A linear cone complementarity problem LCCP($M$, $K$, $q$) with unknown $\lambda$ is defined as $K^* \ni \lambda \perp M\lambda + q \in K$, $M \in \mathbb{R}^{m \times m}$, $q \in \mathbb{R}^m$, $K \subseteq \mathbb{R}^m$ is a nonempty closed convex cone, SOL(LCCP($M$, $K$, $q$)) denotes its solution set, and a linear complementarity problem LCP($M$, $q$) is equal to LCCP($M$, $\mathbb{R}_+^m$, $q$). Let $M \in \mathbb{R}^{n \times n}$, it is positive definite $M > 0$ (resp. semidefinite $M \ge 0$), not necessarily symmetric [38, Definition 2.2.15], if $x^\top M x > 0$ for all $x \neq 0$ (resp. $x^\top M x \ge 0$ for all $x$), its Moore-Penrose pseudo-inverse is denoted $M^\dagger$, $M_\bullet$ is its $j$th column, $M_\bullet$ is its $i$th row. Its range is $\text{Im}(M)$, its kernel is $\text{Ker}(M)$. The closure of a set $C$ is denoted $\bar{C}$, its least-norm element is denoted $[C]^0$, its relative interior as $\text{rint}(C)$, its interior as $\text{int}(C)$. The Lebesgue measure is $dt$, the Dirac measure with atom at $t$ is $\delta_t$, $f(\cdot)$ is AC means it is absolutely continuous. The next properties are closely related to passivity [18].

**Definition 1.** A pair of matrices $(B, C)$ has the $X$-property if there exists $X = X^\top \succ 0$ such that $XB = C^\top$. A triple of matrices $(B, C, D)$ has the $XD$-property if there exists $X = X^\top \succ 0$ such that $\text{Ker}(D + D^\top) \subseteq \text{Ker}(XB - C^\top)$. In both cases the square root of $X$ is denoted as $R$, with $R = R^\top \succ 0$.

Clearly $(B, C)$ has the $X$-property $\Leftrightarrow (-B, -C)$ has the $X$-property, and $(B, C, D)$ has the $XD$-property $\Leftrightarrow (-B, -C, D)$ has the $XD$-property $\Leftrightarrow (-B, -C, -D)$ has the $XD$-property $\Leftrightarrow (B, C, -D)$ has the $XD$-property. It is noteworthy that both properties are closely related to passivity [18], which has been largely used to prove well-posedness of LCS, together with the closely related property of maximal monotonicity, see, e.g., [10, 16, 17, 21, 13, 19, 14, 15, 25, 22, 28, 29, 27, 79, 54, 53, 63].

## 2 Dynamical System

Following [14], let us consider the following DALCS in semi-explicit form:

$$
\begin{align}
(a) \quad & P_I \dot{x}(t) = A_I x(t) + B_I z(t) + \bar{B}_I \lambda(t) + E_I(t) \nonumber \\
(b) \quad & 0 = A_{II} x(t) + B_{II} z(t) + \bar{B}_{II} \lambda(t) + E_{II}(t) \tag{2} \\
(c) \quad & K^* \ni \lambda(t) \perp w(t) = C_I x(t) + C_{II} z(t) + D \lambda(t) + F(t) \in K, \nonumber
\end{align}
$$

with the differential state $x(t) \in \mathbb{R}^{n_1}$, the algebraic state $z(t) \in \mathbb{R}^{n_2}$, the multiplier $\lambda(t) \in \mathbb{R}^m$, $w(t) \in \mathbb{R}^m$, $P_I \in \mathbb{R}^{n_1 \times n_1}$ is full-rank, $K \subseteq \mathbb{R}^m$ is a closed nonempty convex cone, $K^*$ is its dual cone, $\bar{B}_{II} \in \mathbb{R}^{p \times m}$, $A_{II} \in \mathbb{R}^{p \times n_1}$, $B_{II} \in \mathbb{R}^{p \times n_2}$, $C_I \in \mathbb{R}^{m \times n_1}$, $C_{II} \in \mathbb{R}^{m \times n_2}$, $D \in \mathbb{R}^{m \times m}$, $B_I \in \mathbb{R}^{n_1 \times n_2}$, $A_I \in \mathbb{R}^{n_1 \times n_1}$, $\bar{B}_I \in \mathbb{R}^{n_1 \times m}$, and $E_I: \mathbb{R}_+ \to \mathbb{R}^{n_1}$, $E_{II}: \mathbb{R}_+ \to \mathbb{R}^p$, $F: \mathbb{R}_+ \to \mathbb{R}^m$ are right-continuous bounded functions of time. Thus $p$ is the number of equality constraints, $m$ is the number of complementarity constraints. In (2), (a) has to be satisfied $dt$-almost-everywhere, (b) and (c) $dt$-everywhere. Time-continuous solutions are analyzed in [14] under strict-passivity and strong monotonicity conditions of some operators, with constant $K$. However an example in [19, section 6.1]
shows that the algebraic state $z(\cdot)$ can jump while $x(\cdot)$ is continuous and $\lambda$ is a time-function: this is a kind of discontinuity without impulsive terms (Dirac measure), or impulse-free jump. Such phenomena are well-known in time-varying and nonlinear DAE [66, 31]. Let us rewrite (2) (c) equivalently as:

$$ 
\begin{aligned}
\lambda(t) & \in \begin{aligned}[t]
& -\partial\Psi_K(C_I x(t) + C_{II} z(t) + D\lambda(t) + F(t)) \\
& = -\partial\Psi_{K-F(t)}(C_I x(t) + C_{II} z(t) + D\lambda(t)) \\
& = -\partial\Psi_{K-C_I x(t)-F(t)}(C_{II} z(t) + D\lambda(t))
\end{aligned} \\
\Leftrightarrow \lambda(t) & \in \begin{aligned}[t]
& (D + \partial\Psi_{K-F(t)}^{-1})(-\lambda(t)) \ni C_I x(t) + C_{II} z(t) \\
& (D + \partial\Psi_{K-C_I x(t)-F(t)}^{-1})(-\lambda(t)) \ni C_{II} z(t) \\
& -(D + \partial\Psi_{K-F(t)}^{-1})^{-1}(C_I x(t) + C_{II} z(t)) \\
& = -(D + \partial\sigma_{K-F(t)})^{-1}(C_I x(t) + C_{II} z(t)) \\
& = -(D + \partial\sigma_{K-C_I x(t)-F(t)})^{-1}(C_{II} z(t)).
\end{aligned}
\end{aligned} 
\quad (3) $$

Therefore (2) is rewritten equivalently as:

$$ 
\begin{aligned}
(a) \quad & P_I \dot{x}(t) \in A_I x(t) + B_I z(t) + E_I(t) + \bar{B}_I \lambda(t) \\
(b) \quad & 0 = A_{II} x(t) + B_{II} z(t) + E_{II}(t) + \bar{B}_{II} \lambda(t) \\
(c) \quad & \lambda(t) \in -(D + \partial\sigma_{K(t)-F(t)})^{-1}(C_I x(t) + C_{II} z(t)) \\
(c') \quad & \lambda(t) \in -(D + \partial\sigma_{K(t)-C_I x(t)-F(t)})^{-1}(C_{II} z(t))
\end{aligned} 
\quad (4) $$

We see that in general (4) is a differential inclusion (DI) with a set-valued constraint. Assume that $D > 0$, then (4) is equivalent to:

$$ 
\begin{aligned}
(a) \quad & P_I \dot{x}(t) = A_I x(t) + B_I z(t) + E_I(t) - \bar{B}_I (D + \partial\sigma_{K-F(t)})^{-1} (C_I x(t) + C_{II} z(t)) \\
(b) \quad & 0 = A_{II} x(t) + B_{II} z(t) + E_{II}(t) - \bar{B}_{II} (D + \partial\sigma_{K-F(t)})^{-1} (C_I x(t) + C_{II} z(t))
\end{aligned} 
\quad (5) $$

where the operator $(D + \partial\sigma_{K-F(t)})^{-1}(\cdot)$ is single-valued Lipschitz continuous, using [16, Proposition 1]. Thus the DALCS is a nonlinear time-varying DAE. As shown in this article, its algebraic state can have discontinuities even if $F(t) = 0$ and $E_{II}(t) = 0$ for all $t$. Assume now $D = 0$, then (4) is equal to:

$$ 
\begin{aligned}
(a) \quad & P_I \dot{x}(t) \in A_I x(t) + B_I z(t) + E_I(t) - \bar{B}_I \mathcal{N}_{K-F(t)}(C_I x(t) + C_{II} z(t)) \\
(b) \quad & 0 \in A_{II} x(t) + B_{II} z(t) + E_{II}(t) - \bar{B}_{II} \mathcal{N}_{K-F(t)}(C_I x(t) + C_{II} z(t))
\end{aligned} 
\quad (6) $$

which is a singular DI. Under the X-property of $\left(\begin{pmatrix} P_I^{-1} \bar{B}_I \\ \bar{B}_{II} \end{pmatrix}, (C_I, C_{II})\right)$, (6) is a differential algebraic FOSwP (DAFOSwP), using a variable change as in [10, 21, 20]:

$$ R \begin{pmatrix} I_{n_1} & 0 \\ 0 & 0 \end{pmatrix} R^{-1} \begin{pmatrix} \dot{x}(t) \\ \dot{z}(t) \end{pmatrix} \in R \begin{pmatrix} P_I^{-1} A_I & P_I^{-1} B_I \\ A_{II} & B_{II} \end{pmatrix} R^{-1} \begin{pmatrix} \tilde{x}(t) \\ \tilde{z}(t) \end{pmatrix} + \begin{pmatrix} P_I^{-1} E_I(t) \\ E_{II}(t) \end{pmatrix} - \mathcal{N}_{\Phi(t)} \begin{pmatrix} \tilde{x}(t) \\ \tilde{z}(t) \end{pmatrix}, \quad (7) $$

where $\Phi(t) \triangleq \{(x, z) | C_I x + C_{II} z \in K - F(t)\}$, $\begin{pmatrix} \tilde{x}(t) \\ \tilde{z}(t) \end{pmatrix} = R \begin{pmatrix} x(t) \\ z(t) \end{pmatrix}$. If $X = \text{diag}(X_I, X_{II})$, then X-property of $\left(\begin{pmatrix} P_I^{-1} \bar{B}_I \\ \bar{B}_{II} \end{pmatrix}, (C_I, C_{II})\right)$ is equivalent to X-property of $\begin{pmatrix} P_I^{-1} \bar{B}_I \\ C_I \end{pmatrix}$ and $(\bar{B}_{II}, C_{II})$ (which will be used later to study state jumps). In such a case $\tilde{z}(\cdot)$ is the algebraic state of the DAFOSwP and $\tilde{x}(\cdot)$ its differential state.
# 3 State-jumps Characterization

Let us make basic assumptions.

**Assumption 1.** The state variables are right-continuous, they possess a left and a right limits at $t$, denoted $x(t^+)$, $x(t^-)$ and $z(t^+)$, $z(t^-)$, and they may jump at time $t$, i.e., $x(t^+) \neq x(t^-)$ or $z(t^+) \neq z(t^-)$.

Hence Assumption 1 allows the algebraic state to jump while the differential state is continuous, and vice-versa.

**Assumption 2.** The multiplier is a measure with atom at $t$, i.e., $\lambda = \nu_t \delta_t + \lambda(t)dt$ for some $\nu_t \ge 0$ and a piecewise right-continuous AC function $\lambda(\cdot)$.

This allows for the time-function $\lambda(\cdot)$ to possess discontinuities (at junction times when constraints are activated or deactivated).

**Assumption 3.** The discontinuity time $t$ is isolated, i.e., there exists $\eta > 0$ such that the state variables are AC on $(t - \eta, t)$ and $(t, t + \eta)$.

Assumption 2 means that we do not consider solutions and multipliers which are higher degree distributions, which may be considered in such systems [1, 35, 36, 68]. Moreover we do not provide conditions which guarantee that multipliers are at most measures (as it is the case for passive LCS [21, 26, 29]). Assumption 3 does not preclude the existence of finite accumulations of discontinuity times. Jumps may occur when $E_{II}(\cdot)$ jumps at $t$ so that the equality constraint (2) (b) does not hold on the right of $t$, or, similarly, if $F(\cdot)$ jumps at $t$ so that the inclusion in (2) (c) does not hold on the right of $t$ if state continuity is enforced (including $t = 0$, i.e., initial discontinuities). But, state jumps may occur also while both $E_{II}(\cdot)$ and $F(\cdot)$ are continuous, as shown later. There are basically three issues when dealing with state jumps in dynamical systems:

1. Given data at a time $t$, is a jump necessary to continue the integration on the right of $t$ (this is a mathematical problem)?

2. Determine a state-jump law (this is a modeling problem).

3. Prove the system's well-posedness with this state-jump model (this is a mathematical problem).

In this article we essentially focus on item 1 for DALCS (2). Similarly to [14], we mainly focus on the properties of the constraints (2) (b) and (c). Notice that considering varying $K(t)$ is possible because the analyses which follow in the next section are done for fixed time $t$. In general well-posedness of (2) when components are time-varying is a complex issue, even for LCS [63].

## 3.1 General Considerations

In view of Assumption 2, the dynamics at the jump time $t$ is given by the algebraic equations/inclusion: there exists $\nu_t$ such that

$$ 
\begin{align} 
(a) \quad & P_I(x(t^+) - x(t^-)) = \bar{B}_I \nu_t \nonumber \\ 
(b^\pm) \quad & 0 = A_{II}x(t^\pm) + B_{II}z(t^\pm) + E_{II}(t^\pm) + \bar{B}_{II}\lambda(t^\pm) \tag{8} \\ 
(c) \quad & \nu_t \in -(D + \partial\sigma_{K-F(t^+)})^{-1}(C_I x(t^+) + C_{II}z(t^+)) \nonumber \\ 
(c') \quad & K^* \ni \nu_t \perp C_I x(t^+) + C_{II}z(t^+) + D\nu_t + F(t^+) \in K \nonumber 
\end{align} 
$$
where (c) ⇔ (c'). The problem (8) (a) (b$^+$) (c') is a mixed LCCP with unknowns $x(t^+$), $z(t^+$), $\lambda(t^+$), $\nu_t$. Notice that Assumption 1 implies $x(t^+) = x(t)$ and $z(t^+) = z(t)$, however for the sake of clarity the superscript is written explicitly. Both equalities in (8) (b) are equalities of functions at the left and the right of $t$: ($b^-$) is assumed to hold automatically because the system has been integrated until $t$, ($b^+$) is imposed so that (2) (b) is satisfied after a state jump (this is necessary for Assumption 3 to hold). Notice also that the inclusion in (8) (c) is a kind of modeling assumption, mimicking usual setting for FOSwP, LCS and SOSwP [25, 58, 59, 60, 56, 79]. Equality (2) (b) has to be satisfied everywhere in time. Hence the equality

$$0 = A_{II}\dot{x}(\tau) + B_{II}\dot{z}(\tau) + \bar{B}_{II}\dot{\lambda}(\tau) + \dot{E}_{II}(\tau) \quad (9)$$

holds at all times $\tau$ where differentiability holds ($dt$-almost-everywhere if the functions are AC). At a jump time, from (8) (b$^+$):

$$0 = A_{II}(x(t^+) - x(t^-)) + B_{II}(z(t^+) - z(t^-)) \\ + \bar{B}_{II}(\lambda(t^+) - \lambda(t^-)) + E_{II}(t^+) - E_{II}(t^-), \quad (10)$$

which merely corresponds to subtracting both equalities in (8) (b). The multiplier $\lambda(t^+)$ satisfies (2) (c) on the right of $t$, so it satisfies the same inclusion as $\nu_t$ in (8) (c). The following holds true:

**Lemma 4.** We have:

$$\nu_t \in \ker(\bar{B}_{II}). \quad (11)$$

*Proof.* Consider (2) (b) with $\lambda = \nu_t\delta_t + \lambda(t)dt$, then equalling measures of same degree it follows that $0 = \bar{B}_{II}\nu_t$ (indeed (2) (b) writes $0 = (A_{II}x(t) + B_{II}z(t) + E_{II}(t))dt + \bar{B}_{II}\lambda$ in terms of measures). $\square$

Lemma 4 is similar to [66, Theorem 3.1]. Therefore (8) (c) is rewritten as:

$$\nu_t \in -(D + \partial\sigma_{K-F(t^+)})^{-1}(C_I x(t^+) + C_{II} z(t^+)) \cap \ker(\bar{B}_{II}). \quad (12)$$

From (8) (a), $x(\cdot)$ continuous at $t$ implies $\nu_t \in \ker(\bar{B}_I)$. If $E_{II}(t^+)$ is such that $A_{II}x(t^-) + B_{II}z(t^-) + \bar{B}_{II}\lambda(t^-) + E_{II}(t^+) \neq 0$, then a jump has to occur in $x(\cdot)$, in $z(\cdot)$ or in $\lambda(\cdot)$ to get $A_{II}x(t^+) + B_{II}z(t^+) + \bar{B}_{II}\lambda(t^+) + E_{II}(t^+) = 0$.

**Example 5.** To start with and motivate the analysis, let us consider the circuit example in [19, section 6.1], see Fig. 4 (right circuit), and details in [19], which writes as [19, equation (6.3)]:

$$\begin{align} (a) \quad \dot{x}_1(\tau) &= -2.12 \ 10^3 x_1(\tau) - 3 \ 10^{-3} \lambda(\tau) + 0.301 \ 10^4 u_1(\tau) \nonumber \\ (b) \quad \dot{x}_2(\tau) &= z(\tau) \nonumber \\ (c) \quad 0 &= x_2(\tau) + 10^{-4} \lambda(\tau) + 10^{-4} u_2(\tau) \nonumber \\ (d) \quad 0 \le \lambda(\tau) \perp w(\tau) &= -0.707 x_1(\tau) - z(\tau) \ge 0. \tag{13} \end{align}$$

We have: $A_I = \begin{pmatrix} -2.12 & 10^3 \\ 0 & 0 \end{pmatrix}$, $B_I = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $\bar{B}_I = \begin{pmatrix} -3 & 10^{-3} \\ 0 & 0 \end{pmatrix}$, $A_{II} = (0 \ 1)$, $E_I(\tau) = \begin{pmatrix} 0.301 & 10^4 u_1(\tau) \\ 0 & 0 \end{pmatrix}$, $B_{II} = 0$, $\bar{B}_{II} = 10^{-4}$, $E_{II}(\tau) = 10^{-4} u_2(\tau)$, $C_I = (-0.707 \ 0)$, $C_{II} =$
$-1$, $D = 0$, $F(\tau) = 0$, $u_1(\tau) = \sin(\tau)$, $u_2(\tau) = \cos(10\tau)$. The dynamics (13) fits with (7), with constant set $\Phi$, and each pair $(\bar{B}_I, C_I)$ and $(-\bar{B}_{II}, C_{II})$ has the X-property. Assumption 7 holds so that $x(\cdot)$ is continuous at $t$. This dynamics stems from a special Weierstrass canonical form for passive descriptor-variable systems [41, 28, 24], and from [28], it possesses Lipschitz continuous solutions $x_1(\cdot)$ and $x_2(\cdot)$ with uniqueness, while $\lambda(\cdot)$ is AC. Numerical simulations (see Fig. 1) with an implicit Euler scheme of order 1, show that there exists a time $t \approx 5.9s$ such that $z(t^+) \neq z(t^-)$, $w(t^+) \neq w(t^-)$, $\lambda(t^-) = 0$, while all other variables are continuous at $t$. This is a kind of impulse-free state jump, which is coherent with (12). Thus (10) reduces to $0 = 0$, while $\nu_t = 0$. The GE (12) does not provide enough information to conclude about the necessity of a jump in $z(\cdot)$, which is due to an interplay between complementarity and dynamics. A transformation is proposed in [28], which allows to rewrite (13) as:

$$
\begin{align}
(a) \quad \dot{x}_1(\tau) &= -2.12 \ 10^3 x_1(\tau) - 3 \ 10^{-3} \lambda(\tau) + 0.301 \ 10^4 u_1(\tau) \nonumber \\
(b) \quad 10^{-4} \dot{\lambda}(\tau) &= 0.707 x_1(\tau) - 10^{-4} \dot{u}_2(\tau) + w(\tau) \nonumber \\
(c) \quad x_2(\tau) &= -10^{-4} \lambda(\tau) - 10^{-4} u_2(\tau) \nonumber \\
(d) \quad z(\tau) &= -10^{-4} \dot{\lambda}(\tau) - 10^{-4} \dot{u}_2(\tau) \nonumber \\
(e) \quad 0 \le \lambda(\tau) \perp w(\tau) \quad (=-0.707 x_1(\tau) - z(\tau)) \ge 0. \tag{14}
\end{align}
$$

This is obtained by differentiating once (13) (c) and using (13) (d). In (14) the multiplier is seen as a state variable and $w(\cdot)$ is seen as a multiplier (or slack variable). Thus (14) is a LCS with state $(x_1, \lambda)$, multiplier $w$. If $\dot{\lambda}(\cdot)$ jumps at $t$ (because in (14) (b) the new multiplier $w(\cdot)$ jumps at $t$), then the jump in $z(\cdot)$ follows from (14) (d). Since $w(\tau) \in \mathcal{N}_{\mathbb{R}_+}(\lambda(\tau))$, we can rewrite equivalently (14) (a) (b) (e) as the differential inclusion:

$$
\begin{align}
\dot{x}_1(\tau) &= -2.12 \ 10^3 x_1(\tau) - 3 \ 10^{-3} \lambda(\tau) + 0.301 \ 10^4 u_1(\tau) \tag{15} \\
10^{-4} \dot{\lambda}(\tau) &\in 0.707 x_1(\tau) - 10^{-4} \dot{u}_2(\tau) - \mathcal{N}_{\mathbb{R}_+}(\lambda(\tau)), \nonumber
\end{align}
$$

or as

$$
\begin{align}
\dot{x}_1(\tau) &\in -2.12 \ 10^3 x_1(\tau) - 3 \ 10^{-3} \lambda(\tau) + 0.301 \ 10^4 u_1(\tau) \nonumber \\
&\quad - \mathcal{N}_{\mathbb{R}}(\lambda(\tau)) \tag{16} \\
10^{-4} \dot{\lambda}(\tau) &\in 0.707 x_1(\tau) - 10^{-4} \dot{u}_2(\tau) - \mathcal{N}_{\mathbb{R}_+}(\lambda(\tau)), \nonumber
\end{align}
$$

Thus (16) has a set-valued right-hand side $\mathcal{N}_{\mathbb{R} \times \mathbb{R}_+}(\lambda, \lambda)$ which defines a maximal monotonite operator. It follows from classical arguments that the solution is AC (provided $u_1$ and $u_2$ satisfy basic regularity conditions [4]). However it may happen that the selection $w(\cdot)$ inside $\mathcal{N}_{\mathbb{R}_+}(\lambda)$ jumps. How can this be calculated? Assume that $\lambda(s) > 0$, $w(s) = 0$, for all $s \in (t - \eta)$, $\eta > 0$, and $t$ is a junction time at which $\lambda(t) = 0$. Then on $s \in (t + \epsilon)$, for some $\epsilon > 0$, the contact LCP $0 \le \dot{\lambda}(s) \perp w(s) \ge 0$ holds, equivalently using (14) (b): $0 \le 0.707x_1(s) - 10^{-4}\dot{u}_2(s) + w(s) \perp w(s) \ge 0$. Consequently $w(s) = \max[0; -0.707x_1(s) + 10^{-4}\dot{u}_2(s)]$. In case $-0.707x_1(s) + 10^{-4}\dot{u}_2(s) > 0$ on $(t + \epsilon)$, then $w(\cdot)$ jumps at $t$ from $w(t^-) = 0$ to $w(t^+) > 0$, and necessarily so does $z(\cdot)$, using (14) (d) and (b): $z(t^+) = -w(t^+) - 0.707x_1(t) + 10^{-4}\dot{u}_2(t) - 10^{-4}\dot{u}_2(t) = -w(t^+) - 0.707x_1(t)$. Thus $z(t^+)$ is the solution of a minimization problem:

$$
z(t^+) = \min[0; 0.707x_1(t) - 10^{-4}\dot{u}_2(t)] - 0.707x_1(t). \quad (17)
$$

Since $t \approx 5.9s$ and $x_1(t) \approx -0.4$, we obtain: $-0.707x_1(t) + 10^{-4}\dot{u}_2(t) \approx -0.707(-0.4) - 10^{-4} \cdot 10 \sin(59) \approx 0.32 > 0$, thus $w(t^+) \approx 0.32$ (recall that $w(t^-) = 0$) and $z(t^+) = 0$, $\dot{\lambda}(t^+) \approx 0$ so that $\lambda(\cdot)$ stays null on the right of $t \approx 5.9s$ (all these approximate numerical
<figure><img src="image_1.png" /></figure>

<figure><img src="image_2.png" /><figcaption>Figure 1: Backward Euler simulation of (13) ([19, Example 1, Fig. 2, 3]).</figcaption></figure>
values fit with Fig. 1). This example shows that some subtle mechanisms can occur in DALCS which involve jumps in the algebraic state, while it is known that LCS usually undergo state jumps. In the sequel we shall focus on precise characterization of such events.

## 3.2 Jumps in $z(\cdot)$ with Continuous $x(\cdot)$

In this section problem (8) is analyzed using equivalent GEs that take the form of inclu-
sions.

### 3.2.1 The Generalized Equation (GE) at State Jump Times

**Lemma 6.** Let Assumptions 1, 2, 3 hold and $x(t^{-}) = x(t^{+})$. Then $z(t^{+})$ satisfies the GE:

$$ 
\begin{aligned}
0 \in & B_{II}z(t^{+}) - B_{II}z(t^{-}) + E_{II}(t^{+}) - E_{II}(t^{-}) - \bar{B}_{II}\lambda(t^{-}) \\
& -(A_{II}P_{I}^{-1}\bar{B}_{I} + \bar{B}_{II})(D + C_{I}P_{I}^{-1}\bar{B}_{I} + \partial\sigma_{K-C_{I}x(t^{-})-F(t^{+})})^{-1}(C_{II}z(t^{+})) \\
& -A_{II}P_{I}^{-1}\bar{B}_{I}((D + C_{I}P_{I}^{-1}\bar{B}_{I} + \partial\sigma_{K-C_{I}x(t^{-})-F(t^{+})})^{-1}(C_{II}z(t^{+}))) \cap \text{Ker}(\bar{B}_{II})).
\end{aligned} 
\quad (18) $$

Conversely, any solution $z(t^+)$ to (18) satisfies (8) for some $\nu_t$ and $\lambda(t^+)$.

*Proof.* Follows from (4) (c) and (8) and Lemma 4, inserting (a) into (c'), using (b) and (94). □

Notice that both sets $A_{II}P_I^{-1}\bar{B}_I((D+C_I P_I^{-1}\bar{B}_I+\partial\sigma_{K-C_I x(t^-)-F(t^+)})^{-1}(C_{II}z(t^+)))$ in the right-hand side of (18) do not play the same role, since the first one contains $\lambda(t^+)$ while the second one contains $\nu_t$. Let us make the following:

**Assumption 7.** The matrix $\bar{B}_{II}$ has full column rank $m$.

Hence $\text{Ker}(\bar{B}_{II}) = \{0\}$ and from Lemma 4 this assumption implies that $\nu_t = 0$ so that $x(t^+) = x(t^-)$. This is satisfied in (13). From now on we shall adopt the notation: $A_z \triangleq B_{II}$, $B_z \triangleq -(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})$, $D_z \triangleq D + C_I P_I^{-1}\bar{B}_I$, $C_z \triangleq C_{II}$, $S_z(t, x(t^-)) \triangleq K - C_{I}x(t^-) - F(t^+)$ is convex nonempty for each $t$, $x(t^-)$, $z(t^-)$, $f_z(t, z(t^-), \lambda(t^-)) \triangleq -B_{II}z(t^-) + E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-)$, so that (18) is rewritten equivalently as the GE with unknown $z(t^+)$:

$$ 0 \in A_z z(t^+) + f_z(t, z(t^-)) + B_z(D_z + \partial\sigma_{S_z(t, x(t^-))})^{-1}(C_z z(t^+)), \quad (19) $$

The third term in (19) is a Lur'e operator [16, 20]. Notice that the signs can be changed in the right-hand side of (18) since they can be changed in (2) (b). It is also noteworthy that (19) does not involve jumps in $x(\cdot)$ but only post-jump quantities.

**Example 8** (Example 5 continued). In (13) we have $A_{II} = (0 \ 1)$, $P_I = I_2$, $\bar{B}_I = (-3.10^{-3} \ 0)^{\top}$, $B_{II} = 0$, $\bar{B}_{II} = 10^{-4}$ (hence $x(t^+) = x(t^-)$), $C_I = (-0.707 \ 0)$, $C_{II} = -1$, $D = 0$, $F = 0$, and $E_{II}(\cdot)$ is time continuous, $S_z(t, x(t^-)) = \mathbb{R}_+ + 0.707x_1(t^-)$. Therefore (18) is:

$$ 0 \in (C_I \bar{B}_I + \partial\sigma_{\mathbb{R}^+ + 0.707x_1(t^-)})^{-1}(-z(t^+)) \quad (20) $$

Since $D_z = C_I \bar{B}_I > 0$, the operator in (20) is maximal monotone single-valued Lipschitz continuous [16, Proposition 1]. The LCP in (8) (c') yields: $0 \le \nu_t \perp C_I x(t^-) + C_I \bar{B}_I \nu_t -$
$z(t^+) \ge 0$. So the value of $\nu_t$ depends on $C_I x(t^-) = -0.707 x_1(t^-)$. If $x_1(t^-) < 0$ then $\nu_t = 0$ and $z(t^+) \approx 0$ solve this LCP: intuitively this is the reason why $x(\cdot)$ does not jump at $t$ ($\approx 5.9s$) in [19, section 6.1]. Assume $z(t^+) = z(t^-) \approx 0.4$, then $C_I x(t^-) - z(t^+) \approx 0.707 \times 0.4 - 0.4 \approx -0.117 < 0$ so that necessarily $\nu_t > 0$: a jump has to occur in $x(\cdot)$ at $t$, which is contradicted by the numerical data in [19, Figures 2, 3] and by Lemma 4: $z(\cdot)$ jumps at $t \approx 5.9s$ but $x(\cdot)$ is continuous. This conclusion is confirmed later, see Example 13.

### 3.2.2 Existence and Uniqueness of $z(t^+)$

Let us state conditions which guarantee existence and uniqueness of solutions to the GE in (19).

**Proposition 9.** Let Assumption 7 hold. The following assertions are true:

1. Assume that: (i) $D_z \ge 0$, (ii) $p = n_2$ and $(B_z, C_z)$ (resp. $(-B_z, C_z)$) has the X-property, (iii) $RA_z R^{-1} \ge 0$ (resp. $RA_z R^{-1} \le 0$), (iv) $\mathrm{Im}(C_z) \cap \mathrm{rint}(\mathrm{Im}(D_z + \partial\sigma_{S_z(t,x(t))})) \ne \emptyset$. Then the GE (19) has a closed convex set of solutions.

2. Let (i) $D_z = 0$, (ii) and (iv) in item 1 hold, and $RA_z R^{-1} = R^{-1}A_z^\top R \ge 0$ (resp. $RA_z R^{-1} = R^{-1}A_z^\top R < 0$) in item (iii). Then:

$$z(t^+) = R^{-1}\mathrm{proj}_{RA_z R^{-1}}[\bar{S}_z(t, x(t)); Rz(t^-) - RA_z^{-1}(E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-))]. \quad (21)$$

$$\text{with } \bar{S}_z(t, x(t)) = \{\zeta \mid C_z R^{-1} \zeta \in S_z(t, x(t))\}.$$

3. Let conditions (i)-(iv) of item 1 hold, with $D_z = D_z^\top$, and $K = \{w \in \mathbb{R}^m \mid Hw \ge 0\}$ for some matrix $H \in \mathbb{R}^{m \times m}$. Consider the set:

$$\mathcal{S} \stackrel{\Delta}{=} (\mathrm{dom}(g_t))_{\infty} \cap \mathrm{Ker}(RA_z R^{-1} \cap R^{-1}A_z R) \cap \{w \in \mathbb{R}^{n_2} \mid R^{-1}A_z Rw \in \Gamma^*\}, \quad (22)$$

where $g_t(\cdot)$, $(\mathrm{dom}(g_t))_{\infty}$ and the convex polyhedral cone $\Gamma \subseteq \mathbb{R}^{n_2}$ are given in the proof. Then:

(a) If $\mathcal{S} = \{0\}$ the GE (19) has at least one solution.

(b) If $\mathcal{S} \ne \{0\}$ the GE (19) has at least one solution if there exists $\xi_0 \in \mathrm{dom}(g_t)$ such that:

$$\langle Rf_z(t, z(t^-)) - RA_z R^{-1}\xi_0, v \rangle > 0, \text{ for all } v \in \Gamma, v \ne 0. \quad (23)$$

(c) If $\mathrm{rank}(A_z) = n_2$ the solution is unique.

4. Assume that: (i) K is polyhedral, (ii) $p = n_2$, $(B_z, C_z, D_z)$ has the XD-property and $D_z = D_z^\top \ge 0$, (iii) for all $t \ge 0$, $\mathrm{Im}(C_z) \cap \mathrm{rint}(\mathrm{Im}(D_z) + S_z(t, x(t))) \ne \emptyset$, (iv) for all $t \ge 0$ and each $v \in \mathrm{Im}(C_z) \cap (\mathrm{Im}(D_z) + S_z(t, x(t)))$, it holds that $\mathrm{Im}(D_z + D_z^\top) \cap (D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(v) \ne \emptyset$, (v) $F(\cdot)$ is AC. Let $\xi = Rz(t^+)$. Then the GE (19) is equivalent to:

$$0 \in R(A_z R^{-1}\xi + f_z(t, z(t^-)) - (R^{-1}C_z^\top - RB_z)\lambda^{im}(t, \xi) + \mathcal{F}(t, \xi), \quad (24)$$

where: $\lambda^{im}(t, \xi) = \mathrm{proj}[\mathrm{Im}(D_z+D_z^\top), \lambda] = [\Phi(t, C_z R^{-1}\xi)]^0$, for each $\lambda \in \Phi(t, C_z R^{-1}\xi) = (D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z R^{-1}\xi)$, and $\mathcal{F}(t, \xi) = R^{-1}C_z^\top \Phi(t, C_z R^{-1}\xi)$. (v) Assume there exists $M \ge 0$ such that $RA_z R^{-1} - M \ge 0$ and the single-valued mapping:
M : ξ ↦ − (R⁻¹Cz − RBz)λim(t, ξ) + Mξ is (maximal) monotone for each t. Then (24) takes the form (1) for each t, with A(η) ≜ R(AzR⁻¹ξ + fz(t, z(t⁻)) − (R⁻¹Cz − RBz)λim(t, ξ), B(η) ≜ ℘(t, η). If A(·) is strongly monotone, the GE (19) has a unique solution.

Proof. 1. From (i) it follows that the mapping (Dz+∂σS_z(t,x(t)))⁻¹(·) is maximal monotone for each t [5, Corollary 24.4, Proposition 20.22]. Let ζ = Rz(t⁺), (Bz, Cz) has the X-property and RAzR⁻¹ ≍ 0. Then (18) is equivalent to:

$$
\begin{align}
0 \in \mathrm{RA}_z R^{-1}\zeta + R^{-1}XB_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z R^{-1}\zeta) &+ Rf_z(t, x(t), z(t^{-})) \tag{25} \\
&= \mathrm{RA}_z R^{-1}\zeta + R^{-1}C_z^{\top}(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z R^{-1}\zeta) + Rf_z(t, x(t), z(t^{-})) \nonumber
\end{align}
$$

Using [73, Theorem 12.43] and the condition (iv), it is inferred that the mapping ζ ↦
R⁻¹C_zᵀ(D_z + ∂σ_{S_z(t,z(t⁻),x(t))})⁻¹(C_z R⁻¹ζ)) is maximal monotone. Therefore the map-
ping in the right-hand side of (25) is maximal monotone. The assertion follows from [5,
Proposition 23.39]. The case (−B_z, C_z) has the X-property follows same lines.

2. The result is a consequence of (25) and (95).

3. The proof of (a) (b) and (c) is direct from Proposition 34, but it relies on the various sets in (22), details on which are given now. Let us first rewrite (25) under the form (91), with **M** = RA<sub>z</sub>R<sup>-1</sup>, **q** = Rf(t, z(t<sup>-</sup>)). We have (D<sub>z</sub> + ∂σ<sub>S<sub>z</sub>(t,x(t))</sub>)(x) = ∂f<sub>t</sub>(x), with f<sub>t</sub>(x) = 1/2 x<sup>&top;</sup> D<sub>z</sub>x + σ<sub>S<sub>z</sub>(t,x(t))</sub>(x) [72, Theorem 23.8]. The function f<sub>t</sub>(·) is proper convex LSC, and (∂f<sub>t</sub>)<sup>-1</sup>(·) = ∂f<sub>t</sub><sup>*</sup>(·). Then R<sup>-1</sup>C<sub>z</sub>∂f<sub>t</sub><sup>*</sup>(C<sub>z</sub>R<sup>-1</sup>)(·) = ∂g<sub>t</sub>(·), with g<sub>t</sub>(·) = f<sub>t</sub><sup>*</sup> ∘ C<sub>z</sub>R<sup>-1</sup>(·), and since rint(S<sub>z</sub>(t, x(t)) ≠ , hence rint dom(∂σ<sub>S<sub>z</sub>(t,x(t))</sub>) ≠  since S<sub>z</sub>(t, x(t)) is convex valued, and [72, Theorem 23.9] applies. From [72, Theorem 5.7] g<sub>t</sub>(·) is convex. Therefore for each fixed t, (25) is equivalently rewritten as:

$$
\langle \mathbf{M}\zeta + \mathbf{q}, v - u \rangle + g_t(v) - g_t(u) \ge 0, \text{ for all } v \in \mathbb{R}^m. \quad (26)
$$

which fits with (91). It is necessary to calculate (dom(g_t))_∞ and (dom(g_t,∞))^*. Let us first calculate dom(g_t) = {w ∈ R^m | f_t ∘ C_{II}R^{-1}(w) ≠ ∅} = {w ∈ R^m | C_z R^{-1}w ∈ dom(f_t^*)}. Using [3, Theorem 2, Chapter 10, section 3]: dom(∂f_t^*) ⊆ dom(f_t^*) ⊆ dom(∂f_t^*). Also dom(∂f_t^*) = Im(∂f_t). This range can be characterized using Proposition 36, from which it is inferred that dom(∂f_t^*) = S_z(t, x(t)) + Im(D_z) = dom(f_t^*), and dom(g_t) = {w ∈ R^m | C_z R^{-1}w ∈ S_z(t, x(t)) + Im(D_z)} is closed convex and nonempty from (iv). In a second step let us calculate (dom(g_t))_∞, the recession cone of dom(g_t) = {w ∈ R^m | C_z R^{-1}w ∈ S_z(t, x(t)) + Im(D_z)}.
Three cases are considered:

$$
(\alpha) D_z = 0: \text{ then } \mathrm{dom}(g_t) = \{w \in \mathbb{R}^m \mid H(C_z R^{-1}w + C_I x(t) + F(t^+)) \ge 0\} \text{ and} \\
(\mathrm{dom}(g_t))_\infty = \{w \in \mathbb{R}^m \mid HC_z R^{-1}w \ge 0\}.
$$

$$
(\beta) D_z \succcurlyeq 0: \text{ then } \mathrm{dom}(g_t) = \{w \in \mathbb{R}^m \mid C_z R^{-1} w \in \mathbb{R}^m\} = \mathbb{R}^m, (\mathrm{dom}(g_t))_\infty = \mathbb{R}^m.
$$

$$
(\gamma) D_z \succcurlyeq 0: \text{ then } (\mathrm{dom}(g_t))_{\infty} = \{w \in \mathbb{R}^m \mid C_z R^{-1} w \in K + \mathrm{Im}(D_z)\}.
$$

Items (α) and (β) are a direct consequence of the assumptions and recession cones prop-
erties for polyhedral sets [44]. This provides the first set needed to calculate ℘ in (22).
Let us now compute the set in (92). First g<sub>t,∞</sub>(·) = (f<sub>t</sub><sup>*</sup> ∘ C<sub>z</sub>R<sup>-1</sup>)<sub>∞</sub>(·) has to be cal-
culated, where g<sub>t</sub>(·) = f<sub>t</sub><sup>*</sup> ∘ C<sub>z</sub>R<sup>-1</sup>(·). The developments are similar to those in the
proof of [13, Lemma 2, Corollary 1], see [13, Appendix D], and are therefore omitted
here. Let us merely summarize these results. Assume that D<sub>z</sub> is unitarily similar to
D<sub>z</sub> = (D̄<sub>z</sub> 0), with unitary matrix U, and D̄<sub>z</sub> ≍ 0. Suppose that rank(H) = m. Let
$\Gamma \stackrel{\Delta}{=} \{\xi \in \mathbb{R}^{n_2} | UHC_z R^{-1} \xi \ge 0\}$. Then $g_{t,\infty}(\xi) = \Psi_{\Gamma}(\cdot)$, and $(\mathrm{dom}(g_{t,\infty}))^* = \Gamma^* = \{\xi \in \mathbb{R}^{n_2} | w = \sum \alpha_i (UHC_z R^{-1})_{i\bullet}^{\top}, \alpha_i \ge 0\}$ [44, Example 15]. Therefore (a) (resp. (b)) follows from Proposition 34 (a) (resp. (b)). $\square$

4. Using Proposition 36 and (i), we have $\mathrm{Im}(D_z + \partial\sigma_{S_z(t,x(t))}) = \mathrm{Im}(D_z) + S_z(t, x(t))$. Then relying on (ii)-(iv) the proof of (24) follows developments in [79, section 3.2], see in particular [79, Lemmata 3, 4, equations (21), (22)]. (v) The last result follows from [5, Corollary 23.35], since the sum of a strongly maximal monotone and a maximal monotone operators is strongly maximal monotone. $\square$

The case when $A_z$ is not full rank can be analysed with item 3 in Proposition 9. If $D_z \neq D_z^\top$ and $K = \mathbb{R}_+^m$, we can use (98) to characterize $\mathrm{Im}(D_z + \partial\sigma_{S_z(t,x(t))})$ in item 4 (iii) and (iv). See Example 19 for an application of (24) in a particular case. Notice that if $(B_z, C_z)$ has the X-property, the term premultiplying $\lambda^{im}(t, \xi)$ in (24) vanishes. Then if $A_z > 0$ the GE (24) has a unique solution for any $f_z(t, z(t^{-}))$. The maximal monotonicity of $\mathcal{F}(\cdot)$ can also be studied using [17, Proposition 5.5, section 6.1]. Let us study the maximal monotonicity of $\mathcal{M}(\cdot)$ in item 4.

**Lemma 10.** Consider item 4 in Proposition 9. Then there exists $M \ge 0$ such that the operator $\mathcal{M}(\cdot)$ is maximal monotone.

*Proof.* It follows from [79, Lemma 3] that $\lambda^{im}(t, \xi)$ is Lipschitz continuous in $\xi$ for each $t$, hence from Rademacher's Theorem its Jacobian $\frac{\partial\lambda^{im}}{\partial\xi}(t, \xi)$ is bounded almost everywhere, and the mapping is hypomonotone (see also [17, Remark 5.3]). From the Lemma's assumption on piecewise linearity, the result follows. $\square$

If $K = \mathbb{R}_+^m$ and $D_z$ is a P-matrix then the LCCP (which becomes the LCP: $0 \le \lambda \perp D_z\lambda + C_I x(t^-) + F(t^+) + C_{II} R^{-1}\xi \ge 0$) solutions are known to be piecewise-linear functions of $C_I x(t^-) + F(t^+) + C_{II} R^{-1}\xi$ [38]. Thus Lipschitz continuity of $\lambda^{im}$ holds. The next example (see also Example 19) shows that Proposition 9 and (19) do not furnish in general a unique admissible solution $z(t^+)$.

**Example 11** (Example 5 continued). Consider (20) in the light of Proposition 9. We have $A_z = B_{II} = 0$, $D_z = C_I \bar{B}_I = 0.707 \times 3.10^{-3} > 0$, $B_z = -10^{-4}$, $C_z = -1$, $X = 10^4$, $R = 10^2$, $S_z(t, x(t)) = \mathbb{R}_+ + 0.707x_1(t)$, $x_1(t) \approx -0.4$, $f_z(t, z(t^{-})) = 0$, $p = n_2 = m = 1$, $n_1 = 2$. Let us apply item 3: $H = 1$, $U = 1$, $R^{-1} = 10^{-2}$, and $\Gamma^* = \{w | w = -\alpha, \alpha \ge 0\} = \{0\}$, $(\mathrm{dom}(g_t))_\infty = \mathbb{R}$ from (β) in the proof of item 3. Hence in (22): $\mathcal{S} = \mathbb{R} \cap \mathbb{R} \cap \mathbb{R}$. We infer that neither item 3 (a) nor item 3 (b) apply. Clearly $M$ as in item 4 does not exist since $A_z = 0$ (though Lemma 10 applies). The GE in (24) boils down to (20): $0 = (2.121 \times 10^{-3} + \partial\sigma_{S_z(t,x_1(t))})^{-1}(-z(t^+))$ since the right-hand side is single-valued. Equivalently, $z(t^+) \in \partial\sigma_{S_z(t,x_1(t))}(0) = (-\infty, 0.707 \times 0.4]$, which is not sufficient to calculate $z(t^+)$ uniquely. Calculations in Example 5 show that $z(t^+)$ satisfies a minimization condition. This is analysed in Examples 8 and 13.

*Proof of existence of solutions to (19) can also be tackled using [5, Corollaries 24.22, 24.23]. It relies on 3\* monotonicity [5] of both operators in (19). It requires similar conditions to those of Proposition 9. The proposition relies on $p = n_2$: there are as many algebraic states as equality constraints, $B_{II}$ is square. Notice that [40, Corollary 2.2.5] cannot be used here because it requires compactness of $S_z(t, x(t))$. The next corollary analyses (19) when $z$ is continuous at $t$.*
**Corollary 12.** Let Assumption 7 hold. Assume that $D_z \succ 0$. Then:

1. If $E_{II}(t^+) = E_{II}(t^-)$ and $\bar{B}_{II}\lambda(t^-) = 0$, then $z(\cdot)$ is continuous at $t$ only if
$$ \mathrm{Ker}(B_z) \cap \mathrm{SOL}(\mathrm{LCCP}(D_z, K, C_z z(t^-) + C_I x(t) + F(t^+))) \neq \emptyset. \quad (27) $$

2. Let $D_z = D_z^\top \succ 0$, $z(t^+) = z(t^-) = z(t)$, and $E_{II}(t^+) = E_{II}(t^-)$. The GE (19) (equivalently (30)) has a unique solution if and only if $\mathrm{proj}_{D_z}[K^*; -D_z^{-1}(C_z z(t^-) + C_I x(t) + F(t^+))] \in \mathrm{Ker}(B_z)$. This holds if and only if $K^* \subseteq \mathrm{Ker}(B_z)$.

3. Let $D_z = 0$, $z(t^+) = z(t^-) = z(t)$ and $E_{II}(t^+) = E_{II}(t^-)$. Then any $z(t) \in \{w \in \mathbb{R}^{n_2} | C_z w \in S_z(t, x(t))\}$ is a solution to (30).

4. Let $D_z = 0$, $z(t^+) = z(t^-) = z(t)$, $(B_z, C_z)$ has the X-property, and $\mathrm{rint}(S_z(t, x(t))) \cap \mathrm{Im}(C_z) \neq \emptyset$. Then (30) has a solution if and only if

$$ z(t) \in \partial \sigma_{\tilde{S}_z(t,x(t))}(X(E_{II}(t^-) - E_{II}(t^+))), \quad (28) $$

where $\tilde{S}_z(t, x(t)) = \{w \in \mathbb{R}^{n_2} | C_z w \in S_z(t, x(t))\}$. This inclusion holds only if $z(t) \in \tilde{S}_z(t, x(t))$.

5. Let all the assumptions stated in Proposition 9 item 4 hold. Then: (i) $z(\cdot)$ is continuous at $t$ only if

$$ -R(E_{II}(t^+) - E_{II}(t^-)) \in -(R^{-1}C_z^\top - RB_z)[\Phi(t, C_z R^{-1}\xi)]^0 + \mathcal{F}(t, \xi), \quad (29) $$

is solvable, with $\Phi(t, v)$ and $\mathcal{F}(t, \xi)$ as in Proposition 9 item 4, $\xi = Rz(t) = Rz(t^-)$. (ii) The inclusion (29) always holds if $0 \in \Phi(t, C_z R^{-1}\xi)$ and $E_{II}(\cdot)$ is continuous at $t$. (iii) The inclusion (29) is solvable only if $-X((E_{II}(t^+) - E_{II}(t^-)) + (C_z^\top - XB_z)[\Phi(t, C_z R^{-1}\xi)]^0 \in C_z^\top(\mathrm{Im}(D_z) + S_z(t, x(t)))$. (iv) If in addition the pair $(B_z, C_z)$ has the X-property, the latter reduces to $-(E_{II}(t^+) - E_{II}(t^-)) \in B_z(\mathrm{Im}(D_z) + S_z(t, x(t)))$.

**Proof.** First notice that under stated conditions like continuity of $z(\cdot)$, GE (19) is equivalent to:

$$ 0 \in B_z(D_z + \partial \sigma_{S_z(t,x(t))})^{-1}(C_z z(t)) + E_{II}(t^+) - E_{II}(t^-) \quad (30) $$

Then continuity of $z(\cdot)$ at $t$ is equivalent to existence of a solution $z(t)$ to (30).

1. The first part follows from (30), which is equivalent to $B_z\xi = 0$, $\xi \in (D_z+\partial\sigma_{S_z(t,x(t))})^{-1}(C_z z(t))$. We have:

$$ \begin{aligned} \xi \in (D_z + \partial \sigma_{S_z(t,x(t))})^{-1}(C_z z(t)) &\Leftrightarrow C_z z(t) \in D_z \xi + \partial \sigma_{S_z(t,x(t))}(\xi) \\ &\Leftrightarrow C_z z(t) - D_z \xi \in \partial \sigma_{S_z(t,x(t))}(\xi) \Leftrightarrow \xi \in \mathcal{N}_{S_z(t,x(t))}(C_z z(t) - D_z \xi) \\ &\Leftrightarrow \xi \in \mathcal{N}_K(C_z z(t) - D_z \xi + C_I x(t) + F(t^+)) \\ &\Leftrightarrow K^* \ni -\xi \perp C_z z(t) - D_z \xi + C_I x(t) + F(t^+) \in K. \end{aligned} \quad (31) $$

where closedness and convexity of $S_z(t, x(t))$ and (94), (95) were used. The result is proved.

2. The result follows from $\xi \in \mathcal{N}_K(C_z z(t) - D_z \xi + C_I x(t) + F(t^+)) \Leftrightarrow C_z z(t) - D_z \xi + C_I x(t) + F(t^+) \in \mathcal{N}_{-K^*}(\xi)$ [73, Example 11.4] and using (94) and (95). The last assertion is straightforward.

3. Then (30) is $0 \in B_z \mathcal{N}_{S_z(t,x(t))}(C_z z(t))$. For all $z(t)$ in the set $C_z^{-1}(S_z(t, x(t)))$, the normal cone is nonempty and thus it contains $\{0\}$.
4. Then (30) is equivalent to

$$X(E_{II}(t^{-}) - E_{II}(t^{+})) \in C_{z}^{\top} \mathcal{N}_{\tilde{S}_{z}(t,x(t^{-}))}(C_{z}z(t)) = \mathcal{N}_{\tilde{S}_{z}(t,x(t^{-}))}(z(t)),$$

using the Chain Rule [72, Theorem 23.9] and given the item's assumptions. Inverting the inclusion the first result is proved. Using mapping inversion we get: $\mathrm{Im}(\partial\sigma_{\tilde{S}_z(t,x(t^{-}))}) = \mathrm{dom}(\mathcal{N}_{\tilde{S}_z(t,x(t^{-}))}) = \tilde{S}_z(t, x(t^{-})), $ which ends the proof.

5. (i) Notice that (29) is equivalent to (19). Then same lines as in item 4 of Proposition 9 to obtain (24) are followed. (ii) . (iii) Follows from Proposition 36. (iv) Immediate from Definition 1. $\square$

Notice that [2, Proposition 3] can be used to guarantee that the Lur'e operator is single-valued, which restricts conditions of solvability for (30). The next example demonstrates that if (30) is not satisfied then $z(\cdot)$ cannot be continuous at $t$.

**Example 13 (Example 5 continued).** Calculations with the numerical values taken from Fig. 5 [19, Fig. 2, 3], i.e., $\lambda(t^{-}) = 0$, $x_1(t^{-}) \approx -\frac{7}{9}$ and $z(t^{-}) \approx \frac{2}{3}$ at $t \approx 5.9$s, yield $C_z z(t^{-}) + C_I x(t) \approx -0.1167 \notin K = \mathbb{R}_+$: conditions in Corollary 12 item 2 are not satisfied, since $B_z = -10^{-4}$ and $K^* = \mathbb{R}_+ \not\subset \mathrm{Ker}(B_z) = \{0\}$. Also

$$\mathrm{Ker}(B_z) \cap \mathrm{SOL}(\mathrm{LCCP}(D_z, K, C_I x(t) + F(t^+))) = \{0\} \cap \mathrm{SOL}(\mathrm{LCP}(2.1 \times 10^{-3}, C_I x(t) + C_z z(t^{-}))),$$

and since $C_I x(t) + C_z z(t^{-}) \approx (-0.707 \times \frac{-7}{9}) - \frac{2}{3} \approx -0.1167 < 0$, we have that $\mathrm{SOL}(\mathrm{LCP}(2.1 \times 10^{-3}, C_I x(t) + C_z z(t^{-}))) > 0$ so that the necessary condition (27) is not satisfied: the algebraic state variable $z(\cdot)$ cannot be continuous at $t \approx 5.9$s in this example. This occurs while $E_{II}(\cdot)$ and $F(\cdot)$ are both continuous, see (13). But, as explained in Example 5, $\lambda(t^{+}) \neq \lambda(t^{-})$ and $w(t^{+}) \neq w(t^{-})$ at $t \approx 5.9$s. See Example 19 for an application of item 2 of Corollary 12.

Let us reexamine Corollary 12 using the assumptions in Proposition 9, in terms of the minimization of a function $h_t(\cdot)$. This may be of practical importance for calculations.

**Corollary 14.** Let Assumption 7 hold. Let $D_z = D_z^{\top} \succ 0$, and (i)-(iv) in item 1 Proposition 9 hold.

1. Then $z(\cdot)$ is continuous at $t$ only if $z(t)$ minimizes the proper convex LSC function $h_t: \zeta \mapsto (E_{II}(t^{+}) - E_{II}(t^{-}))^{\top}\zeta + g_t(\zeta)$, with

$$g_t(\zeta) = \inf_{w \in S_z(t,x(t))} \begin{cases} \frac{1}{2}(C_z R^{-1}\zeta - w)^{\top} D_z^{\dagger} (C_I R^{-1}\zeta - w) & \text{if } C_z R^{-1}\zeta - w \in \mathrm{Im}(D_z) \\ +\infty & \text{if } C_z R^{-1}\zeta - w \notin \mathrm{Im}(D_z), \end{cases} \quad (32)$$

at $z(t^{-})$.

2. Let $D_z = 0$, then $h_t(\cdot)$ attains a minimum at $\zeta^*$ if and only if

$$\zeta^* = \arg\min_{w \in \tilde{S}_z(t,x(t))} (RE_{II}(t^+) - RE_{II}(t^-))^{\top} w, \quad (33)$$

$$\text{with } \tilde{S}_z(t, x(t)) = \{\zeta \in \mathbb{R}^{n_2} | C_z R^{-1} \zeta \in S_z(t, x(t))\}.$$
3. Let $D_z \succ 0$, then $h_t(\cdot)$ is single-valued, Lipschitz continuous, proper and convex, with

$$g_t(\cdot) = \operatorname{proj}_{D_z^{-1}}[S_z(t, x(t)); C_z R^{-1}\cdot]. \quad (34)$$

It attains its minimum at $\zeta^*$ if and only if $E_{II}(t^-) - E_{II}(t^+) \in \partial p'_t(\zeta^*)$, where $p_t(\cdot)$ is proper convex continuous, its derivative $p'_t(\cdot)$ is single-valued monotone Lipschitz continuous convex.

**Proof.** 1. The form of $h_t(\cdot)$ follows from item 3 in Proposition 9. Using item 1.(ii) in this proposition, the GE in (30) is rewritten equivalently as $0 \in R^{-1}C_z(D+\partial\sigma_{S_z(t,x(t))}^{-1}(C_zR^{-1}\zeta)+E_{II}(t^+) - E_{II}(t^-)$. Then we have $g_t(\cdot) = f_t^* \circ C_z R^{-1}(\cdot)$ (see item 3 in the proof of Proposition 9), and (30) is equivalent to $0 \in \partial h_t(\zeta)$, since $\zeta = Rz(t^+) = z(t)$ and $h_t(\cdot)$ is proper convex LSC. Let us calculate $f_t^*(\cdot)$. From [50, Theorem 2.3.1, p.227]:

$$ \begin{aligned} f_t^*(y) &= \left(\frac{1}{2}x^\top D_z x + \sigma_{S_z(t,x(t))}(x)\right)^*(y) = \operatorname{cl}\left(\frac{1}{2}(x^\top D_z x)^*(y) \vee^+ \sigma_{S_z(t,x(t))}^*(y)\right) \\ &= \left(\frac{1}{2}(x^\top D_z x)^*(y) \vee^+ \sigma_{S_z(t,x(t))}^*(y)\right) = \inf_{w \in \mathbb{R}^m} \left(\Psi_{S_z(t,x(t))}(y) + q(y-w)\right) \quad (35) \\ &= \inf_{w \in S_z(t,x(t))} q(y-w), \end{aligned} $$

where $\vee^+$ is the infimal convolution, it was used that both functions are closed since they are convex proper LSC, and [50, Definition 2.3.1, p.92]. Using [50, Example 1.1.4, p.212-213]:

$$ q(w-y) = \begin{cases} \frac{1}{2}(y-w)^{\top} D_z^{\dagger}(y-w) & \text{if } y-w \in \operatorname{Im}(D_z) \\ +\infty & \text{if } y-w \notin \operatorname{Im}(D_z) \end{cases} \quad (36) $$

It follows that $g_t(\cdot)$ is given in (32). From Corollary 12 it is inferred that $0 \in \partial(h_t \circ R)(z(t))$, which ends the proof of the first result.

2. Let $D_z = 0$, and the conditions of Proposition 9 hold, then (30) boils down to:

$$ \begin{aligned} & 0 \in B_z \quad \mathcal{N}_{\tilde{S}_z(t,x(t))}(C_z z(t)) + E_{II}(t^+) - E_{II}(t^-) \\ \Leftrightarrow & 0 \in R^{-1} C_z^\top \mathcal{N}_{\tilde{S}_z(t,x(t))}(C_z R^{-1}\zeta) + RE_{II}(t^+) - RE_{II}(t^-) \\ \Leftrightarrow & 0 \in \tilde{\mathcal{N}}_{\tilde{S}_z(t,x(t))}(\zeta) + RE_{II}(t^+) - RE_{II}(t^-) \\ &= \partial \left( \Psi_{\tilde{S}_z(t,x(t))}(\zeta) + (RE_{II}(t^+) - RE_{II}(t^-))^{\top} \zeta \right) = \partial h_t(\zeta), \end{aligned} \quad (37) $$

where $\tilde{S}_z(t, x(t)) \subseteq \mathbb{R}^{n_2}$ is nonempty by assumption (iv) in Proposition 9 and use is made of [5, Corollary 24.4]. Notice that (37) also follows from (32). It is inferred that $h_t(\cdot)$ attains a minimum at $\zeta$ if and only if $\zeta = \arg\min_{w \in \tilde{S}_z(t,x(t))}(RE_{II}(t^+) - RE_{II}(t^-))^{\top}w$ (this is a conic programming problem).

3. Let $D_z \succ 0$ (not necessarily symmetric), then the map $(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(\cdot)$ is single-valued, Lipschitz continuous, see Proposition 35, and maximal monotone. Then $z(t)$ is continuous at $t$ only if $B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z z(t)) = E_{II}(t^+) - E_{II}(t^-)$. It is deduced from (32) that

$$ \begin{aligned} g_t(\zeta) &= \inf_{w \in S_z(t,x(t))} \frac{1}{2}(C_z R^{-1}\zeta - w)^{\top} D_z^{-1} (C_z R^{-1}\zeta - w) \\ &= \operatorname{proj}_{D_z^{-1}}[S_z(t, x(t)); C_z R^{-1}\zeta], \end{aligned} \quad (38) $$

which is single-valued, Lipschitz continuous and maximal monotone since $S_z(t, x(t))$ is closed convex [73, Example 2.25, Corollary 12.20]. We also know that $g_t(\cdot)$ is convex since $g_t(\cdot) = f_t^* \circ CR^{-1}(\cdot)$ (see the proof of Proposition 9 item 3). Thus it is even
maximal cyclically monotone [73, p.548], and consequently there exists a closed proper convex LSC function $p_t(\cdot)$ such that $g_t(\cdot) = \partial p_t(\cdot)$ [72, Theorem 24.9]. Therefore $h_t(\zeta) = (E_{II}(t^+) - E_{II}(t^-))^\top\zeta + \partial p_t(\zeta)$. From the Lipschitz continuity of $g_t(\cdot)$, it is deduced that $\partial p_t(\cdot)$ is single-valued monotone Lipschitz continuous, and we can denote it as $p'_t(\cdot)$. It is inferred that $h_t(\zeta) = (E_{II}(t^+) - E_{II}(t^-))^\top\zeta + p'_t(\zeta)$, is a Lipschitz continuous proper monotone convex function which possesses a minimum at $\zeta^*$ if only if $0 \in \partial h_t(\zeta^*) = E_{II}(t^+) - E_{II}(t^-) + \partial p'_t(\zeta^*)$. $\square$

**Example 15.** Let us illustrate item 3, with $K = \mathbb{R}_+$, $m = p = n_2 = n_1 = 1$, $B_z = C_z = D_z = X = R = 1$, $S_z(t, x(t)) = \mathbb{R}_+ - C_{II}x(t) - F(t^+)$. The functions $g_t(\cdot)$, $p_t(\cdot)$ and $\partial p'_t(\cdot)$ are depicted in Figure 2. It is inferred that if $E_{II}(t^-) - E_{II}(t^+) \notin [0, 1]$, then $0 \in \partial h_t(\zeta^*)$ has no solution $\zeta^*$, hence (30) has no solution, and $z(t^+) \neq z(t^-)$.

In the course of integration the value of $z(t^-)$ is given, so one has to check if there is a minimum at $\zeta^* = Rz(t^-)$. For instance, if $z(t^-)$ is such that $\zeta^* > a$, then necessarily $E_{II}(t^+) - E_{II}(t^-) = 1$. If $z(t^-)$ is such that $\zeta^* < a$, then necessarily $E_{II}(t^+) - E_{II}(t^-) = 0$. Consider once again (13), where $B_z = -10^{-4}$, $D_z = 0.707 \times 3.10^{-3}$, $C_z = -1$, $X = 10^4$, $R = 10^2$, $z(t^-) \approx 0.33$, $\zeta^* \approx 33$, $x_1(t) \approx -\frac{7}{9}$, $C_I x(t) = -0.707x_1(t) \approx 0.5498$. $F(\cdot) = 0$ (numerical data are taken from [19, Figure 2]). Then $g_t(\zeta) = \text{proj}_{D_z^{-1}}[\mathbb{R}_+ - 0.318; -10^{-2}\zeta]$. In this example $E_{II}(\cdot) = 10^4 u_2(\cdot)$ is continuous. It is concluded that the GE in (30) cannot be satisfied, which is in agreement with the jump in $z(\cdot)$ at the considered time instant.

<figure><img src="image_3.png" /><figcaption>Figure 2: (a) $g_t(\cdot)$, (b) $p_t(\cdot)$, (c) $\partial p'_t(\cdot)$ (the set-valued step function).</figcaption></figure>

Let us examine (18) (or (19)) when $C_{II} = 0$ (i.e., the algebraic state variable does not appear in the complementarity constraint). In such a case, a condition like in Proposition 9 item 1 (ii), cannot hold when $B_z \neq 0$, since it implies $XB_z = 0 \Leftrightarrow B_{z,\bullet,j} \in \text{Ker}(X) \Leftrightarrow B_z = 0$. The next lemma examines this case.

**Lemma 16.** Let Assumption 7 hold. Consider the GE in (19) with $C_{II} = 0$ and $D_z \succ 0$. Then $z(t^+)$ satisfies

$$A_z z(t^+) \in -f_z(t, z(t^-)) - B_z(D_z + \partial \sigma_{S_z(t,x(t))})^{-1}(0) \quad (39)$$

1. Assume that (i) rank($A_z$) = $n_2$, $0 \in \text{Im}(D_z) + \text{int}(S_z(t, x(t)))$, (ii) $K_\infty \cap \text{Ker}(D_z + D_z^\top) \cap \{w \in \mathbb{R}^m \mid D_z w \in (K_\infty)^*\} = \{0\}$, (iii) $\text{Im}(D_z + D_z^\top) \subset \text{Ker}(B_z)$; or assume that (i) holds with (iv) $D_z \succ 0$. Then $z(t^+)$ is uniquely defined as $z(t^+) = (A_z^\top A_z)^{-1} A_z^\top (-f_z(t, z(t^-)) - B_z(D_z + \partial \sigma_{S_z(t,x(t))})^{-1}(0))$.
2. Assume that $D_z = D_z^\top$, $K = \mathbb{R}_+^m$, then the set $(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0)$ is nonempty if and only if $C_I x(t) + F(t^+) \in Q_{D_z}^*$, where $Q_{D_z} = \{\xi \in \mathbb{R}^m | 0 \le \xi \perp D_z \xi \ge 0\}$.

Proof. 1. (i) and Proposition 36 guarantee that the right-hand side of (39) is nonempty. Indeed $\{0\} \in \text{dom}(D_z+\partial\sigma_{S_z(t,x(t))})^{-1} = \text{Im}(D_z+\partial\sigma_{S_z(t,x(t))}) \simeq S_z(t, x(t)) + \text{Im}(D_z)$. From (19) the inclusion (39) is obtained. From (ii) and (iii) and using [16, Proposition 3] it follows that the mapping $B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(\cdot)$ is single-valued, Lipschitz continuous, with domain $\mathbb{R}^{n_2}$. Same conclusions hold with (i) and (iv) [16, Proposition 1]. Hence the $\in$ in (39) can be replaced by equality. Using (i) the result follows from [6, Proposition 8.1.9]. 2. We have $\xi \in (D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0) \Leftrightarrow 0 \in D_z\xi + \partial\sigma_{S_z(t,x(t))}(\xi) \Leftrightarrow 0 \le -\xi \perp D_z(-\xi) + C_I x(t^-) + F(t^+) \ge 0$. The result follows from [38, Definition 2.2.15, Corollaries 3.8.10 and 3.8.12, Theorem 3.8.6, Exercise 3.12.1]: since $D_z$ is a copositive plus symmetric matrix, there is equivalence between the LCP solvability and the inclusion in item 2, see Appendix B. $\square$

Using Proposition 36: if $K$ is polyhedral, then the interior in item 1 (i) can be replaced by $S_z(t, x(t))$. Let $C_{II} = 0$, $\text{rank}(B_z) = m$, and define $\zeta = -B_z^\top(A_z z(t^+) + f_x(t, z(t^-))$, then using (19) we obtain:

$$0 \in D_z \zeta + \partial \sigma_{S_z(t,x(t))}(\zeta) \qquad (40)$$

Thus once $\zeta$ is known, calculation of $z(t^+)$ boils down to solving a linear problem (this does not apply to Example 11 where $A_z = 0$). Clearly (40) has a nonempty set of solutions only if $\text{Im}(D_z) \cap S_z(t, x(t)) \neq \emptyset$. Proposition 34 applies to (40) with $M = D_z$, $q = 0$, $\varphi(\cdot) = \sigma_{S_z(t,x(t))}(\cdot)$, as shown in the next proposition.

**Proposition 17.** Let Assumption 7 hold. Let $C_{II} = 0$ and $\text{rank}(B_z) = m$. Consider the GE in (40). Then:

1. Let $D_z \succ 0$, then (40) has a unique solution $\zeta$. Let in addition $D_z = D_z^\top$. Then the solution $\zeta = [D_z^{-1} S_z(t, x(t))]^0$.

2. Let $D_z \succcurlyeq 0$, assume that $0 \in S_z(t, x(t))$, and define the set $\mathcal{S} \triangleq \{w \in \mathbb{R}^m | D_z w \in -(S_z(t, x(t)))_\infty \cap -(S_z(t, x(t)))_\infty^* \cap \text{Ker}(D_z + D_z^\top)$, then:

(a) If $\mathcal{S} = \{0\}$, the GE (40) has at least one solution.

(b) If $\mathcal{S} \neq \{0\}$, the GE (40) has at least one solution if there exists $w_0 \in \text{dom}(\sigma_{S_z(t,x(t))}) = -(S_z(t, x(t)))_\infty^*$ such that $\langle -D_z^\top w_0, v \rangle + \sigma_{S_z(t,x(t)),\infty}(v) > 0$ for any $v \in \mathcal{S}$, $v \neq 0$.

(c) Assume that $\text{Im}(D_z + D_z^\top) \subseteq \text{Ker}(B_z)$, and $\mathcal{S} = \{0\}$. Then $A_z z(t^+)$ is unique as a solution to (19), and the unique solution to (40) is given by $\zeta = (D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0)$.

Proof. 1. Since $D_z \succ 0$ it is just needed to check that $\{0\}$ belongs to the sets $(\text{dom}(\varphi))_\infty$ and $\mathcal{K}(M, \varphi)$ in Proposition 34, where $\varphi(\cdot)$ is given above. Then from item c) in that proposition uniqueness follows. The set $S_z(t, x(t))$ is closed convex nonempty, so $\sigma_{S_z(t,x(t))}(\cdot)$ is proper convex LSC (being the conjugate of $\Psi_{S_z(t,x(t))}$), and $\partial\sigma_{S_z(t,x(t))}(\cdot)$ is maximal monotone [5, Theorem 20.40] and 3* monotone [5, Example 24.9], hence it is cocoercive [5, Proposition 24.12], and it is the Yosida approximation of a maximal monotone mapping $A(\cdot)$ [5, Proposition 23.20]. The domain of the Yosida approximation of $A(\cdot)$ is equal to the domain of its resolvent [5, Proposition 23.2], which is
$\mathbb{R}^m$ (either from Proposition 35 of from [73, Theorem 12.12]). Thus $\text{dom}(\partial\sigma_{S_z(t,x(t))})$ is a closed convex nonempty set, and $(\text{dom}(\partial\sigma_{S_z(t,x(t))}))_\infty$ is a closed convex cone [44, Proposition 5] which contains $\{0\}$ [5, Proposition 6.48]. The support function $\sigma_{S_z(t,x(t))}(\cdot)$ is proper convex, hence $(\sigma_{S_z(t,x(t))})_\infty(\cdot)$ is proper convex, with closed convex domain, and $(\text{dom}(\sigma_{S_z(t,x(t))})_\infty)^*$ is a closed convex cone which contains $\{0\}$. Hence the set $\mathcal{K}(M, \varphi)$ in (92) contains zero, and it is inferred that the set $(\text{dom}(\varphi))_\infty \cap \ker\{M + M^\top\} \cap \mathcal{K}(M, \varphi) = \{0\}$. Hence from Proposition 34 a) and c), the first result follows. For the second assertion, using mapping inversion, (40) is equivalent to $\zeta \in \mathcal{N}_{S_z(t,x(t))}(-D_z\zeta)$. Equivalently $D_z(-\zeta) \in -D_z\mathcal{N}_{S_z(t,x(t))}(D_z(-\zeta)) = \mathcal{N}_{D_z^{-1}S_z(t,x(t))}(-\zeta)$ by the Chain Rule and symmetry. Using (95) it follows that $\zeta = \text{proj}_{D_z}[D_z^{-1}S_z(t, x(t)); 0]$, which defines the minimal norm element $[D_z^{-1}S_z(t, x(t))]^0$.

2. It is necessary to calculate $(\text{dom}(\sigma_{S_z(t,x(t))})_\infty$ and $(\text{dom}(\sigma_{S_z(t,x(t))})_\infty)^*$ to calculate the sets needed in Proposition 34. We have $\text{dom}(\sigma_{S_z(t,x(t))}) = \text{bar}(S_z(t, x(t)))$ [72, p.112], and also $(\text{bar}(S_z(t, x(t)))^\circ = S_z(t, x(t))_\infty$ [5, Proposition 6.48]. Since $0 \in S_z(t, x(t))$, it follows that $(\text{bar}(S_z(t, x(t)))^{\circ\circ} = \text{bar}(S_z(t, x(t))) = (S_z(t, x(t))_\infty)^\circ = -(S_z(t, x(t))_\infty)^*$, where [72, Theorem 14.5] is used. Thus $(\text{dom}(\sigma_{S_z(t,x(t))})_\infty = -(\sigma_{S_z(t,x(t))})_\infty = -(S_z(t, x(t))_\infty)^*$ since the dual is a nonempty convex cone [5, Corollary 6.49]. Now, we have for any proper convex LSC $f(\cdot): f_\infty(\cdot) = \sigma_{\text{dom}(f^*)}(\cdot)$ [73, Theorem 11.5]. Let $f(\cdot) = \sigma_{S_z(t,x(t))}(\cdot)$, then $f^*(\cdot) = \Psi_{S_z(t,x(t))}(\cdot)$, $\text{dom}(f^*) = S_z(t, x(t))$ and $f_\infty(\cdot) = \sigma_{S_z(t,x(t))}(\cdot)$. Therefore

$$ \text{dom}(\sigma_{S_z(t,x(t))}, \infty) = \text{dom}(\sigma_{S_z(t,x(t))}) = \text{bar}(S_z(t, x(t))), \quad (41) $$

and consequently

$$ (\text{dom}(\sigma_{S_z(t,x(t))}, \infty)^* = (\text{bar}(S_z(t, x(t)))^* = -(S_z(t, x(t)))_\infty \quad (42) $$

since $S_z(t, x(t))$ is closed [5, Proposition 6.48]. Hence in (92):

$$ \mathcal{K}(D_z, \sigma_{S_z(t,x(t))}) = \{w \in \mathbb{R}^m \mid D_z w \in -(S_z(t, x(t)))_\infty\}. \quad (43) $$

Item a) and b) follow from Proposition 34. Item c): From [16, Proposition 3] and the item's assumptions, the operator $B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z\cdot)$ is single-valued, Lipschitz continuous with domain $\mathbb{R}^{n_2}$. Hence $B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0)$ is a singleton, and (19) becomes $-A_z z(t^+) - f_z(t, z(t^-)) = B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0) \Rightarrow -(B_z^\top B_z)^{-1} B_z^\top (A_z z(t^+) + f_z(t, z(t^-))) = (D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(0)$. $\square$

<figure><img src="image_4.png" /></figure>

Various sets used to compute $\mathcal{S}$ in Proposition 17 item 2 can be characterized when $K$ is polyhedral.

**Lemma 18.** Assume that $K = \{w \in \mathbb{R}^m \mid Hw \ge 0\}$ for some $H \in \mathbb{R}^{m \times m}$. Then:

1. $S_z(t, x(t))_\infty = \{w \in \mathbb{R}^m \mid Hw \ge 0\} = K$,

2. $(S_z(t, x(t))_\infty)^* = \{w \in \mathbb{R}^m \mid w = \sum_{i=1}^m \alpha_i(H)_{i\bullet}^\top, \alpha_i \ge 0\}$,

3. $\mathcal{K}(D_z, \sigma_{S_z(t,x(t))}) = \{w \in \mathbb{R}^m \mid -HD_zw \ge 0\}$.

*Proof.* 1. We have $S_z(t, x(t)) = K - C_I x(t) - F(t^+)$, hence $S_z(t, x(t)) = \{w \in \mathbb{R}^m \mid H(w + C_I x(t) + F(t^+)) \ge 0\}$, the result follows from [7, Proposition 1.4.8]. 2. Follows from [44, Example 15]. 3. Follows from the developments at the end of the proof of item 2 in Proposition 17. $\square$

<figure><img src="image_5.png" /></figure>
### 3.2.3 Relaxing Assumption 7

Now let us consider the next example, which does not satisfy Assumption 7, while $x(\cdot)$ is still continuous at $t$. The main difference with (13) is that $D \succ 0$ (while $D = 0$ in (13)), however both systems yield $D_z \succ 0$.

**Example 19.** Let us consider the example in [19, Section 6.3]:

$$ egin{cases} 
(a) \left\{ \begin{array}{l} \dot{x}_1(\tau) = -2x_1(\tau) + 2\lambda_1(\tau) - \lambda_2(\tau) \\ \dot{x}_2(\tau) = z(\tau) \end{array} \right. \\ 
(b) \quad 0 = x_2(\tau) + \lambda_1(\tau) - 2\lambda_2(\tau) \\ 
(c) \quad 0 \le \begin{pmatrix} \lambda_1(\tau) \\ \lambda_2(\tau) \end{pmatrix} \perp \begin{pmatrix} w_1(\tau) \\ w_2(\tau) \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} x_1(\tau) + \begin{pmatrix} 1 \\ 3 \end{pmatrix} x_2(\tau) + \begin{pmatrix} -1 \\ 2 \end{pmatrix} z(\tau) \\ 
\qquad + \begin{pmatrix} 2 & -1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \lambda_1(\tau) \\ \lambda_2(\tau) \end{pmatrix} \ge 0. 
\end{cases} 
\quad (44) $$

We have $A_{II} = (0 \ 1)$, $B_I = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $\bar{B}_I = \begin{pmatrix} 2 & -1 \\ 0 & 0 \end{pmatrix}$, $B_{II} = 0$, $\bar{B}_{II} = (1 \ -2)$, $C_I = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$, $C_{II} = (-1 \ 2)^\top$, $D = \begin{pmatrix} 2 & -1 \\ 0 & 1 \end{pmatrix} \succ 0$, $F(t) = 0$, $E_{II}(t) = 0$, $K = K^* = \mathbb{R}_+ \times \mathbb{R}_+$. Similarly to Example 5, numerical simulations demonstrate a jump in $z(\cdot)$ at $t \approx 0.5$s (see Fig. 3), while $\lambda_1(\cdot)$, $\lambda_2(\cdot)$, $x_1(\cdot)$ and $x_2(\cdot)$ are continuous, $\dot{x}_2(\cdot)$, $\dot{\lambda}_1(\cdot)$ and $\dot{\lambda}_2(\cdot)$ jump (hence $z(\cdot)$ jumps also from (44) (a)). Examination of Fig. 3 reveals that $z(t^-) \approx 6.0$ and $z(t^+) \approx 1.5$, $x_1(t) \approx 3$, $x_2(t) \approx 0$, $\lambda_1(t) = \lambda_2(t) \approx 0$. The mapping $(D + \partial\sigma_{K-F(t^+)})^{-1}(\cdot)$ in (8) (c) is single valued, Lipschitz continuous, with domain $\mathbb{R}^m$, and (maximal) monotone for each $t$. Therefore $(D+\partial\sigma_{K-F(t^+)})^{-1}(C_I x(t^+) + C_{II} z(t^+)) \cap \text{Ker}(\bar{B}_{II})$ is at most a singleton, and if nonempty then (8) (a) defines $x(t^+)$ uniquely. The right-hand side of (18) is single valued as well. In the particular case where $K = \mathbb{R}^m$ we can solve directly the LCP $0 \le \xi \perp D\xi + C_I x(t^+) + C_{II} z(t^+) \ge 0$ which has unique solutions since $D \succ 0$. Using the above values gives $\xi = (0 \ 0)^\top$, hence $(D + \partial\sigma_{K-F(t^+)})^{-1}(C_I x(t^+) + C_{II} z(t^+)) \cap \text{Ker}(\bar{B}_{II}) = \{(0 \ 0)^\top\}$. It is inferred from (8) (c) that $\nu_t = 0$ and $x(\cdot)$ is continuous at $t$. This is of course not constructive in general since post-jump values are unknowns.

Let us examine the evolution of the variables from the left to the right of $t$: is a jump in $z(\cdot)$ necessary, sufficient, or both, or neither one? Notice first that $\dot{\lambda}_1(t^-) < 0$ and $\lambda_1(t) = 0$ so necessarily $\dot{\lambda}_1(t^+) \ge 0$: there must be a jump in $\dot{\lambda}_1(\cdot)$ at $t$. Also $\dot{\lambda}_2(t^-) = 0$ and $\lambda_2(t) = 0$, hence $\dot{\lambda}_2(t^+) \ge 0$. We have $\dot{x}_2(t^+) - \dot{x}_2(t^-) = z(t^+) - z(t^-) = -\dot{\lambda}_1(t^+) + \dot{\lambda}_1(t^-) + 2\dot{\lambda}_2(t^+)$, with $-\dot{\lambda}_1(t^+) + \dot{\lambda}_1(t^-) < 0$ and $\dot{\lambda}_2(t^+) \ge 0$. At this stage it remains possible that $z(t^+) = z(t^-)$ (if $2\dot{\lambda}_2(t^+)$ compensates for $-\dot{\lambda}_1(t^+) + \dot{\lambda}_1(t^-)$), or that $z(t^+) \ne z(t^-)$ (if $2\dot{\lambda}_2(t^+)$ does not compensate for $-\dot{\lambda}_1(t^+) + \dot{\lambda}_1(t^-)$). The analysis is continued in Example 32.

Let us study the GE in (19). Notice that inserting (44) (b) into (c) to eliminate $x_2$, modifies the LCP matrix $D$ to $\tilde{D} = D + (1 \ 3)^\top(-1 \ 2)$, but this does not bring anything new to the analysis at jumps since (11) holds. Item 2 in Corollary 12 states the necessary and sufficient condition $\mathbb{R}_+ \times \mathbb{R}_+ \subseteq \text{Ker}((-1 \ 2)) = \{z = (z_1 \ z_2)^\top \mid z_1 = 2z_2\}$, which does not hold. Hence this criterion shows that $z(\cdot)$ cannot be continuous at $t$. Let us now compute the set of solutions $z(t^+)$ to the GE (18). Since $\nu_t = 0$, the inclusion (18) reduces to $0 = (-1 \ 2)(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}((-z(t^+) \ 2z(t^+)^\top)$, with $D_z = \begin{pmatrix} 6 & -3 \\ -2 & 2 \end{pmatrix} \succ 0$, $C_z = (-1 \ 2)^\top = B_z^\top = -\bar{B}_{II}^\top$, thus $X = R = 1$ (this system has both the $X$ and the
<figure><img src="image_6.png" /><figcaption>Example 3: $\lambda_1(t)$, $\lambda_2(t)$ time plots</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3: Backward Euler simulation of (44) [19, Example 3, Fig. 7].</figcaption></figure>
XD-proPERTIES OF DEFINITION 1). The right-hand side is single-valued Lipschitz continuous, and monotone (by the chain rule). We are led to find z(t^+) such that ξ = (ξ₁ ξ₂)ᵀ = (Dz + ∂σS_z(t,x(t^-)))⁻¹((-z(t^+) 2z(t^+)ᵀ) and ξ ∈ Ker(Bz) = {ξ ∈ R² | ξ₁ = 2ξ₂}.

Thus ξ satisfies the LCP: 0 ≤ ξ ⊥ Dzξ + C_Ix(t^-) + C_IIz(t^+) ≥ 0, i.e., 0 ≤ (ξ₁/ξ₂) ⊥ (6ξ₁ - 3ξ₂ + 6 - z(t^+)) ≥ 0. For each z(t^+) this LCP has a unique solution ξ since Dz > 0. There are four LCP-modes (to lighten notation we denote z^+ = z(t^+)): 1) 6 - z^+ ≥ 0 and -3 + 2z^+ ≥ 0, 2) 6 - z^+ ≥ 0 and -3 + 2z^+ < 0, 3) 6 - z^+ < 0 and -3 + 2z^+ ≤ 0, 4) 6 - z^+ < 0 and -3 + 2z^+ ≥ 0.

1. Then $\frac{3}{2} \le z^+ \le 6$ and $\xi_1 = \xi_2 = 0$.

2. Then $z^+ < \frac{3}{2}$, $-2\xi_1 + 2\xi_2 > 0$ so $0 \le \xi_1 < \xi_2$, and $\xi_1 = \xi_2 = 0$ is impossible. Assume that $\xi_1 = 0$ and $\xi_2 > 0$ ($\xi_1 > 0$ and $\xi_2 = 0$ is not allowed). Moreover we must have $-\xi_1 + 2\xi_2 = 0$ (kernel condition), a contradiction. Assume that $0 < \xi_1 < \xi_2$. Then $6\xi_1 - 3\xi_2 + 6 - z(t^+) = -2\xi_1 + 2\xi_2 - 3 + 2z(t^+) = 0$. This yields $\xi_1 \le \frac{1}{2}\xi_2$ and $\xi_1 \le \xi_2 \Rightarrow \xi_1 \le \frac{1}{2}\xi_2$. The kernel condition yields $2 \le \frac{1}{2}$, a contradiction. Thus this case is impossible.

3. Then $z^+ > 6$ and $z^+ \le \frac{3}{2}$: this case is impossible.

4. Then $z^+ > 6$, and $\xi_1 = \xi_2 = 0$ is impossible. Both cases $\xi_1 = 0$ and $\xi_2 > 0$ or $\xi_2 = 0$ and $\xi_2 = 0$ yield a solution $\xi \notin \ker(B_z)$. Let $\xi_1 > 0$ and $\xi_2 > 0$. For same reasons as in item 2, the kernel condition cannot be satisfied. This case is impossible.

It is inferred that the only solution of the LCP is $\xi_1 = \xi_2 = 0$ with $z^+ \in [\frac{3}{2}, 6]$ (which is the vertical segment $[z(t^+), z(t^-)]$ in Fig. 3). Therefore the time-stepping event-capturing numerical scheme used in [19] computes the least norm element in the set of admissible $z(t^+)$ (as indicated in [19, Section 6.3] the numerical scheme indeed computes the least norm element of an undetermined LCP with unknowns the multipliers). However the continuous solution with $z(t^+) = 6$ is also a solution of the GE, in agreement with the above analysis.

Example 19 shows again that in general the GE (19) does not have unique solutions, and a jump rule (possibly stemming from modeling) has to be added to get uniqueness. Another question is whether or not all values $z^+ \in [\frac{3}{2}, 6]$ allow to continue the integration. Let us state the following result.

**Proposition 20.** Let Assumptions 1, 2, 3 hold.

1. Let $D = D^\top > 0$ and $D_z = D_z^\top > 0$. Then $x(\cdot)$ is continuous at $t$, and $z(t^+)$ is a solution of the equation $0 = A_z z(t^+) + f_z(t, z(t^-), \lambda(t^-)) + B_z(D_z + \partial\sigma_{S_z(t,x(t))})^{-1}(C_z z(t^+))$.

2. Let $D_z \succcurlyeq 0$, ($B_z, C_z$) have the X-property, $0 \in S_z(t, x(t^-))$, K be polyhedral, and conditions of item 4 in Proposition 9 hold true. Then $z(\cdot)$ is discontinuous at $t$ only if $X(E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-)) \notin -C_z^\top K^*$.

*Proof.* 1. Using (94) and (95), we deduce that $\lambda(t) = \text{proj}_D[K^*; C_I x(t) + C_{II} z(t) + F(t^+)]$. Thus (2) is equal to the nonlinear nonsmooth DAE:

$$ \begin{cases} (a) \ P_I \dot{x}(t) = A_I x(t) + B_I z(t) + \bar{B}_I \text{proj}_D[K^*; C_I x(t) + C_{II} z(t) + F(t^+)] + E_I(t) \\ (b) \ 0 = A_{II} x(t) + B_{II} z(t) + \bar{B}_{II} \text{proj}_D[K^*; C_I x(t) + C_{II} z(t) + F(t^+)] + E_{II}(t), \end{cases} \tag{45} $$
From (45) (a) we infer that $x(\cdot)$ is continuous, being the solution of an ODE with Lipschitz continuous in $x$ rigth-hand side. Therefore $\nu_t = 0$, and (19) holds. Since $D_z \succ 0$, (19) holds with equality. 2. Inclusion (24) becomes $0 \in -RA_z(z(t^+) - z(t^-)) - R\bar{B}_{II}\lambda(t^-) + R(E_{II}(t^+) - E_{II}(t^-)) + R^{-1}C_z^\top(D_z + \partial\sigma_{S_z(t,x(t^-))}(C_zz(t^+))).$ Multiplying on the left by $R$ and setting $z(t^+) - z(t^-) = 0$, it follows that $0 \in X(E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-)) + C_z^\top(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(C_zz(t)).$ We have $\mathrm{Im}((D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}) = \mathrm{dom}(D_z + \partial\sigma_{S_z(t,x(t^-))}) = \mathrm{Dom}(\partial\sigma_{S_z(t,x(t^-))}).$ From [72, pp.112, 227] we have $\mathrm{dom}(\partial\sigma_{S_z(t,x(t^-))}) \subseteq \mathrm{dom}(\sigma_{S_z(t,x(t))}) = \mathrm{bar}(S_z(t, x(t^-))$ (the barrier cone), $(\mathrm{bar}(S_z(t, x(t^-)))^\circ = S_z(t, x(t^-))_\infty = K_\infty$ (see Lemma 18), and using [72, Theorem 14.5] we have $(\mathrm{bar}(S_z(t, x(t^-)))^\circ = \mathrm{bar}(S_z(t, x(t^-))) = (K_\infty)^\circ = -(K_\infty)^* = -K^*$. $\square$

An application example is given in section 4.2. Notice that item 2 in Proposition 20 does not apply to Examples 5 and 19 where the condition $0 \in S_z(t, x(t^-))$ is not satisfied. Assumption 7 suppresses half of the right-hand side in (18). Another case is as follows:

**Assumption 21.** $K$ is polyhedral, $0 \in S_z(t, x(t^-))$, and $-K^* \subseteq \mathrm{Ker}(\bar{B}_{II})$.

Assumption 21 does not imply that $x(\cdot)$ is continuous, contrarily to Assumption 7. Following same steps as in the proof of Proposition 17 item 2, Assumption 21 implies that $\mathrm{Im}((D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{S_z(t,x(t^-))})^{-1}) \subseteq \mathrm{Ker}(\bar{B}_{II})$. Indeed we have $\mathrm{Im}((D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{S_z(t,x(t^-))})^{-1}) = \mathrm{Dom}(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{S_z(t,x(t^-))}) = \mathrm{Dom}(\partial\sigma_{S_z(t,x(t^-))}) \subseteq \mathrm{dom}(\sigma_{S_z(t,x(t^-))}) = \mathrm{bar}(S_z(t, x(t^-))) = (K_\infty)^\circ = K^\circ = -K^*$, where we recall that $S_z(t, x(t^-)) = K - C_I x(t^-) - F(t^+)$. Hence under Assumption 21 the GE in (18) becomes:

$$0 \in \begin{aligned} & B_{II}z(t^+) - B_{II}z(t^-) + E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-) \\ & + B_z(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(C_{II}z(t^+)) \\ & + (B_z + \bar{B}_{II})(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(C_{II}z(t^+)). \end{aligned} \quad (46)$$

If $(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(\cdot)$ is single-valued, or if $(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(w)$ is a cone for each $w \in \mathrm{Im}(D_z + \partial\sigma_{S_z(t,x(t^-))})$, then (46) simplifies to:

$$0 \in \begin{aligned} & B_{II}z(t^+) - B_{II}z(t^-) + E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-) \\ & - A_{II}P_I^{-1}\bar{B}_I(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(C_{II}z(t^+)). \end{aligned} \quad (47)$$

The GE (47) can be analysed in a similar way as (19). For the sake of briefness this is not pursued.

## 3.3 Jumps in $x(\cdot)$ with Continuous $z(\cdot)$

The foregoing section focuses on jumps in the algebraic state $z(\cdot)$. The variable $x(\cdot)$ has a jump only if $\nu_t \neq 0$ and Assumption 7 does not hold. Instances when $x(\cdot)$ has to jump are if $C_{II} = 0$, $D = 0$ and $C_I x(t^-) + F(t^+) \notin K$ (i.e., $F(t^+)$ is such that $w(t^+) \notin K$ in (2) (c) if $x(t^+) = x(t^-)$), see Example 27. In Lemma 16 it is assumed that $x(t^-)$ satisfies the complementarity constraints, which insures that the right-hand-side of (39) is nonempty. However this may not be the case.

**Lemma 22.** Let $K$ be polyhedral, the differential state $x(\cdot)$ can have a discontinuity at $t$ only if

$$K^\circ \cap \mathrm{Ker}(\bar{B}_{II}) \neq \{0\}. \quad (48)$$
Proof. From (12) it follows that $\nu_t \neq 0$ only if the right-hand side contains another element than the zero vector. Since $\mathrm{Im}((D + \partial\sigma_{K-F(t^+)})^{-1} \circ C_I) \subseteq \mathrm{Im}((D + \partial\sigma_{K-F(t^+)})^{-1})$, this is possible only if $\mathrm{Im}((D + \partial\sigma_{K-F(t^+)})^{-1}) \cap \mathrm{Ker}(\bar{B}_{II}) \supset \{0\}$. Now $\mathrm{Im}((D + \partial\sigma_{K-F(t^+)})^{-1} = \mathrm{dom}(D + \partial\sigma_{K-F(t^+)})) = \mathrm{dom}(\partial\sigma_{K-F(t^+)}) = \mathrm{bar}(K - F(t^+)) = ((K - F(t^+))_\infty)^\circ = (K_\infty)^\circ = K^\circ$, where Lemma 18, [72, p.112] and [5, Propositions 6.48, 6.49] are used. $\square$

Clearly $\mathrm{rank}(\bar{B}_{II}) = m \Rightarrow x(t^+) = x(t^-)$. In (13), $\bar{B}_{II} = 10^{-4}$ so $x(\cdot)$ is continuous. In (44), condition (48) holds with $\mathbb{R}^2 \cap \mathrm{Ker}((-1\ 2)) \neq \{0\}$. However (48) states a necessary condition for state jumps, so this does not contradict continuity of $x(\cdot)$ in (44). Moreover item 1 in Proposition 20 applies to (48) (in both examples this result is coherent with the well-posedness results obtained in [28]). Let us consider the problem in (12).

**Lemma 23.** Assume that (48) holds and $z(t^-) = z(t^+)$. Then $x(t^+)$ solves (8) if and only if it satisfies:

$$0 \in x(t^+) - x(t^-) + P_I^{-1}\bar{B}_I((D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I x(t^+)) \cap \mathrm{Ker}(\bar{B}_{II})), \quad (49)$$

and

$$0 \in A_{II}x(t^+) - A_{II}x(t^-) + E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I x(t^+)), \quad (50)$$

$$\text{where } S_x(t, z(t^-)) \stackrel{\Delta}{=} K - C_{II}z(t^-) - F(t^+).$$

The proof of (49) and (50) follows directly from (8) (a) (b) (c) and (11). The next proposition shows that conditions of existence and uniqueness of $x(t^+)$ with continuous $z(\cdot)$ are similar to those for $z(t^+)$ with continuous $x(\cdot)$.

**Proposition 24.** Let $D = D^\top \succcurlyeq 0$, $x(t^-)$ and $z(t^-)$ be given, $z(t^+) = z(t^-)$. Let $\mathrm{Im}(C_I) \cap \mathrm{rint}(\mathrm{Im}(D + \partial\sigma_{S_x(t,z(t))})) \neq \emptyset$.

1. Assume that $(P_I^{-1}\bar{B}_I, C_I)$ has the X-property and $\bar{B}_{II} = 0$. Then (49) has a unique solution $x(t^+)$. Let $\xi = Rx$. This is a solution of (49) (50) only if

$$E_{II}(t^+) - E_{II}(t^-) \in A_{II}P_I^{-1}\bar{B}_I(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}(\mathbf{I}_d + \partial f)^{-1}(\xi(t^-))). \quad (51)$$

$$\text{where } f(\zeta) = (D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}\zeta) \text{ and } \xi = Rx(t^+).$$

2. Let Assumption 21 hold (replacing $S_z(t, x(t))$ by $S_x(t, z(t)))$. Then: (i) (49) (50) imply that

$$E_{II}(t^+) - E_{II}(t^-) \in B_z(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I x(t^+)), \quad (52)$$

where we recall that $B_z = -(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})$, and $x(t^+)$ solves (49). (ii) If $(P_I^{-1}\bar{B}_I, C_I)$ has the X-property, such $x(t^+)$ is unique. (iii) Let $K$ be polyhedral, $(P_I^{-1}\bar{B}_I, C_I, D)$ have the XD-property, for all $t \ge 0$, $\mathrm{Im}(C_I) \cap \mathrm{rint}(\mathrm{Im}(D)+S_x(t, z(t))) \neq \emptyset$, for all $t \ge 0$ and each $v \in \mathrm{Im}(C_I) \cap (\mathrm{Im}(D)+S_x(t, z(t)))$, it holds that $\mathrm{Im}(D) \cap (D+\partial\sigma_{S_x(t,z(t))})^{-1}(v) \neq \emptyset$, $F(\cdot)$ is AC, $\xi = Rx(t^+)$. Then (49) is equivalently rewritten as:

$$0 \in \xi - x(t^-) - (R^{-1}C_I^\top - R\bar{B}_I)\lambda^{im}(t, \xi) + \mathcal{F}(t, \xi), \quad (53)$$

where we have $\lambda^{im}(t, \xi) = \mathrm{proj}[\mathrm{Im}(D), \lambda] = [\Phi(t, C_I R^{-1}\xi)]^0$, for each $\lambda \in \Phi(t, C_I R^{-1}\xi) = (D+\partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}\xi)$, and $\mathcal{F}(t, \xi) = R^{-1}C_I^\top\Phi(t, C_I R^{-1}\xi)$. Assume that there exists $M \prec I_{n_1}$ such that $-(R^{-1}C_I^\top - R\bar{B}_I)\lambda^{im}(t, \xi) + M \succcurlyeq 0$. Then the GE (53) has a unique solution $\xi$.
3. Assume that conditions of item 2 (iii) are satisfied, excepted that ($\bar{B}_{II}$, $C_I$, $D$) has the XD-property. Then the GE (50) is equivalent to:

$$0 \in -R(A_{II}R^{-1}\xi + f_x(t, x(t^{-})) - (R^{-1}C_I^{\top} - R\bar{B}_{II})\lambda^{im}(t, \xi) + \mathcal{F}(t, \xi), \quad (54)$$

with $f_x(t, x(t^{-})) = A_{II}x(t^{-}) - E_{II}(t^{+}) + E_{II}(t^{-})$. Assume there exists $M \ge 0$ such that $-RA_{II}R^{-1} - M \ge 0$, and the single-valued mapping: $\mathcal{M}: \xi \mapsto -(R^{-1}C_I - R\bar{B}_{II})\lambda^{im}(t, \xi) + M\xi$ is (maximal) monotone for each $t$. Then (54) takes the form (1) for each $t$, with $A(\eta) \stackrel{\Delta}{=} -R(A_{II}R^{-1}\xi + f_x(t, x(t^{-})) - (R^{-1}C_I - R\bar{B}_{II})\lambda^{im}(t, \xi)$, $B(\eta) \stackrel{\Delta}{=} \mathcal{F}(t, \eta)$. If $A(\cdot)$ is strongly monotone, the GE (54) has a unique solution.

*Proof.* 1. Then (49) is equivalent to

$$0 \in R(x(t^{+}) - x(t^{-})) + R^{-1}C_I^{\top}(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}R x(t^{+})). \quad (55)$$

From [73, Theorem 12.43] it follows that the mapping

$$Rx(t^{+}) \mapsto R^{-1}C_I^{\top}(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}Rx(t^{+})) \quad (56)$$

is maximal monotone. The mapping $Rx(t^{+}) \mapsto R(x(t^{+}) - x(t^{-}))$ is strongly maximal monotone, so the sum of both mappings is (strongly) maximal monotone [5, Corollary 24.4]. From [5, Corollary 23.37] the GE (49) has a unique solution. Last assertion follows from inserting (49) into (50). Notice that (49) can be rewritten equivalently under the conditions of item 1, using a resolvant as $\xi(t^{+}) = (I_d + \partial f)^{-1}(\xi(t^{-}))$, where $\xi = Rx$, $f(\zeta) = (D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I R^{-1}\zeta)$. Thus the second inclusion in item 1 is rewritten as (51).

2. (i) Under modified Assumption 21, $\mathrm{Im}((D + \partial\sigma_{S_x(t,z(t))})^{-1}) \subseteq \mathrm{Ker}(\bar{B}_{II})$. The result follows inserting (49) into (50). (ii) First transform the GE as done in (25), then use the fact that the identity mapping is 1-strongly monotone. (iii) Follows same lines as the proof of (24).

3. Since (50) and (19) have the same structure, the proof mimics the proof of Proposition 9 item 4. □

Let the assumptions in item 1 hold, and $D = 0$. Then using (94) (95) and a reasoning similar to [21, Section 3.1], we obtain $x(t^{+}) = \mathrm{proj}_R[\tilde{S}_x(t, z(t)); x(t^{-})]$ is the unique solution of (49), where $\tilde{S}_x(t, z(t)) = \{\xi \in \mathbb{R}^{n_1} | C_I R^{-1}\xi \in S_x(t, z(t))\}$. Then the necessary condition in item 1 is equal to $E_{II}(t^{+}) - E_{II}(t^{-}) \in A_{II}P_I^{-1}\bar{B}_I \mathcal{N}_{S_x(t,z(t))}(C_I \mathrm{proj}_R[\tilde{S}_x(t, z(t)); x(t^{-})])$. When $A_{II}P_I^{-1}\bar{B}_I = 0$ (as in Examples 5 and 19, though the other conditions of the Proposition are not satisfied neither in (13) nor in (44)), Proposition 24 item 1 tells us that $E_{II}(\cdot)$ must be continuous at $t$. Item 1 shows that the jump in $x(\cdot)$ and the discontinuity in $E_{II}(\cdot)$ are related. In practice $x(t^{+})$ can be calculated from (49), then (50) can be tested, or the reverse. Condition $\bar{B}_{II} = 0$ is satisfied in Example (76)-(77). It is noteworthy that matrices $A_{II}$ and $B_{II}$ play a major role in the GEs (54) and (24).

**Corollary 25.** Assume that Assumption 21 holds (replacing $S_z(t, x(t^{-}))$ by $S_x(t, z(t^{-})))$, and that $z(t^{+}) = z(t^{-}) = z(t)$ and $x(t^{+}) = x(t^{-}) = x(t)$. Then (49) is equivalent to:

$$0 \in \bar{B}_I(D + \partial\sigma_{S_x(t,z(t))})^{-1}(C_I x(t)). \quad (57)$$
(i) This GE has a solution if and only if

$$ \mathrm{Ker}(\bar{B}_I) \cap \mathrm{SOL}(\mathrm{LCCP}(D, K, C_I x(t) + C_{II} z(t) + F(t^+))) \neq \emptyset. \quad (58) $$

(ii) Assume $D = D^\top \succ 0$, then (57) has a solution if and only if $\mathrm{proj}_D[K^*; -D^{-1}(C_I x(t) + C_{II} z(t) + F(t^+))] \in \mathrm{Ker}(\bar{B}_I)$. In turn this holds if $K^* \subseteq \mathrm{Ker}(\bar{B}_I)$, or if $-D^{-1}(C_I x(t) + C_{II} z(t^-) + F(t^+)) \in K^* \cap \mathrm{Ker}(\bar{B}_I)$.

(iii) Let $K = \mathbb{R}_+^m$ and $D = D^\top \succ 0$, then (58) fails if $C_I x(t) + C_{II} z(t) + F(t^+) \notin Q_D^*$.

*Proof.* First the modified Assumption 21 implies that $\mathrm{Im}(D + \partial\sigma_{S_x(t,z(t))})^{-1} \subseteq \mathrm{Ker}(\bar{B}_I)$, hence (49) $\Leftrightarrow$ (57). (i) Notice that (57) is equivalent to the existence of $\xi \in \mathbb{R}^m$ such that $\bar{B}_I\xi = 0$, $\xi \in (D + \partial\sigma_{S_x(t,z(t^-),x(t))})^{-1}(C_I x(t))$. The last inclusion is equivalent to the LCCP: $K^* \ni \xi \perp D\xi + C_I x(t) + C_{II} z(t) + F(t^+) \in K$. (ii) Let us rewrite (57) as in the proof of (i). Using (94) and (95), it follows that $\xi$ is given by the projection. The sufficient conditions are a direct consequence. (iii) This follows from Appendix B. $\square$

If (58) fails, then $x(\cdot)$ must jump at $t$.

**Example 26 (Example 5 continued).** Let us examine the corollary at $t \approx 5.9s$. Condition (53) is equal to $\mathrm{Ker}((-3.10^{-3} 0)^\top) \cap \mathrm{SOL}(\mathbb{R}_+ \ni \xi \perp -z(t^-) \in \mathbb{R}_+) = \emptyset$ since $z(t^-) > 0$. Therefore (58) is not satisfied, and the GE (57) has no solution. Thus the Corollary's assumptions do not hold (either $z(\cdot)$ or $x(\cdot)$ must be discontinuous, and we know that the former jumps while the latter is continuous at the considered time).

**Example 27 (Case of LCS).** For LCS we have $A_{II} = 0$, $B_{II} = 0$, $\bar{B}_{II} = 0$, $E_{II} = 0$, $B_I = 0$, $C_{II} = 0$, $P_I = I_{n_1}$, $K = \mathbb{R}_+^m$, $S_x(t, z(t)) = \mathbb{R}_+^m - F(t^+)$. Condition (48) holds which allows for state jumps. Concerning (58): $\mathrm{SOL}(\mathrm{LCP}(D, \mathbb{R}_+^m, C_I x(t) + F(t^+))) \neq \emptyset$ if $C_I x(t) + F(t^+) \in Q_D^*$ (where $Q_D = \{\xi \in \mathbb{R}^m \mid 0 \le \xi \perp D\xi \ge 0\}$) [38, Theorem 3.8.6] (equivalence holds if $D = D^\top$, see Appendix B). If this condition does not hold, $x(\cdot)$ has to jump to get $C_I x(t^+)) + F(t^+) \in Q_D^*$ [25, 29, 21, 20]. The GE (49) reduces to: $0 \in x(t^+) - x(t^-) + \bar{B}_I((D + \partial\sigma_{S_x(t)})^{-1}(C_I x(t^+)))$. Under passivity-like conditions as in Definition 1, then if $D = 0$ the unique solution is a projection, if $D \succ 0$ it can be calculated solving a LCP. It can be checked that the state jump law for LCS proposed and analysed in [26, 21, 46] (see [20, Definition 2.2, Lemma 2.3]) solves this GE.

**Example 28 (Case of DAE).** Semi-explicit DAEs are recovered by setting $K = \mathbb{R}^m$ ($\Rightarrow K^* = \{0\} \Rightarrow \lambda = 0$). Then $\mathrm{Im}(D + \partial\sigma_{S_z(t,x(t^-))})^{-1} = \mathrm{dom}(D + \partial\sigma_{S_z(t,x(t^-))}) = \mathrm{dom}(D + \partial\sigma_{\mathbb{R}^m}) = \{0\}$, since $\mathrm{dom}(\sigma_{\mathbb{R}^m}) = \mathrm{bar}(\mathbb{R}^m) = \{0\}$. Thus (19) boils down to $0 = B_{II}z(t^+) - B_{II}z(t^-) + E_{II}(t^+) - E_{II}(t^-)$. In the above theoretical framework, we also have $x(t^+) = x(t^-)$. It is inferred that when particularized to DAEs, the proposed framework is limited to jumps of algebraic states triggered by discontinuous exogenous terms.

If condition (53) does not hold, then either $x(\cdot)$ or $z(\cdot)$ has to jump (as in Example 26). If it is guaranteed that $z(\cdot)$ does not jump, then $x(\cdot)$ has to jump. Let us end this section by noting the following fact. Assume that $C_{II} = 0$, $D = 0$, and that $w(t^-) = x(t^-) + F(t^+) \notin K$. Then if condition (48) is not true, solutions cannot be continued after $t$. This shows the significant interplay between equality and inequality constraints in DALCS. Conditions under which both states jump are tackled in section 3.5.
### 3.4 Jump in $z(\cdot)$ with Continuous $x(\cdot)$ and $\lambda(\cdot)$: Analysis via Derivatives

Example 5 motivates us to consider such cases. Also, Example 8 shows how (18) may be used to calculate $z(t^+)$ in a particular case, when all other variables are time-continuous.

Notice that such state jumps satisfy
$$ \begin{pmatrix} x(t^+) \\ z(t^+) \end{pmatrix} - \begin{pmatrix} x(t^-) \\ z(t^+) \end{pmatrix} = \begin{pmatrix} 0 \\ z(t^+) - z(t^-) \end{pmatrix} \in \text{Ker}(\text{diag}(P_I, 0)) $$

which is the impulse-free jump condition in time-varying DAE [66, 31] (Assumption 7 guarantees an impulse-free solution). The functions equality in (9) is assumed to hold dt-almost everywhere and using Assumptions 1–3 it can be extended to measures equality as

$$ 0 = A_{II}dx + B_{II}dz + \bar{B}_{II}d\lambda + dE_{II}, \quad (59) $$

Notice that $d\lambda$ in (59) is different in general from $\nu_t\delta_t$ at an atom of $d\lambda$, which corresponds to a discontinuity in $\lambda(\cdot)$ at $t$, hence an impulsive $\dot{\lambda}$. In other words, $d\lambda = (\lambda(t^+) - \lambda(t^-))\delta_t + \dot{\lambda}(t)dt$, while $\lambda = \nu_t\delta_t + \lambda(t)dt$.

**Lemma 29.** Consider the equality in (9), supposed to hold on an open interval $I$.

1. Assume that $E_{II}(\cdot)$, $x(\cdot)$ and $\lambda(\cdot)$ are AC on $I$, $t \in I$ is a time of discontinuity of $z(\cdot)$, then:

$$ z(t^+) - z(t^-) \in \text{Ker}(B_{II}). \quad (60) $$

2. If $E_{II}(t^+) \neq E_{II}(t^-)$ then $z(t^+) = B_{II}^\dagger B_{II} z(t^-) + B_{II}^\dagger (E_{II}(t^-) - E_{II}(t^+)) + (I_{n_2} - B_{II}^\dagger B_{II})\zeta$ for some $\zeta \in \mathbb{R}^{n_2}$, and necessarily $E_{II}(t^+) - E_{II}(t^-) \in \text{Im}(B_{II})$.

*Proof.* 1. Follows from (59) which reduces to $0 = B_{II}(z(t^+) - z(t^-))$. 2. Follows from $0 = B_{II}(z(t^+) - z(t^-)) + E_{II}(t^+) - E_{II}(t^-)$ and applying [6, Proposition 8.1.9]. $\square$

How can the jump $z(t^+) - z(t^-)$ occurring on such an interval $I$ be characterized? This is done in Example 5, see (17), exploiting the special structure of the system (13). From Example 5, we see that the jump in $z(\cdot)$ is due to the jump in $\dot{\lambda}(\cdot)$ (see (14) (d)) and differentiating once (14) (b) the jump in $\dot{\lambda}(\cdot)$ is equivalent to the existence of a jump in $\dot{x}_2(\cdot)$. From (2) we know that outside jumps it holds dt-almost-everywhere:

$$ \begin{aligned} \bar{B}_{II} \dot{\lambda}(\tau) = & -A_{II} P_I^{-1} A_I x(\tau) - A_{II} P_I^{-1} B_I z(\tau) - A_{II} P_I^{-1} E_I(\tau) - \dot{E}_{II}(\tau) \\ & -A_{II} P_I^{-1} \bar{B}_I \lambda(\tau) - B_{II} \dot{z}(\tau) \end{aligned} \quad (61) $$

Let $\zeta = (\lambda^\top, z^\top)^\top$, then we obtain:

$$ \begin{aligned} (a) \quad (\bar{B}_{II}, B_{II}) \dot{\zeta}(\tau) = & (-A_{II} P_I^{-1} \bar{B}_I, -A_{II} P_I^{-1} B_I) \zeta(\tau) - A_{II} P_I^{-1} A_I x(\tau) \\ & -A_{II} P_I^{-1} E_I(\tau) - \dot{E}_{II}(\tau) \end{aligned} \quad (62) $$

$$ (b) \quad P_I \dot{x}(\tau) = A_I x(\tau) + (\bar{B}_I, B_I) \zeta(\tau) + E_I(\tau) $$

Let us premultiply (62) (a) by $(\bar{B}_{II}^\top \quad B_{II}^\top) X$, for some $X \in \mathbb{R}^{p \times p}$. This yields:

$$ \tilde{P} \begin{pmatrix} \dot{\zeta}(\tau) \\ \dot{x}(\tau) \end{pmatrix} = \tilde{A} \begin{pmatrix} \zeta(\tau) \\ x(\tau) \end{pmatrix} + \tilde{E}(\tau) + \begin{pmatrix} w(\tau) \\ 0 \\ 0 \end{pmatrix} \quad (63) $$

$$ K^* \ni \lambda(\tau) = (I_m, 0, 0) \begin{pmatrix} \zeta(\tau) \\ x(\tau) \end{pmatrix} \perp w(\tau) \in K, $$
$$ \text{with: } \tilde{P} = \begin{pmatrix} \bar{B}_{II}^{\top} X \bar{B}_{II} & \bar{B}_{II}^{\top} X B_{II} & 0 \\ B_{II}^{\top} X \bar{B}_{II} & B_{II}^{\top} X B_{II} & 0 \\ 0 & 0 & P_I \end{pmatrix}, \quad \tilde{E}(\tau) = \begin{pmatrix} -\bar{B}_{II}^{\top} X (A_{II} P_I^{-1} E_I(\tau) + \dot{E}_{II}(\tau)) - F(\tau) \\ -B_{II}^{\top} X (A_{II} P_I^{-1} E_I(\tau) + \dot{E}_{II}(\tau)) \\ E_I(\tau) \end{pmatrix}, $$

$$ \tilde{P} \in \mathbb{R}^{(n_1+n_2+m) \times (n_1+n_2+m)}, \quad \tilde{A} = \begin{pmatrix} -\bar{B}_{II}^{\top} X A_{II} P_I^{-1} \bar{B}_I - D & -\bar{B}_{II}^{\top} X A_{II} P_I^{-1} B_I - C_{II} & -\bar{B}_{II}^{\top} X A_{II} P_I^{-1} A_I - \bar{B}_{II}^{\top} X A_{II} P_I^{-1} \bar{B}_I \\ -B_{II}^{\top} X A_{II} P_I^{-1} \bar{B}_I & -B_{II}^{\top} X A_{II} P_I^{-1} B_I & -B_{II}^{\top} X A_{II} P_I^{-1} A_I \\ \bar{B}_I & B_I & A_I \end{pmatrix} $$

$\tilde{A} \in \mathbb{R}^{(n_1+n_2+m) \times (n_1+n_2+m)}$. Now $w$ plays the role of a multiplier in (63), just as it does in (14) (interchanging the slack variables' roles in complementarity problems and systems is customary). Thus (63) is a descriptor-variable linear complementarity system (DVLCS [13]) with exogeneous signals, which is assumed to be well-posed on $I \setminus \{t\}$ with $AC(\lambda(\cdot))$ and $x(\cdot)$. Matrix $\tilde{P}$ reflects the couplings between $\lambda$ and $z$ inside (2) (b). To simplify expression of $\tilde{P}$ let us adopt the notation $\bar{P} = \bar{B}_{II}^{\top} X \bar{B}_{II} \in \mathbb{R}^{m \times m}$, $P = B_{II}^{\top} X B_{II} \in \mathbb{R}^{n_2 \times n_2}$, $\bar{P} = \bar{B}_{II}^{\top} X B_{II} \in \mathbb{R}^{m \times n_2}$. The relative degree between $w$ and $\lambda$ is denoted as $r_{w\lambda} \in \mathbb{R}^m$.

**Proposition 30.** Let $K = \mathbb{R}_+^m$ and $X = X^\top$. Assume that one of the next statements holds:

1. $\text{rank}(P) = n_2$, $M = M^\top \triangleq \bar{P} - \bar{P} P^{-1} \bar{P}^\top$ and $\text{rank}(M) = m$.

2. $\text{rank}(\bar{P}) = m$, $\text{rank}(P - \bar{P}^\top \bar{P}^{-1} \bar{P}) = n_2$, $N = N^\top \triangleq \bar{P}^{-1} \bar{P} (P - \bar{P}^\top \bar{P}^{-1} \bar{P})^{-1} \bar{P}^\top \bar{P}^{-1} + \bar{P}^{-1}$ and $\text{rank}(N) = m$.

3. $B_{II} = 0$ and $\text{rank}(\bar{P}) = m$.

(i) Then $r_{w\lambda} = (1 \dots 1)^\top$. (ii) Moreover if Assumptions in item 1 (resp. item 2, resp. item 3) above hold and $M \succ 0$ (resp. $N \succ 0$, resp. $\bar{P} \succ 0$), then the LCP: $0 \le \frac{d\lambda}{d\tau}(\tau) \perp w(\tau) \ge 0$ always has a unique solution $w(\tau)$, given as the solution of a minimization problem.

**Proof.** (i) Using matrix inversion for the partitioned submatrix of $\tilde{P}: \begin{pmatrix} \bar{P} \\ \bar{P}^\top \end{pmatrix} [6$, Fact 3.22.3] and (63), allows us to conclude for items 1 and 2. Item 3 follows by inverting directly the first line in the dynamical part of (63). (ii) Follows from the fact that $M$, $N$ and $\bar{P}$ are symmetric full rank, and symmetric P-matrices are positive definite. In each case the matrix of the obtained LCP is equal to $M$ (resp. $N$, resp. $\bar{P}$), and the LCP writes: $0 \le M^{-1}w(\tau) + q_1(x(\tau), z(\tau), \lambda(\tau), \tau) \perp w(\tau) \ge 0$ (resp. $0 \le N^{-1}w(\tau) + q_2(x(\tau), z(\tau), \lambda(\tau), \tau) \perp w(\tau) \ge 0$, resp. $0 \le \bar{P}^{-1}w(\tau) + q_3(x(\tau), z(\tau), \lambda(\tau), \tau) \perp w(\tau) \ge 0$), for some $q_i(x(\tau), z(\tau), \lambda(\tau), \tau), i \in \{1, 2, 3\}$, linear in $x, z, \lambda$ and $\tilde{E}$ from (63). The last assertion follows from the fact that all three LCP matrices are symmetric [38, p.5]. $\square$

The DVLCS (63) may be seen as a relative degree/index reduction. In our setting it aims at analyzing the system when a jump in $z(\cdot)$ occurs at $t$. At a jump time in $z(\cdot)$ we know that the DVLCS still makes sense, as noted above, and (61) gives $\bar{B}_{II} \dot{\lambda}(t) = -A_{II} P_I^{-1} A_I x(t) - A_{II} P_I^{-1} B_I z(t) - A_{II} P_I^{-1} E_I(t) - \dot{E}_{II}(t) - A_{II} P_I^{-1} \bar{B}_I \dot{\lambda}(t)$.

$$ \textbf{Example 31.} \quad (\text{Example 5 continued}) \quad \text{Let us choose } X = 10^4, \text{ then: } \tilde{A} = \begin{pmatrix} 0 & 0 & 0.707 & 0 \\ 0 & 0 & 0 & 0 \\ -3 \ 10^{-3} & 0 & -2.12 \ 10^3 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}, $$
$$ \tilde{P} = \begin{pmatrix} 10^{-4} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & I_2 \end{pmatrix}, \quad \tilde{E} = \begin{pmatrix} -10^{-4} \dot{u}_2 \\ 0 \\ 0.301 \ 10^4 u_1 \\ 0 \end{pmatrix}, \quad P = 0, \bar{P} = 0, \bar{\bar{P}} = 10^{-4}. \text{ Since } m = 1 \text{ item} $$

3 in Proposition 30 applies. Due to the choice of X, system (63) is equal to (14) (a) (b) (e):

$$ \begin{pmatrix} 10^{-4} & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \dot{\lambda}(\tau) \\ \dot{x}_1(\tau) \end{pmatrix} = \begin{pmatrix} 0 & 0.707 \\ -2.12 \ 10^3 & -3 \ 10^3 \end{pmatrix} \begin{pmatrix} \lambda(\tau) \\ x_1(\tau) \end{pmatrix} + \begin{pmatrix} w(\tau) \\ 0 \end{pmatrix} + \begin{pmatrix} -10^{-4} \dot{u}_2(\tau) \\ 0.301 \ 10^4 u_1(\tau) \end{pmatrix} \\ 0 \le \lambda(\tau) \perp w(\tau) \ge 0. \tag{64} $$

The analysis made in Example 5 using the LCP: $0 \le \frac{d\lambda}{d\tau}(\tau) \perp w(\tau) \ge 0$, reveals that $w(\cdot)$ jumps at the junction time $t$, so that both $\dot{\lambda}(\cdot)$ and $z(\cdot)$ (see (14) (d)) jump at $t$. Notice that the input and output matrices of the LCS (64) have the X-property, so that its well-posedness (existence and uniqueness of AC solutions) follows from [21] or [19, Section 5.3].

$$ \textbf{Example 32.} \ (\text{Example 19 continued}) \ We \ have \ p=1, \ \tilde{A} = \begin{pmatrix} -2 & 1 & X-1 & -2 & -1 \\ 0 & -1 & -2X+2 & 1 & -3 \\ 0 & 0 & 0 & 0 & 0 \\ 2 & -1 & 0 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}, $$

$$ \tilde{P} = \begin{pmatrix} X & -2X & 0 & 0 \\ -2X & 4X & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad \tilde{E} = (0 \ 0 \ 0 \ 0)^\top, \quad \bar{\bar{P}} = X \begin{pmatrix} 1 & -2 \\ -2 & 4 \end{pmatrix}. \text{ Choosing } X=1 \text{ allows} $$

to suppress the dependency of $\tilde{A}$ on $z$. However Proposition 30 item 3 does not apply since $\text{rank}(\bar{\bar{P}}) = 1$ and $m = 2$. It is noteworthy that Examples 5 and 19 differ in that the “equivalent” form as proposed in [28] takes the form

$$ \underbrace{\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & -2 & 4 \end{pmatrix}}_{\triangleq P_e} \begin{pmatrix} \dot{x}_1(\tau) \\ \dot{\lambda}_1(\tau) \\ \dot{\lambda}_2(\tau) \end{pmatrix} = \underbrace{\begin{pmatrix} -2 & 2 & -1 \\ -2 & -1 & -1 \\ 1 & 3 & -7 \end{pmatrix}}_{\triangleq A_e} \begin{pmatrix} x_1(\tau) \\ \lambda_1(\tau) \\ \lambda_2(\tau) \end{pmatrix} + \underbrace{\begin{pmatrix} 0 \\ w_1(\tau) \\ w_2(\tau) \end{pmatrix}}_{\triangleq B_e w(\tau)} \quad (65) \\ 0 \le w(\tau) \perp C_e x_e(\tau) = B_e^\top x_e = \lambda(\tau) \ge 0, $$

for (44), where $\bar{\bar{P}}$ appears in the left-hand side matrix. Using $x_2 = -\lambda_1 + 2\lambda_2$ (see (44) (b)) allows us to recover (65) from (63) with $X = 1$. From (65) and using the Moore-Penrose pseudo inverse of $(\begin{pmatrix} 1 \\ -2 \end{pmatrix})$, we find that $\dot{\xi} \triangleq \dot{\lambda}_1 - 2\dot{\lambda}_2 = \frac{1}{5}(-7\lambda_1 + 13\lambda_2 - 4x_1 + w_1 - 2w_2)$, hence $\dot{\xi}(t^-) = \frac{1}{5}(-12 + w_1(t^-) - 2w_2(t^-)) = -\dot{x}_2(t^-) = -z(t^-) = -6$. Also we have: $\dot{\xi} + 2(-2\dot{\xi}) = 5\lambda_1 - 15\lambda_2 + w_1 + 2w_2$, hence $\dot{\xi}(t^-) = -\frac{1}{3}w_1(t^-) - \frac{2}{3}w_2(t^-) = -z(t^-)$. It is inferred that $w_2(t^-) = 9$ and $w_1(t^-) = 0$ (as expected from the numerical values in Fig. 3). Now $z(t^+) = \frac{3}{2} = 2x_1(t) - w_1(t^+) = 6 - w_1(t^+) \Rightarrow w_1(t^+) = \frac{9}{2}$, and $2z(t^+) = x_1(t) + w_2(t^+) = 3 + w_2(t^+) = 3 \Rightarrow w_2(t^+) = 0$ (these pre- and post-discontinuity values satisfy (44) (c)). Notice that $z(t^+) = x_1(t) - w_1(t^+) \Rightarrow w_1(t^+) = 6 - z(t^+) \ge 0$ and $2z(t^+) = x_1(t) + w_2(t^+) \Rightarrow w_2(t^+) = 2z(t^+) - 3 \ge 0$. It is inferred that $z(t^+) \in [\frac{3}{2}, 6]$:
the interval obtained in Example 19 from the analysis of the GE (19) is recovered using a different analysis (remind that it is the vertical segment in Fig. 3). The multiplier is computed in [19, section 6.3] by discretizing (65) and then using Lemke's algorithm to solve the LCP: $0 \le \lambda_{k+1} \perp \begin{pmatrix} 1 & -2 \\ -2 & 4 \end{pmatrix} \lambda_{k+1} + q_k(h, x_{1,k}, \lambda_k) \ge 0$, and it provides $z(t^+) = \frac{3}{2}$: we conclude that it corresponds to choosing a particular state-jump rule for $z(\cdot)$, that provides an admissible post-jump value to continue the integration. The next question is: are all $z(t^+) \in [\frac{3}{2}, 6]$ admissible for (44), i.e., locally AC solutions can be continued on $(t, t + \epsilon)$, $\epsilon > 0$? This is outside the scope of this article which focuses solely on characterizing state jumps.

From [28, Theorems 2, 3], both (13) and (44) enjoy the existence and uniqueness of AC solutions $x_1(\cdot)$ and $x_2(\cdot)$, with AC signal $\bar{B}_{II}\lambda(\cdot) = (\lambda_1 - 2\lambda_2)(\cdot)$. The major discrepancy between Example 5 and Example 19 is that the “equivalent” system is the DVLCS in (65) for the latter, while it is a regular LCS (64) for the former. Let $w$ be the input and $\lambda$ be the output, from [18, Theorem 2.22], the transfer matrix of the quintuple $(P_e, A_e, B_e, C_e, 0)$ is positive real, but it is not minimal (the fifth condition in [19, Theorem G.1] does not hold), hence it cannot be transformed into the canonical form (??), and the well-posedness results in [28, Theorems 2, 3] do not apply to (65). However using the chain rule of Convex Analysis, (65) is equivalently rewritten as the singular DI:

$$P_e \dot{x}_e(t) \in - (A_e x_e(t) + \mathcal{N}_{\Phi_e}(x_e(t)) \quad (66)$$

with $\Phi_e = \{x_e \in \mathbb{R}^3 | B_e^\top x_e \ge 0\}$. Since $A_e \prec 0$, the mapping $x_e \mapsto -A_e x_e + \mathcal{N}_{\Phi_e}(x_e)$ is maximal monotone [72]. Since $P_e = P_e^\top \succcurlyeq 0$, [28, Lemmas 2, 3] apply to (66). Using

the unitary matrix $U = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ 0 & -\frac{2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \end{pmatrix}$, we obtain $UP_eU^\top \dot{\xi}_e(t) \in - (UA_eU^\top \xi_e(t) + U\mathcal{N}_{\Phi_e}(U^\top \xi_e(t))$, with $\xi_e = U^\top x_2 = Ux_e = (x_1 \frac{\lambda_1+2\lambda_2}{\sqrt{5}} - \frac{2\lambda_1+\lambda_2}{\sqrt{5}})^\top$, $UP_eU^\top = \text{diag}(1, 5, 0)$.

Using the chain rule of Convex Analysis (⇒ $U\mathcal{N}_{\Phi_e}(U^\top \xi_e) = \mathcal{N}_{\tilde{\Phi}_e}(\xi_e)$) and [28, Lemmas 2, 3], it is inferred that solutions of (66) satisfy $x_1(\cdot)$ and $(\lambda_1+2\lambda_2)(\cdot)$ are AC with uniqueness (so that $\lambda_1(\cdot)$ and $\lambda_2(\cdot)$ are AC since $(\lambda_1 - 2\lambda_2)(\cdot)$ is AC), while $(-2\lambda_1 + \lambda_2)(\tau)$ is an AC selection of a convex valued operator.

Examples 31 and 32 and Proposition 30 teach us that examining (63) is essential to characterize jumps in $z(\cdot)$. The major difference between (13) and (44) is that $\bar{P}$ is full rank in the former, hence Proposition 30 applies.

**Remark 1.** According to [66, Remark 3.2], if $-A_{II}P_I^{-1}E_I(\cdot) - \dot{E}_{II}(\cdot)$ and $E_I(\cdot)$ are time-continuous then the solution $(\zeta, x)$ of (62) has no self-generated jumps, provided the DAE has index 1. The index is related to the rank of $\bar{P}$, i.e., to the rank of $(\bar{B}_{II}, B_{II}) \in \mathbb{R}^{p \times (m+n_2)}$. When $p = m + n_2$ (the matrix is square) the index is determined from the Weierstrass canonical form [31, Remarks 3.2, 3.4]. Thus this implies that $m = 0$ if $p = n_2$ as in Proposition 9, which is clearly impossible.

## 3.5 Simultaneous Jumps in $x(\cdot)$ and $z(\cdot)$

The foregoing sections focus on separated jumps in $x(\cdot)$ and $z(\cdot)$. Let $K = \mathbb{R}_+^m$, $C_{II} = 0$, $D = 0$. Further cases when states have to jump are when in spite of the fact that $w(t^-) \ge 0$, we have $C_I x(t^-) + F(t^+) < 0$ so that necessarily $x(t^+) \neq x(t^-)$, and
$A_{II}x(t^+) + B_{II}z(t^-) + E_{II}(t^+) \neq 0$: necessarily we must have $z(t^+) \neq z(t^-)$ to get (8) (b$^+$). Lemma 16 with (39) and Corollary 25 with (58) motivate us to analyse possible simultaneous discontinuities in $z(\cdot)$ and $x(\cdot)$, in case where $(D_z+\partial\sigma_{S_z(t,x(t^-))})^{-1}(0) = \emptyset$, or when SOL(LCCP($D, K, C_Ix(t)+F(t^+)$)) = $\emptyset$, due to nonadmissible $x(t^-)$. The foregoing analyses do not apply to this case. To simplify let us consider $D=0$, at a jump time we obtain the GE with unknowns $x(t^+)$ and $z(t^+)$:

$$ (a) \quad 0 \in x(t^+) - x(t^-) + P_I^{-1}\bar{B}_I (\mathcal{N}_{K-F(t^+)}(C_I x(t^+) + C_{II} z(t^+)) \cap \text{Ker}(\bar{B}_{II})) \quad (67) $$

$$ (b) \quad 0 \in -A_{II}x(t^+) - B_{II}z(t^+) - E_{II}(t^+) + \bar{B}_{II}\mathcal{N}_{K-F(t^+)}(C_I x(t^+) + C_{II} z(t^+)), $$

which reduces to (49)-(50) if $z(t^+) = z(t^-)$. It is noteworthy that $\text{Ker}(\bar{B}_{II}) \neq \{0\}$ since otherwise $x(\cdot)$ is continuous (see Lemma 4). Mimicking Propositions 9 and 24, let us propose the following.

**Proposition 33.** (State jumps in FOSwP) Assume that: (i) $K^* \subseteq \text{Ker}(\bar{B}_{II})$, $K$ is a polyhedral convex cone, $0 \in K - F(t^+)$, (ii) the pair $(\begin{pmatrix} P_I^{-1}\bar{B}_I \\ \bar{B}_{II} \end{pmatrix}, (C_I, C_{II}))$ has the X-property, (iii) $\text{Im}((C_I, C_{II})) \cap \text{rint}(K - F(t^+)) \neq \emptyset$, (iv) $Q \triangleq \begin{pmatrix} I_{n_1} & 0 \\ -A_{II} & -B_{II} \end{pmatrix} > 0$, and $RQR^{-1} = R^{-1}Q^\top R > 0$. Then the GE (67) has a unique solution $(x(t^+), z(t^+))$ for any data $x(t^-), F(t^+), E_{II}(t^+)$, given by:

$$ \begin{pmatrix} x(t^+) \\ z(t^+) \end{pmatrix} = R^{-1} \text{proj}_{RQR^{-1}} \left[ \tilde{K}(t); R \begin{pmatrix} Rx(t^-) \\ E_{II}(t^+) \end{pmatrix} \right], \quad (68) $$

$$ \text{where } \tilde{K}(t) = \{\eta \in \mathbb{R}^n \mid (C_I, C_{II})R^{-1}\eta \in K - F(t^+)\}. $$

*Proof.* (i) is same as Assumption 21 and allows us to rewrite (67) (a) without $\text{Ker}(\bar{B}_{II})$. (ii) and (iii) allows us to transform (67) similarly to what we did to obtain (21) or (55). Then (iv) allows us to obtain (68) using (94) (95). $\square$

If $0 \neq D \geq 0$, then the GE can be analysed similarly as in Proposition 9 item 4 or Proposition 24 item 3. As pointed out in section 2, (ii) and $D=0$ allow us to transform the DALCS into the DAFOSwP in (7). Thus Proposition 33 characterizes jumps in DAFOSwP. Notice that (59) means that the derivative of (2) (b) vanishes at atoms of the measures $dx, dz, d\lambda, dE_{II}$. Proposition 33 applies when $\bar{B}_{II} = 0$ so that (67) (a) reduces to $0 \in x(t^+) - x(t^-) + P_I^{-1}\bar{B}_I \mathcal{N}_{K-F(t^+)}(C_I x(t^+) + C_{II} z(t^+))$. Proposition 33 is a one-shot criterion for both states. Another approach is to solve first (67) (a) to obtain a unique $x(t^+)$ as a function of $z(t^+)$, then solve (67) (b) for $z(t^+)$. However this does not bring advantage in the calculation compared to (68).

It is noteworthy that (18) is generic since it is obtained without assuming that $x(t^+) = x(t^-)$. Assume that the set $(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{K-C_Ix(t^-)-F(t^+)})^{-1}(C_{II}z(t^+))$ in (18) is convex, then by [72, Theorem 3.2] we have

$$ \begin{align*} & -(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{K-C_Ix(t^-)-F(t^+)})^{-1}(C_{II}z(t^+)) \\ & -(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{K-C_Ix(t^-)-F(t^+)})^{-1}(C_{II}z(t^+)) \\ & \qquad = -2(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{K-C_Ix(t^-)-F(t^+)})^{-1}(C_{II}z(t^+)). \end{align*} $$

Therefore if (i) in Proposition 33 holds, the GE in (18) reads as:

$$ \begin{align} 0 \in B_{II}z(t^+) - B_{II}z(t^-) + E_{II}(t^+) - E_{II}(t^-) - \bar{B}_{II}\lambda(t^-) \tag{69} \\ -2(A_{II}P_I^{-1}\bar{B}_I + \bar{B}_{II})(D + C_I P_I^{-1}\bar{B}_I + \partial\sigma_{K-C_Ix(t^-)-F(t^+)})^{-1}(C_{II}z(t^+)). \notag \end{align} $$
which is (compare to (19)):

$$0 \in A_z z(t^+) + f_z(t, z(t^-)) + 2B_z(D_z + \partial\sigma_{S_z(t,x(t^-))})^{-1}(C_z z(t^+)). \quad (70)$$

Thus jump in $x(\cdot)$ implies a bigger set-valued right-hand side. All results obtained for (19) in section 3.2 apply to (70), replacing $B_z$ by $2B_z$.

*Remark 2 (Jumps at Initial Time).* State discontinuities at $t = 0$ have been well identified in both LCS [26, 29, 21, 46, 20] and DAEs or descriptor-variable systems [35, 36, 66, 68, 67, 69]. They are due to nonadmissible (inconsistent is used in DAE theory) initial data which do not satisfy the constraints, and which have to jump to admissible values. They occur even in the absence of exogenous signals (the novelty in DALCS being that jumps in $z(\cdot)$ can occur at $t > 0$ even if no exogenous signals are present, see Example 19). The above framework handles jumps at initial times. Initial states are admissible if and only if there exists $\lambda(0)$ such that:

$$\begin{cases} (b) \quad 0 = A_{II}x_0 + B_{II}z_0 + \bar{B}_{II}\lambda(0) + E_{II}(0) \\ (c) \quad K^* \ni \lambda(0) \perp w(0) = C_{I}x_0 + C_{II}z_0 + D\lambda(0) + F(0) \in K, \end{cases} \quad (71)$$

with $x(0) = x_0$, $z(0) = z_0$. Equivalently, the GE

$$0 \in A_{II}x_0 + B_{II}z_0 - \bar{B}_{II}(D + \partial\Psi_{K-F(0)}^{-1})^{-1}(C_{I}x_0 + C_{II}z_0) + E_{II}(0) \quad (72)$$

is solvable with $x_0$ and $z_0$. As shown above, initial consistency does not preclude existence of state jumps at later times. This fact is already present in nonlinear DAEs with impasse points [33, 74, 51, 64, 65, 66, 67, 69, 37]. Algebraic state jumps solely due to system's internal dynamics (no exogenous excitations) may be considered as stemming from same reasons as DAE impasse points do (impossibility of continuing further the integration without imposing a state jump). They involve other mechanisms, however.

*Remark 3.* Relative degree in LCS and index in DAE are defined (when they exist) from structural properties. However they also influence the nature of solutions as Schwartz' distributions with maximum degree [69, Theorem 20.2] [1]. For instance, LCS with relative degree 0 (resp. 1, resp. 2) have continuously differentiable (resp. continuous, resp. measure) solutions. One path for their extension to DALCS is to start from the solution sets for $z(\cdot)$ and $x(\cdot)$. For instance, Example 5 shows that when $D = 0$ but $D_z > 0$, it is possible that $z(\cdot)$ jumps while $\nu_t = 0$ and $x(\cdot)$ is continuous. In this example one has $C_{I}x(t^{-}) + C_{II}z(t^{-}) + F(t^{+}) = 0.707 \times \frac{7}{9} - 0.33 \approx 0.2198 > 0$: the unilateral constraint is satisfied at $t$. But $A_{II}x(t^{-}) + B_{II}z(t^{-}) + E_{II}(t^{+}) = 0 + 0 + 10^{-4}\cos(10 \times 5.9) \approx 0.51 \times 10^{-4} \neq 0$. Hence it is the equality constraint which triggers the jump in $z(\cdot)$. In such a case we may say that the system has index 2 and relative degree 1. A sound and general notion extending index and relative degree to DALCS is still missing.

# 4 Examples

Various examples from circuits, switching DAEs, constrained mechanical systems, the class of passive DALCS are given to illustrate the foregoing developments.
<figure><img src="image_8.png" /><figcaption>Figure 4: RLC circuits with ideal diode.</figcaption></figure>

## 4.1 Circuit with Ideal Diode

Let us consider the circuit studied in [19, Section 6.2], depicted in Fig. 4, (left circuit), see details in [19]. Its dynamics reads as [19, Equation (6.8)]:

$$ egin{cases} (a) \dot{x}_1(t) = \frac{-1}{RC}x_1(t) - \frac{1}{C}x_2(t) - \frac{1}{C}z(t) + \frac{1}{C}\lambda_1(t) \\ (b) \dot{x}_2(t) = \frac{1}{L}x_1(t) + \frac{u_1(t)}{L} \\ (c) 0 = \lambda_2(t) + u_c(t) + x_1(t) \\ (d) K^* \ni \lambda(t) \perp w(t) = \begin{pmatrix} 1 \\ 0 \end{pmatrix} x_1(t) + \begin{pmatrix} 0 \\ 1 \end{pmatrix} z(t) \in K, \end{cases} \quad (73) $$

with $u(t) = u_c(t)$, $x_1 = u_D = e(t)$, $x_2 = i_1$, $z = i_5$, $\lambda_1 = i_4$, $\lambda_2 = u - u_c$, $K = \mathbb{R}^+ \times \mathbb{R} \Rightarrow K^* = \mathbb{R}^+ \times \{0\}$. We have $B_I = \begin{pmatrix} \frac{-1}{C} \\ 0 \end{pmatrix}$, $B_{II} = 0$, $A_{II} = (1 \ 0)$, $\bar{B}_I = \begin{pmatrix} \frac{1}{C} & 0 \\ 0 & 0 \end{pmatrix}$, $\bar{B}_{II} = (0 \ 1)$, $D = 0$, $C_{II} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $C_I = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $P_I = I_2$, $n_1 = m = 2$, $p = n_2 = 1$, $S_x(t, z(t^{-})) = (\mathbb{R}_+ \times \mathbb{R}) \setminus \begin{pmatrix} 0 \\ z(t^{-}) \end{pmatrix}$, hence $\{(0,0)\} \in S_x(t, z(t^{-}))$. Both pairs $(\bar{B}_{II}, C_{II})$ and $(\bar{B}_I, C_I)$ have the X-property. Moreover $-K^* \subseteq \text{Ker}(\bar{B}_{II}) = \{(x, y) \in \mathbb{R}^2 | y = 0\}$. Since $D = 0$ we have $\lambda^{im} = 0$ and we can choose $M = 0$ in item 2 of Proposition 24, and Assumption 21 is satisfied. Thus items 1 and 2 of Proposition 24 apply. Proposition 33 does not apply because (iv) does not hold since $B_{II} = 0$. Condition (48) holds true. Let us check (58). We have $\text{Ker}(\bar{B}_I) = \{\eta \in \mathbb{R}^2 | \eta_1 = 0\}$, so it suffices to check if some vectors $(0 \ \lambda_2)^\top$ can solve the LCCP (73) (d). Since $\lambda_2 = 0$, any $z(t)$ can solve the second line of the LCCP. The first line is $0 \le \lambda_1 \perp x_1 \ge 0$. If $x_1(t) < 0$ then $\text{SOL}(\text{LCCP}) = \emptyset$, so (58) does not hold and $x_1(\cdot)$ has to jump (see below conditions on this). If $x_1(t) = 0$ then any $\lambda_1 \ge 0$ is admissible, if $x_1(t) > 0$ then $\lambda_1 = 0$, and (58) holds so that continuous states at $t$ solve the GE (49).

We have $A_z = B_{II} = 0$, $B_z = -(\frac{1}{C} \ 1)$, $C_z = C_{II} = (0 \ 1)^\top$, $D_z = \begin{pmatrix} \frac{1}{C} & 0 \\ 0 & 0 \end{pmatrix} \succ 0$, $f_z(t, x(t^{-}), \lambda(t^{-})) = -\lambda_2(t^{-})$, $S_z(t, x(t^{-})) = K - C_I x(t^{-}) - F(t^{+}) = (\mathbb{R}_+ \times \mathbb{R}) - \begin{pmatrix} x_1(t^{-}) \\ 0 \end{pmatrix} = \begin{pmatrix} \mathbb{R}_+ - x_1(t^{-}) \\ \mathbb{R} \end{pmatrix}$, $E_{II} = u_c$. Therefore GE (18) is (recall that $\lambda_2(t^{-}) = 0$):

$$ 0 \in B_z(D_z + \partial\sigma_{S_z(t,x(t^{-}))}^{-1}(C_z z(t^{+}))) - A_{II}\bar{B}_{II}((D_z + \partial\sigma_{S_z(t,x(t^{-}))}^{-1}(C_z z(t^{+}))) \cap \text{Ker}(\bar{B}_{II})), \quad (74) $$

Clearly item 4 in Proposition 9 does not apply because $M$ in item (v) does not exist. Since $\text{rank}(\bar{B}_{II}) = 1$, Assumption 7 is not satisfied and Corollary 12 does not ap-
ply neither. However since K has a simple form, it is possible in this case to analyse LCCP(Dz, K, Cz z(t+) + C Ix(t−) + F(t+)), since the GE (74) is equivalent to the existence of η and ζ such that Bzη − A$_{II}$ ¯ B$_{II}$ ζ = 0, with −η ∈ SOL(LCCP(Dz, K, Cz z(t+) + C Ix(t−) + F(t+)) and −ζ ∈ SOL(LCCP(Dz, K, Cz z(t+) + C Ix(t−) + F(t+)) ∩ Ker( ¯ B$_{II}$ )), i.e., −ζ ∈ SOL(LCCP(Dz, K, Cz z(t+) + C Ix(t−) + F(t+)) and ζ$_{2}$ = 0. Thus we have R$_{+}$ × { 0 } ∈ ( − η$_{1}$ − η$_{2}$ ) ⊥ ( − 1 C η$_{1}$ 0 ) + ( 0 z ( t $^{+}$) ) + ( x$_{1}$ ( t − ) 0 ) ∈ R$_{+}$ × R , and same for − ζ . Similarly as above we infer that η$_{2}$ = ζ$_{2}$ = 0, z ( t $^{+}$) ∈ R (hence ζ ∈ Ker( ¯ B$_{II}$ ) holds). If x$_{1}$ ( t − ) ≥ 0 then η$_{1}$ = ζ$_{1}$ = 0 and we infer that any z ( t $^{+}$) solves (74) hence (18). Thus we are in a similar situation as Example 19 where (18) does not enjoy uniqueness of solutions. We may however infer the following by inspection, on this particular example: since λ$_{2}$ ( t ) = 0 for all t ≥ 0, x$_{1}$ ( t ) = − u$_{c}$ ( t ), so x$_{2}$ ( t ) is calculated from (73) (b). Thus − z ( t ) + λ$_{1}$ ( t ) = C ˙ x$_{1}$ ( t ) + 1 $_{R}$x$_{1}$ ( t ) + x$_{2}$ ( t ) is given from (73) (a). The existence of jumps in λ$_{1}$ ( · ) and z ( · ) depends on u$_{c}$ ( · ) and u$_{1}$ ( · ). Since we have 0 ≤ λ$_{1}$ ( t ) ⊥ u$_{c}$ ( t ) ≥ 0, it is clear that (73) has global solutions only if u$_{c}$ ( t ) ≥ 0 for all t ≥ 0. Thus if u$_{c}$ (0 − ) = x$_{1}$ (0 − ) < 0, a jump occurs initially.

Remark 4. A remarkable feature of DALCS as illustrated by this circuit, is that existence of solutions may depend directly on exogeneous inputs, due to interplay between equality and complementarity constraints.

4.2 State-dependent-switch DAE

Nonsmooth and switching DAEs are an area of interest [14, 77, 83, 80, 71]. Particular classes of such nonsmooth systems can be studied within the above framework. As noted in [19], since D ≻ 0, (44) is a DAE with nonsmooth Lipschitz continuous vector field and constraint, of the form:

$$
\left\{
\begin{array}{l}
\dot{x}_1 = -2x_1 + PWL(x_1, z) \\
\dot{x}_2 = z \\
0 = x_2 + PWL(x_1, z),
\end{array}
\right.
\tag{75}
$$

where PWL ( · ) generically denotes a piecewise-linear mapping, and we note that x$_{2}$ in (44) (b) can be used in (44) (c) to make λ independent of x$_{2}$ , changing D ≻ 0 to a different feedthrough matrix. Let us consider another example:

$$
\left\{
\begin{array}{l}
\dot{x}_1 = a_1 x_1 + b_1 x_2 + c_1 z + E_{I1}(t) \\
\dot{x}_2 = a_2 x_1 + b_2 x_2 + c_2 z + E_{I2}(t) \\
0 = a_3 x_1 + b_3 x_2 + c_3 z + E_{II,2}(t) & \text{if } x_1 < 0 \\
0 = a_4 x_1 + b_4 x_2 + c_4 z + E_{II,2}(t) & \text{if } x_1 > 0 \\
0 = a_5 x_1 + a_6 x_2 + c_4 z + E_{II,1}(t),
\end{array}
\right.
\tag{76}
$$

for i = 1 , 2, where x$_{1}$ , x$_{2}$ are the differential state variables, z is the algebraic state variable. (76) is rewritten equivalently as the DALCS:

$$
\left\{
\begin{array}{l}
\dot{x}_1 = a_1 x_1 + b_1 x_2 + c_1 z + E_{I1}(t) \\
\dot{x}_2 = a_2 x_1 + b_2 x_2 + c_2 z + E_{I2}(t) \\
0 = a_5 x_1 + a_6 x_2 + c_4 z + E_{II,1}(t) \\
0 = -a_3 \lambda_1 + a_4 \lambda_2 + b_3 x_2 + c_3 z + E_{II,2}(t) \\
0 \le \lambda_1 \perp \lambda_1 + x_1 \ge 0 \\
0 \le \lambda_2 \perp \lambda_2 - x_1 \ge 0.
\end{array}
\right.
\tag{77}
$$
We have $n_1 = 2, n_2 = 1, m = 2, p = 2, D = I_2 > 0, A_I = \begin{pmatrix} A_1 & b_1 \\ a_2 & b_2 \end{pmatrix}, B_I = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}, \bar{B}_I = 0,$
$C_I = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix}, A_{II} = \begin{pmatrix} a_5 & a_6 \\ 0 & b_3 \end{pmatrix}, B_{II} = \begin{pmatrix} c_4 \\ c_3 \end{pmatrix}, C_{II} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \bar{B}_{II} = \begin{pmatrix} 0 & 0 \\ a_3 & a_4 \end{pmatrix}, F(\cdot) = 0.$
Thus $A_z = \begin{pmatrix} c_4 \\ c_3 \end{pmatrix}, B_z = \begin{pmatrix} 0 & 0 \\ a_3 & a_4 \end{pmatrix}, C_z = 0, D_z = I_2, S_z(t, x(t)) = \mathbb{R}_+^2 - \begin{pmatrix} x_1(t^-) \\ -x_1(t^-) \end{pmatrix},$
$f_z(t, z(t^-), \lambda(t^-)) = - \begin{pmatrix} c_4 \\ c_3 \end{pmatrix} z(t^-) + E_{II}(t^+) - E_{II}(t^-) - \begin{pmatrix} 0 & 0 \\ a_3 & a_4 \end{pmatrix} \begin{pmatrix} \lambda_1(t^-) \\ \lambda_2(t^-) \end{pmatrix}.$

From Proposition 20 item 1, $x(\cdot)$ is continuous at $t$, though condition (48) holds true. Since $C_z = 0$, Lemma 16 and Proposition 17 may apply, as we check now. The nonemptyness condition in item 2 of Lemma 16 is satisfied since $Q_{D_z}^* = \mathbb{R}^2$. Assume $c_3 \neq 0$ or $c_4 \neq 0$ so that $\text{rank}(A_z) = 1$. Assume also that $0 \in S_z(t, x(t)) \Rightarrow x_1(t^-) = 0$. From Lemma 16 item 1 (i) and (iv) (and using the comment just after the proof of the Lemma) we have $z(t^+) = z(t^-) + (c_3^2+c_4^2)^{-1}(c_4 c_3) \left( -\begin{pmatrix} E_{II,1}(t^+) - E_{II,1}(t^-) \\ E_{II,2}(t^+) - E_{II,2}(t^-) \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ a_3 & a_4 \end{pmatrix} \lambda(t^-) \right) + \begin{pmatrix} 0 & 0 \\ a_3 & a_4 \end{pmatrix} \text{SOL}(\text{LCP}(I_2, \begin{pmatrix} x_1(t^-) \\ -x_1(t^-) \end{pmatrix})) = z(t^-) - (c_3^2+c_4^2)^{-1}(c_4 c_3) \begin{pmatrix} E_{II,1}(t^+) - E_{II,1}(t^-) \\ E_{II,2}(t^+) - E_{II,2}(t^-) \end{pmatrix}$, since $\lambda(t^-) = \text{SOL}(\text{LCP}(I_2, (x_1(t) - x_1(t))^\top)) = 0$. Thus we are characterizing jumps in $z(\cdot)$ when the switches in (76) occur at $x_1 = 0$, as expected they are solely due to $E_{II}(\cdot)$. System (76) can be made more complex with switching conditions as $0 = a_3x_1+b_3x_2+c_3z+E_{II,2}(t)$ if $\eta x_1 + \kappa x_2 < 0$ and $0 = a_4x_1+b_4x_2+c_3z+E_{II,2}(t)$ if $\eta x_1 + \kappa x_2 > 0$, represented by complementarity conditions $0 \le \lambda_1 \perp \lambda_1 + \eta x_1 + \kappa x_2 \ge 0$, $0 \le \lambda_2 \perp \lambda_2 - \eta x_1 - \kappa x_2 \ge 0$, and equality $0 = \alpha\lambda_1 + \beta\lambda_2 + c_3z + E_{II,2} + \gamma x_1 + \delta x_2$. Constraints have to be imposed on $\alpha, \beta, \gamma, \delta, \kappa$ and $\eta$.

## 4.3 Descriptor-Variable Systems with Relay Feedback

Let us consider Lur'e set-valued systems of the form:

$$ \begin{cases} P\dot{\xi}(t) = A\xi(t) + Bu(t) + f(t) \\ y(t) = C\xi(t) + Gu(t) + g(t) \\ u(t) \in -\text{Sgn}(y(t)), \end{cases} \quad (78) $$

with $\xi(t) \in \mathbb{R}^n, A \in \mathbb{R}^{n \times n}, u(t) \in \mathbb{R}^p, y(t) \in \mathbb{R}^p, C \in \mathbb{R}^{p \times n}, G \in \mathbb{R}^{p \times p}, f(\cdot)$ and $g(\cdot)$ are functions with enough regularity, for any $z \in \mathbb{R}^p$: $\text{Sgn}(z) = (\text{sgn}(z_1), \dots, \text{sgn}(z_p))^\top$, and $\text{rank}(P) < n$. Following [23, 20], (78) is equivalent to:

$$ \begin{cases} P\dot{\xi}(t) = A\xi(t) + f(t) + Be + (-2B \ 0)\lambda \\ 0 \le \begin{pmatrix} \lambda_1(t) \\ \lambda_2(t) \end{pmatrix} \perp \begin{pmatrix} -C\xi(t) \\ 0 \end{pmatrix} + \begin{pmatrix} g(t) - Ge \\ e \end{pmatrix} + \begin{pmatrix} 2G & I_p \\ -I_p & 0 \end{pmatrix} \begin{pmatrix} \lambda_1(t) \\ \lambda_2(t) \end{pmatrix} \ge 0, \end{cases} \quad (79) $$

$e = (1, 1, \dots, 1)^\top \in \mathbb{R}^p, \lambda_1 \in \mathbb{R}^p, \lambda_2 \in \mathbb{R}^p$. The Weiertsrass form of a descriptor-variable system after state space transformation [35] yields us to consider a transformed system with $P = \text{diag}(I_{n_1}, 0)$, $\zeta = (\zeta_1^\top \ \zeta_2^\top)^\top$, $\lambda = (\lambda_1^\top \ \lambda_2^\top)^\top$, and:

$$ \begin{cases} \dot{\zeta}_1(t) = A_1\zeta_1(t) + B_1\lambda(t) + E_1(t) \\ N\dot{\zeta}_2(t) = \zeta_2(t) + B_2\lambda(t) + E_2(t) \\ 0 \le \lambda(t) \perp D\lambda(t) + C_\zeta\zeta(t) + F_\zeta(t) \ge 0, \end{cases} \quad (80) $$
where N is a nilpotent matrix, dimensions of $\zeta_1$ and $\zeta_2$ depend on the degree of the polynomial $\det(sP - A)$. If $N = 0$ (nilpotency degree 1, impulse-free descriptor-variable system) then $z = \zeta_2$ and $x = \zeta_1$. If $N = \begin{pmatrix} 0 & I \\ 0 & 0 \end{pmatrix}$ (nilpotency degree 2, in this section I generically denotes identity matrices with appropriate dimensions) then we obtain $\dot{\zeta}_{22} = \zeta_{21} + B_{21}\lambda + E_{21}(t)$ and $0 = \zeta_{22} + B_{22}\lambda + E_{22}(t)$, and $z = \zeta_{21}$, $x = (\zeta_1^\top \ \zeta_{22}^\top)^\top$. In the second case we may rewrite (78) equivalently as the DALCS:

$$ \left\{ \begin{aligned} \dot{\zeta}_1(t) &= A_1\zeta_1(t) + B_1\lambda(t) + E_1(t) \\ \dot{\zeta}_{22}(t) &= z(t) + B_{21}\lambda(t) + E_{21}(t) \\ 0 &= \zeta_{22}(t) + B_{22}\lambda(t) + E_{22}(t) \\ 0 &\le \lambda(t) \perp D\lambda(t) + C_I x(t) + C_{II} z(t) + F_\zeta(t) \ge 0 \end{aligned} \right. \qquad (81) $$

If $N = \begin{pmatrix} 0 & I & I \\ 0 & 0 & I \\ 0 & 0 & 0 \end{pmatrix}$ (nilpotency degree 3), then we obtain $z = \zeta_{21}$, $x = (\zeta_1^\top \ \zeta_{22}^\top \ \zeta_{23}^\top)^\top$, and the DALCS:

$$ \left\{ \begin{aligned} \dot{\zeta}_1(t) &= A_1\zeta_1(t) + B_1\lambda(t) + E_1(t) \\ \dot{\zeta}_{22} &= z(t) - \zeta_{22}(t) + (B_{21} - B_{22})\lambda(t) + E_{21}(t) - E_{22}(t) \\ \dot{\zeta}_{23} &= \zeta_{22}(t) + B_{22}\lambda(t) + E_{22}(t) \\ 0 &= \zeta_{23}(t) + B_{23}\lambda(t) + E_{23}(t) \\ 0 &\le \lambda(t) \perp D\lambda(t) + C_I x(t) + C_{II} z(t) + F_\zeta(t) \ge 0, \end{aligned} \right. \qquad (82) $$

and so on with matrices N of increasing nilpotency degree. We have $A_I = \begin{pmatrix} A_1 & 0 & 0 \\ 0 & I & 0 \\ 0 & I & 0 \end{pmatrix}$,

$$ B_I = \begin{pmatrix} 0 \\ I \\ 0 \end{pmatrix}, \quad \bar{B}_I = \begin{pmatrix} B_1 \\ B_{21} - B_{22} \\ B_{23} \end{pmatrix}, \quad A_{II} = (0 \ 0 \ I), \quad B_{II} = 0, \quad \bar{B}_{II} = B_{23}, \quad F_\zeta(t) = \begin{pmatrix} g(t) \\ \mathbf{e} \end{pmatrix}, $$

$$ E_I(t) = \begin{pmatrix} E_1(t) \\ E_{21}(t) - E_{22}(t) \\ E_{22}(t) \end{pmatrix}, \quad E_{II}(t) = E_{23}(t). \quad \text{Matrices } C_I \text{ and } C_{II} \text{ need the state space} $$

change to be expressed from (78). One feature of systems (81) and (82) is that $A_z (= B_{II}) = 0$. It is only in the case of nilpotency degree 1 ($N = 0$ in (80)) that $B_{II} = I$. In this case $z$ can be eliminated from the dynamics, however we are interested in determining possible jumps in $\xi$, hence jumps in $z$ are worth studying. For (80) with $N = 0$ we have $\mathcal{S} = \{0\}$ in (22). If the required conditions are satisfied (among these, $(-B_2, C_{II})$ has the X-property), then items 3 or 4 in Proposition 9 can be used to study solvability of the GE (19). Condition (27) gives under Assumption 7 (i.e., $\text{Ker}(B_2) = \{0\}$) the equivalent condition: $\{0\} \in \text{SOL}(\text{LCP}(D+C_I B_1, C_{II} z(t^-) + C_I x(t^-) + F_\zeta(t^+)))$. If this is not satisfied then $z(\cdot)$ must jump at $t$. This reveals conditions under which impulse-free state-jumps occur in impulse-free descriptor-variable systems coupled to relay feedback.

Systems (13) and (44) can be cast into (81). These examples show how non-impulsive state jumps can be triggered in descriptor-variable systems with complementarity constraints. Systems with passive descriptor-variable quintuplet $(P, A, B, C, D)$ and minimal realization [19, Theorem G.1] can be recast in (81), and possess specific properties [24, 41, 28].
## 4.4 Mechanical Systems with Unilateral Springs and Bilateral Constraints

Let us consider a Lagrangian system submitted to holonomic constraints and unilateral springs. We analyse systems with all linear ingredients only, to fit with (2). The dynamics reads as:

$$ egin{cases} (a) M\ddot{q}(t) + K_v\dot{q}(t) + K_p q(t) = \tau(t) + H_s^\top\lambda(t) + H_b^\top\lambda_b(t) \\ (b) 0 = H_b q(t) + f(t) \\ (c) 0 \le \lambda(t) \perp \lambda(t) + C_s q(t) + F_s(t) \ge 0, \end{cases} \quad (83) $$

where $M = M^\top > 0$, $x = (q^\top \dot{q}^\top)^\top$, $q(t) \in \mathbb{R}^{n_1/2}$, $z = \lambda_b \in \mathbb{R}^p$, $\lambda \in \mathbb{R}^m$, $H_b \in \mathbb{R}^{p \times n_1/2}$, $C_s \in \mathbb{R}^{m \times n_1/2}$, $f: \mathbb{R}_+ \to \mathbb{R}^{n_2}$, $F_s: \mathbb{R}_+ \to \mathbb{R}^m$, $\tau(\cdot)$ is a control input. Here $C_s = KH_s$ where $K = \text{diag}(k_i) \in \mathbb{R}^{m \times m}$ is a stiffness matrix, $k_i \ge 0$, $H_s = \text{col}(H_{s,i}) \in \mathbb{R}^{m \times n_1/2}$, $1 \le i \le m$, relates local indentations to the generalized coordinate $q$. Passivity properties as in Definition 1 cannot hold because passivity is a property relating velocities and forces, while the constraints in (83) (b) are in the position $q$. Let us replace (83) (b) by its second derivative:

$$ (b') 0 = H_b M^{-1} H_b^\top \lambda_b(t) + H_b M^{-1} (-K_v \dot{q}(t) - K_p q(t) + \tau(t) + H_s^\top \lambda(t)) + \ddot{f}(t), \quad (84) $$

with $0 = H_b q(0) + f(0)$ and $0 = H_b \dot{q}(0) + \dot{f}(0)$. Then (83) (a) (c) and (84) fit with (2), with: $x = (q^\top \dot{q}^\top)^\top$, $z = \lambda_b$, $A_I = \begin{pmatrix} 0 & I_{n_1/2} \\ -M^{-1}K_p & -M^{-1}K_v \end{pmatrix}$, $B_I = \begin{pmatrix} 0 \\ M^{-1}H_b^\top \end{pmatrix}$, $\bar{B}_I = \begin{pmatrix} 0 \\ M^{-1}H_s^\top \end{pmatrix}$, $A_{II} = (-H_b M^{-1} K_p - H_b M^{-1} K_v)$, $B_{II} = H_b M^{-1} H_b^\top$, $\bar{B}_{II} = H_b M^{-1} H_s^\top$, $C_I = (C_s 0)$, $C_{II} = 0$, $D = I_m$, $E_{II}(t) = H_b M^{-1} \tau(\cdot) + \ddot{f}(\cdot)$. We have $D_z = I_m$, $A_z = H_b M^{-1} H_s^\top$, $B_z = H_b M^{-1} (K_v M^{-1} - I_{n_1/2}) H_s^\top$, $C_z = 0$, $S_z(t, x(t^{-})) = \mathbb{R}_+^m - C_s q(t) - F_s(t^+)$, $f_z(t, x(t^{-}), \lambda(t^{-})) = -H_b M^{-1} H_b^\top z(t^{-}) + E_{II}(t^+) - E_{II}(t^{-}) - H_b M^{-1} H_s^\top \lambda(t^{-})$, where $\lambda(t^{-}) = \text{proj}[\mathbb{R}_+^m; -C_s q(t) - F_s(t^{-})]$ using (94) (95). Thus (19) is equal to:

$$ egin{aligned} 0 \in & H_b M^{-1} H_s^\top z(t^+) - H_b M^{-1} H_b^\top z(t^-) + E_{II}(t^+) - E_{II}(t^-) \\ & - H_b M^{-1} H_s^\top \text{proj}[\mathbb{R}_+^m; -C_s q(t) - F_s(t^-)] \\ & + H_b M^{-1} (K_v M^{-1} - I_{n_1/2}) H_s^\top (I_{n_1/2} + \partial \sigma_{\mathbb{R}_+^m - C_s q(t) - F_s(t^+)}^{-1}(0)), \end{aligned} \quad (85) $$

This class of mechanical systems (83) (a) (c) and (b') in (84), belongs to DALCS (2) only if initial data are chosen correctly, and if equality (84) is satisfied dt-almost everywhere. This implies regularity constraints on both variables $y(t) = H_b q(t) + f(t)$, and $z(t) = H_b \dot{q}(t) + \dot{f}(t)$ which has to be differentiable dt-almost everywhere. From (83) (c), $\lambda(\cdot)$ is a projection, the right-hand side of (83) (a) is Lipschitz continuous in $q$, hence under mild requirement on exogeneous data, $x(\cdot)$ is AC, i.e., $\dot{q}(\cdot)$ is AC and $q(\cdot)$ is differentiable. The next question is whether or not $z(\cdot)$ is continuous. Clearly if bilateral constraints are independent (in the kinetic metric) then this directly depends on continuity of $E_{II}(\cdot)$. Since $C_{II} = 0$, item 1 (i) (iv) in Lemma 16 applies. Proposition 17 can also be used, provided that $\text{rank}(B_z) = m$, which constrains $H_s$ and $H_b$.
## 4.5 Backward Euler Discretization

Let us give few details on backward Euler discretization of (2). It takes the form:

$$ 
\left\{ 
\begin{array}{l}
(a) \ P_I(x_{k+1} - x_k) = \bar{B}_I h \lambda_{k+1} + h(A_I x_k + B_I z_k + E_{I,k+1}) \\
(b) \ 0 = A_{II} x_k + B_{II} z_k + \bar{B}_{II} \lambda_k + E_{II,k} \\
(c) \ 0 = A_{II} x_{k+1} + B_{II} z_{k+1} + \bar{B}_{II} \lambda_{k+1} + E_{II,k+1} \\
(d) \ \lambda_{k+1} \in -(D + \partial \sigma_{K-F_{k+1}})^{-1}(C_I x_{k+1} + C_{II} z_{k+1}) \\
(e) \ K^* \ni \lambda_{k+1} \perp w_{k+1} = C_I x_{k+1} + C_{II} z_{k+1} + D \lambda_{k+1} + F_{k+1} \in K,
\end{array}
\right. 
\qquad (86) 
$$

with $h > 0$ the timestep, $x_{k+1} \approx x(t^+)$, $x_k \approx x(t^-)$, $\lambda_k \approx \lambda(t^-)$, $\lambda_{k+1} \approx \lambda(t^+)$. Notice that $(e) \Leftrightarrow (d)$. The structure of (86) is similar to that of (8), thus $z_{k+1}$ satisfies a GE similar to (19):

$$ 0 \in B_{II} z_{k+1} + f_{z,k} + B_z (D_z + \partial \sigma_{S_{z,k}})^{-1} (C_{II} z_{k+1}), \qquad (87) $$

with $D_z = hC_I P_I^{-1} \bar{B}_I + D$, $S_{z,k} = K - F_{k+1} - C_I x_k - hC_I(A_I x_k + B_{II} z_k + E_{I,k+1})$, $B_z = -hA_{II} P_I^{-1} \bar{B}_I - \bar{B}_{II}$. The results in section 3.2 can be used to analyse the GE (87). When $x(\cdot)$ jumps, $v_t \neq 0$ and $\lambda_{k+1}$ should approximate a Dirac measure as $h \to 0$, which is impossible. Thus it is the bounded “impulse” $\nu_{k+1} = h\lambda_{k+1}$ that has to be computed, with $\nu_{k+1} \approx v_t$. Let $D = 0$, then (86) (a) becomes $P_I(x_{k+1} - x_k) = \bar{B}_I \nu_{k+1} + h(A_I x_k + B_I z_k + E_{I,k+1})$ and (86) (d) becomes $\nu_{k+1} \in -\mathcal{N}_{K-F_{k+1}}(C_I x_{k+1} + C_{II} z_{k+1})$ since the right-hand side is a cone. When $D \neq 0$, the term $\frac{D}{h}$ appears in (86) (d) and (e): this is numerically sound as long as $h > 0$ is not too small. In case of LCS this allows to consistently approximate the jump rule and the impulse [46, Section 4.2].

# 5 Conclusions and Perspectives

This article deals with the characterization of state jumps in a class of singular nonsmooth set-valued systems, named differential-algebraic linear complementarity systems (DALCS). This class of systems extends LCS and DAE. Using Convex Analysis, Complementarity Theory and Variational Inequalities allows us to derive conditions for existence or non-existence of state jumps in both the differential and the algebraic state variable. Impulse-free jumps are shown to exist in DALCS. Several detailed examples (circuits, switching DAE, set-valued descriptor-variable systems) illustrate theoretical findings. Open problems and extensions are numerous. A first extension is to replace complementarity conditions in (2) (c) by an inclusion $\lambda(t) \in -\mathcal{M}(w(t))$, $\mathcal{M}(\cdot)$ a maximal monotone operator. A sound extension of both relative degree (for LCS) and index (for DAE) is lacking (the passivity conditions in [13, 14] can be seen as sufficient conditions for relative degree and index equal to 0 or 1, since they imply no state jumps). Nonlinear ingredients could be introduced (they may occur in circuits with nonlinear components), the central question being: where? A motivation may lie in nonlinear circuits as in Fig. 5 and 6. In these circuits the inductance is assumed to be nonlinear $L(i_3) \Rightarrow v_L = L(i_3) \frac{di_3}{dt}$ [34, Section III] [61, Equ. (4)] [42, Section 4], the charge stored in the capacitor satisfies $q_c = g(v_c)$, and the resistor satisfies $v_R = f(i_2)$ for some functions $L(i_3)$, $g(v_c)$ and $f(i_2)$, see [34, 70] for examples of nonlinearities. Consider the circuit in Fig. 5 (left). Its dynamics reads as
(see Appendix C):

$$
\left\{
\begin{array}{l}
(a) \ L(i_3(t)) \frac{di_3}{dt}(t) = v_{d1}(t) - v_c(t) - u_1(t) \\
(b) \ f(i_3(t) - \dot{q}_c(t)) = v_c(t) - v_{d1}(t) - v_{d2}(t) + u_1(t) - u_2(t) \\
(c) \ 0 = q_c(t) - g(v_c(t)) \\
(d) \ 0 \le v_{d1}(t) \perp w_1(t) = i_1(t) = \dot{q}_c(t) \ge 0 \\
(e) \ 0 \le v_{d2}(t) \perp w_2(t) = -i_2(t) = \dot{q}_c(t) - i_3(t) \ge 0,
\end{array}
\right.
\qquad (88)
$$

hence we have $x = (i_3 q_c)^\top$, $z = v_c$, $\lambda_1 = v_{d1}$, $\lambda_2 = v_{d2}$. If the resistor characteris-

<figure><img src="image_9.png" /><figcaption>Figure 5: Nonlinear RLC circuit with ideal diodes.</figcaption></figure>

tic is invertible, then $\dot{q}_c(t) = -f^{-1}(v_c(t) - v_{d1}(t) - v_{d2}(t) + u_1(t) - u_2(t)) + i_3(t)$: the complementarity system (88) is a differential algebraic nonlinear complementarity system (DANLCS) with nonlinear terms in all its ingredients. Notice that solving for $v_c$ in (88) (c) and substituting in the right-hand side of (88) (a) (b) and in (88) (d) (e) may not be wanted or even possible since the solution may be nonlinear or even nonunique with voltage-controlled capacitors and $g(v_c) = \frac{1}{3}v_c^3 - 3v_c^2 + 8v_c$ [34, Equ. (2.19)]. Nevertheless if substitution is doable then (88) (c) can be eliminated and (88) becomes a nonlinear LCS with nonlinearities inside complementarity, the study of which has not been much tackled, see [21] for one instance. One example of a nonlinear DAFOSwP is depicted in Fig. 5 (right), whose dynamics reads as (see Appendix C):

$$
\begin{array}{l@{\qquad}l@{\qquad}l}
\dot{q}_c(t) = i_2(t) + i(t) & \dot{q}_c(t) \in i(t) - \mathcal{N}_{\phi(t)}(v_c(t)) & \dot{x}(t) \in i(t) - \mathcal{N}_{\phi(t)}(z(t)) \\
0 = q_c(t) - g(v_c(t)) & 0 = q_c(t) - g(v_c(t)) & 0 = x(t) - g(z(t)) \\
0 \le -i_2(t) \perp v_c(t) + u(t) \ge 0 & &
\end{array}
\tag{89}
$$

where we have $x = q_c$, $-i_2 = \lambda$, $v_c = z$, $\phi(t) = \{\zeta | \zeta + u(t) \ge 0\}$. The system (89) is a DAFOSwP. Let us consider the circuit depicted in Fig. 6. Its dynamics reads as:

$$
\left\{
\begin{array}{l}
L(i_3) \frac{di_3}{dt} = v_c + u_2(t) \\
\dot{q}_c = i_1 - i_3 \\
0 = q_c - g(v_c) \\
0 \le i_1 \perp v_d \ge 0 \\
0 = v_c - v_d + u_1
\end{array}
\right.
\quad \Leftrightarrow \quad
\begin{pmatrix} L(x_1) & 0 \\ 0 & 1 \end{pmatrix} \dot{x} = \begin{pmatrix} z + u_2 \\ \lambda - x_1 \end{pmatrix} \in \begin{pmatrix} z + u_2 \\ -x_1 \end{pmatrix} - \mathcal{N}_{\phi(t)}(z),
\qquad (90)
$$

where $x = (i_3 q_c)^\top = (x_1 x_2)^\top$, $z = v_c$, $\lambda = i_1$, $\phi(t) = \{\zeta | \zeta + u_1(t) \ge 0\}$. Once again we obtain a DAFOSwP (compare to (7)).
<figure><img src="image_10.png" /><figcaption>Figure 6: Nonlinear RLC circuit with ideal diode.</figcaption></figure>

When no ideal diode is present, nonlinear circuits as in (88) and (89) are known to exhibit
impasse points. How does the presence of complementarity constraints modify this kind
of behavior? Systems with higher degree distributional solutions, which occur easily in
DAE and in LCS, could be considered. Systems with time-varying parameters may also
be considered (which may occur in circuits with time-varying components). Existence of
solutions is also an important objective, to prove under which conditions Assumptions 1,
2 and 3 hold true.

A Well-posedness of Variational Inequalities

The next proposition stems from [2, Theorem 3, Corollaries 3 and 4]. It concerns variational inequalities (VIs) of the form: Find $u \in \mathbb{R}^n$ such that

$$
\langle \mathbf{M}u + \mathbf{q}, v - u \rangle + \varphi(v) - \varphi(u) \geq 0, \text{ for all } v \in \mathbb{R}^n \quad (91)
$$

where $\mathbf{M} \in \mathbb{R}^{n \times n}$ is a real matrix, $\mathbf{q} \in \mathbb{R}^n$ a vector and $\varphi : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ a proper convex and LSC function. The VI in (91) is equivalent to the inclusion $\mathbf{M}u + \mathbf{q} \in -\partial\varphi(u) \Leftrightarrow u \in (\mathbf{M} + \partial\varphi)^{-1}(-\mathbf{q})$. Writing it as $0 \in \mathbf{M}u + \mathbf{q} + \partial\varphi(u)$ it fits with (1).

The problem in (91) is denoted as VI($\mathbf{M}$, $\mathbf{q}$, $\varphi$) in the next proposition. We also set:

$$
\mathcal{H}(\mathbf{M}, \varphi) = \{x \in \mathbb{R}^n | \mathbf{M}x \in (\mathrm{dom}(\varphi_\infty))^*\}. \quad (92)
$$

Note that $(\mathrm{dom}(\varphi_\infty))^*$ is the dual cone of the domain of the recession function $\varphi_\infty$ while $(\mathrm{dom}(\varphi))^\infty$ (that we may denote also as $\mathrm{dom}(\varphi)^\infty$) is the recession cone of $\mathrm{dom}(\varphi)$.

**Proposition 34.** [2] Let $\varphi : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ be a proper, convex and LSC function with closed domain, $\mathbf{M} \in \mathbb{R}^{n \times n}$, and suppose that $\mathbf{M} \succcurlyeq 0$ (not necessarily symmetric).

a) If $(\mathrm{dom}(\varphi))^\infty \cap \ker\{\mathbf{M} + \mathbf{M}^\top\} \cap \mathcal{H}(\mathbf{M}, \varphi) = \{0\}$ then for each $\mathbf{q} \in \mathbb{R}^n$, problem VI$(\mathbf{M}, \mathbf{q}, \varphi)$ has at least one solution.

b) Suppose that $(\mathrm{dom}(\varphi))^\infty \cap \ker\{\mathbf{M}+\mathbf{M}^\top\} \cap \mathcal{H}(\mathbf{M}, \varphi) \neq \{0\}$. If there exists $x_0 \in \mathrm{dom}(\varphi)$ such that

$$
\langle \mathbf{q} - \mathbf{M}^{\top} x_0, v \rangle + \varphi_{\infty}(v) > 0, \forall v \in (\mathrm{dom}(\varphi))_{\infty} \cap \ker\{\mathbf{M} + \mathbf{M}^{\top}\} \cap \mathcal{H}(\mathbf{M}, \varphi), v \neq 0, (93)
$$
then problem VI(M, q, φ) has at least one solution.

b') If M = M$^{⊤}$ then one can take x₀ = 0 in b).

c) If u₁ and u₂ denote two solutions of problem VI(M, q, φ) then u₁ − u₂ ∈ ker{M+M$^{⊤}$}.

d) If M = M$^{⊤}$, then u is a solution of VI(M, q, φ) if and only if it is a solution of the optimization problem $\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top M x + \langle q, x \rangle + \varphi(x)$.

B Useful Convex Analysis and Complementarity
Theory tools

Let K be a nonempty closed convex cone, then:

$$
K^* \ni x \perp y \in K \Leftrightarrow x \in -\mathcal{N}_K(y) \Leftrightarrow y \in -\mathcal{N}_{K^*}(x). \quad (94)
$$

Let $M = M^{\top} \succ 0$, $x$ and $y$ two vectors, then [40]

$$
M(x - y) \in -\mathcal{N}_K(x) \Leftrightarrow x = \operatorname{proj}_M[K; y] \Leftrightarrow x = \operatorname*{arg\,min}_{z \in K} \frac{1}{2} (z - y)^{\top} M (z - y). \quad (95)
$$

We note that this is a particular case of (91), so that Proposition 34 can be considered as
the characterization of a generalized projection operator VI(M, q, φ). Let f(·) be proper
convex LSC function, then [73, Proposition 11.3]:

$$
x \in \partial f(z) \Leftrightarrow z \in \partial f^*(x), \qquad (96)
$$

hence for any nonempty closed convex set K :

$$
x \in \partial \Psi_K(z) \Leftrightarrow z \in \partial \sigma_K(x). \tag{97}
$$

The next proposition is used also:

**Proposition 35.** [16, Proposition 1] Suppose that the $m \times m$ matrix $D \succ 0$, and $\mathcal{M}:$
$\mathbb{R}^m \to \mathbb{R}^m$ is a maximal monotone operator of the form $\partial\phi(\cdot)$ for a proper convex LSC
function $\phi(\cdot)$. Then the operator $(D+\mathcal{M})^{-1}(\cdot)$ is well-defined (i.e., for each $y \in \mathbb{R}^m$ there
is an $x \in \mathbb{R}^m$ such that $y \in Dx + \mathcal{M}(x))$, single-valued and Lipschitz continuous with
modulus $L = \frac{2}{\lambda_{\min}(D+D^\top)}$.

Under the same conditions, it also holds that (D + M)(·) is maximal monotone [5, Corol-
lary 24.4], strongly with constant λ_min(D+D^T) (using [6, Corollary 10.4.2]), (D+M)^{-1}(·)
is cocoercive with same constant [5, Example 22.6]. Proposition 35 extends to the case
when M(·) is a general maximal monotone operator. At several places in this article the
next result is useful.

**Proposition 36.** (i) Let $D = D^\top \succ 0$, $K$ a closed nonempty convex set. Then $\operatorname{Im}(\partial\sigma_K + D) \simeq K + \operatorname{Im}(D)$. If in addition $K$ is polyhedral then $\operatorname{Im}(\partial\sigma_K + D) = K + \operatorname{Im}(D)$. (ii) Let $D \succ 0$ (not necessarily symmetric), $K = \mathbb{R}_+^m$. Then:

$$
\operatorname{Im}(D + \partial\sigma_K) = \mathbb{R}_+^m + \operatorname{Im}(D + D^\top) + \{D^\top z \mid z \in \mathbb{R}_+^m\}. \quad (98)
$$
This is proved in [62] for (i) and [6, 27, 73] for (ii). The next result is used in the article.
Let $M = M^{\top} \succ 0 \Rightarrow M$ is copositive-plus, and $Q_M \stackrel{\Delta}{=} \{\xi \mid 0 \le \xi \perp M\xi \ge 0\}$. Then:
$q \in Q_M^* \Leftrightarrow \text{LCP}(q, M)$ is feasible $\Leftrightarrow$ $\text{LCP}(q, M)$ is solvable. It is a consequence of Theorem 3.8.6, Corollary 3.8.10, Corollary 3.8.12, and Exercise 3.12.1 in [38]. If $M \ne M^{\top}$ then the second $\Leftrightarrow$ is just $\Rightarrow$.

C Dynamics of Circuits in Fig. 5

Left circuit: this circuit has two voltage sources. Capacitor's law yields $i_1 = \dot{q}_c (= C\dot{v}_c)$, and Kirchhoff's law in current yields $i_3 - i_1 - i_2 = 0$, while Kirchhoff's law in voltage yields $v_c + L(i_3)\frac{di_3}{dt} - v_{d1} + u_1 = 0$ and $v_R(i_2) + L(i_3)\frac{di_3}{dt} + v_{d2} + u_2 = 0$. Hence $v_c = v_{d1} + v_{d2} + v_R(i_2) - u_1 + u_2$, $\dot{q}_c = i_3 - i_2$, and $v_R(i_2) = v_R(i_3 - \dot{q}_c)$. Therefore $L(i_3)\frac{di_3}{dt} = v_{d1} - v_c - u_1$, and $v_R(i_3 - \dot{q}_c) = v_c - v_{d1} - v_{d2} + u_1 - u_2$. Moreover $0 \le v_{d1} \perp i_1 = \dot{q}_c \ge 0$ (diode 1), and $0 \le v_{d2} \perp -i_2 = \dot{q}_c - i_3 \ge 0$ (diode 2).

Right circuit: this circuit has a current and a voltage sources. Capacitor's law yields $i_1 = \dot{q}_c (= C\dot{v}_c)$, and Kirchhoff's law in current yields $i = i_1 + i_2$, while Kirchhoff's law in voltage yields $v_c - v_d + u(t) = 0$. The diode law is $0 \le -i_2 \perp v_d \ge 0$. Equivalences follow from (94).

References

[1] V. Acary, B. Brogliato, and D. Goeleven. Higher order Moreau's sweeping process: Mathematical formulation and numerical simulation. *Math. Program.*, 113(1):133–217, 2008.

[2] K. Addi, B. Brogliato, and D. Goeleven. A qualitative mathematical analysis of a class of linear variational inequalities via semi-complementarity problems: applications in electronics. *Math. Program.*, 126:31–67, 2011.

[3] J.P. Aubin. *Applied Functional Analysis*. John Wiley & Sons, New York, 1979.

[4] J. Bastien, M. Schatzman, and C.H. Lamarque. Study of an elastoplastic model with an infinite number of internal degrees of freedom. *Europ. J. Mechanics A/Solids*, 19(2):199–222, 2002.

[5] H.H. Bauschke and P.L. Combettes. *Convex Analysis and Monotone Operator Theory in Hilbert Spaces*. CMS Books in Mathematics. Canadian Mathematical Society. Société Mathématique du Canada. Springer Science+Business Media, 2011.

[6] D.S. Bernstein. *Scalar, Vector, and Matrix Mathematics. Theory, Facts and Formulas*. Princeton University Press, 3rd edition, 2018.

[7] D.P. Bertsekas. *Convex Optimization Theory*. Athena Scientific, Belmont, 2009.

[8] L.M. Briceno-Arias and D. Davis. Forward-backward-half forward algorithm for solving monotone inclusions. *SIAM Journal on Optimization*, 28(4):2839–2871, 2018.

[9] B. Brogliato. Some perspectives on the analysis and control of complementarity systems. *IEEE Transactions on Automatic Control*, 48(6):918–935, 2003.

[10] B. Brogliato. Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings. *Systems and Control Letters*, 51(5):343–353, 2004.
[11] B. Brogliato. *Nonsmooth Mechanics. Models, Dynamics and Control*. Communications and Control Eng. Springer Int. Pub., Switzerland, third edition, 2016.

[12] B. Brogliato. Non-autonomous higher-order Moreau's sweeping process: Well-posedness, stability and zeno trajectories. *European Journal of Applied Mathematics*, 29(5):941–968, 2018.

[13] B. Brogliato. Analysis of the implicit Euler time-discretization of a class of descriptor-variable linear cone complementarity systems. *Journal of Convex Analysis*, 29(2):481–517, 2022.

[14] B. Brogliato. Analysis of the implicit Euler time-discretization of semi-explicit differential-algebraic linear complementarity systems. *SIAM J. on Control and Optimization*, 60(4):2159–2183, 2022.

[15] B. Brogliato. Dissipative dynamical systems with set-valued feedback loops: Well-posed set-valued Lur'e dynamical systems. *IEEE Control Systems*, 42(3):93–114, June 2022.

[16] B. Brogliato and D. Goeleven. Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems. *Nonlinear Analysis. Theory, Methods and Applications*, 74:195–212, 2011.

[17] B. Brogliato and D. Goeleven. Existence, uniqueness of solutions and stability of non-smooth multivalued Lur'e dynamical systems. *Journal of Convex Analysis*, 20(3):881–900, 2013.

[18] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. *Dissipative Systems Analysis and Control. Theory and Applications*. Communications and Control Engineering. Springer Nature Switzerland AG, Cham, CH, third edition, 2020.

[19] B. Brogliato and A. Rocca. Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems. *Journal of Nonsmooth Analysis and Optimization*, 3:1–39, 2022.

[20] B. Brogliato and A. Tanwani. Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability. *SIAM Review*, 62(1):3–129, 2020.

[21] B. Brogliato and L. Thibault. Existence and uniqueness of solutions for non-autonomous complementarity dynamical systems. *Journal of Convex Analysis*, 17(3-4):961–990, 2010.

[22] M. K. Camlibel, J.S. Pang, and J. Shen. Lyapunov stability of complementarity and extended systems. *SIAM Journal on Optimization*, 17(4):1056–1101, 2007.

[23] M.K. Camlibel. *Complementarity methods in the analysis of piecewise linear dynamical systems*. PhD thesis, Tilburg University, NL, May 2001.

[24] M.K. Camlibel and R. Frasca. Extension of Kalman-Yakubovich-Popov Lemma to descriptor systems. *Syst. and Cont. Lett.*, 58:795–803, 2009.

[25] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. Consistency of a time-stepping method for a class of piecewise linear networks. *IEEE Transactions on Circuits and Systems I, 49(3):349–357, 2002.*

[26] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. On linear passive complementarity systems. *European Journal of Control*, 8(3):220–237, 2002.
[27] M.K. Camlibel, L. Iannelli, and A. Tanwani. Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators. *Math. Program.*, 194:1017–1059, 2022.

[28] M.K. Camlibel, L. Iannelli, A. Tanwani, and S. Trenn. Differential-algebraic inclusions with maximal monotone operators. In *IEEE 55th Conf. Decision and Control*, pages 610–615, Las Vegas, December 2016.

[29] M.K. Camlibel, L. Iannelli, and F. Vasca. Passivity and complementarity. *Math. Program.*, 145:531–563, 2014.

[30] A.R. Champneys and P.L. Várkonyi. The Painlevé paradox in contact mechanics. *IMA Journal of Applied Mathematics*, 81(3):538–588, 07 2016.

[31] Y. Chen and S. Trenn. Impulse-free jump solutions of nonlinear differential-algebraic equations. *Nonlinear Analysis: Hybrid Systems*, 46:101238, 2022.

[32] Y. Chen and S. Trenn. Solution concepts for linear piecewise affine differential-algebraic equations. In *2024 IEEE 63rd Conference on Decision and Control (CDC)*, pages 7804–7809, 2024.

[33] L. Chua and A.C. Deng. Impasse points. Part I: numerical aspects. *Int. J. Circ. Theor. Applic.*, 17:213–235, 1989.

[34] L.O. Chua. Dynamic nonlinear networks: state-of-the-art. *IEEE Trans. Circuits Syst.*, 28(11):1059–1087, 1980.

[35] D. Cobb. On the solution of linear differential equations with singular coefficients. *Journal of Differential Equations*, 46:310–323, 1982.

[36] D. Cobb. Descriptor variable systems and optimal state regulation. *IEEE Transactions on Automatic Control*, 28(5):601–611, 1983.

[37] F. Corinto and M. Forti. Memristor circuits: Bifurcations without parameters. *IEEE Transactions on Circuits and Systems I: Regular Papers*, 64(6):1540–1551, 2017.

[38] R.W. Cottle, J.S. Pang, and R.E. Stone. *The Linear Complementarity Problem*. Computer Science and Scientific Computing. Academic Press, 1992.

[39] J.F. Edmond and L. Thibault. BV solutions of nonconvex sweeping process differential inclusion with perturbation. *Journal of Differential Equations*, 226:135–179, 2006.

[40] F. Facchinei and J.S. Pang. *Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I*. Series in Operations Research. Springer-Verlag, New York, Berlin, Heidelberg, 2003.

[41] R.W. Freund and F. Jarre. An extension of the positive real lemma to descriptor systems. *Optimization Methods and Software*, 19(1):69–87, 2004.

[42] E. Gluskin. A nonlinear resistor and nonlinear inductor using a nonlinear capacitor. *Journal of the Franklin Institute*, 336(7):1035–1047, 1999.

[43] R.K. Goebel. *Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction*. SIAM, Philadelphia, PA, 2024.

[44] D. Goeleven. *Complementarity and Variational Inequalities in Electronics*. Math. Anal. and its Applications. Academic Press, 2017.
[45] K. Gong and L. Zhang. Distributed proximal-correction algorithm for the sum of maximal monotone operators. *IEEE Transactions on Automatic Control*, pages 1–15, 2025.

[46] S. Greenhalgh, V. Acary, and B. Brogliato. On preserving dissipativity properties of linear complementarity dynamical systems with the $\theta$-method. *Numerische Mathematik*, 125:601–637, 2013.

[47] F. Génot and B. Brogliato. New results on Painlevé paradoxes. *European Journal of Mechanics - A/Solids*, 18(4):653–677, 1999.

[48] P.V. Hai and P.T. Vuong. Third order dynamical systems for the sum of two generalized monotone operators. *J. Optim. Theory Appl.*, 202:519–553, 2024.

[49] W. P. M. H. Heemels, J. M. Schumacher, and S. Weiland. Linear complementarity systems. *SIAM Journal on Applied Mathematics*, 60(4):1234–1269, 2000.

[50] J.B. Hiriart-Urruty and C. Lemaréchal. *Fundamentals of Convex Analysis*. Grundlehren Text Editions. Springer-Verlag, Berlin Heidelberg, 2001. Corrected second printing 2004.

[51] I.A. Hiskens and D.J. Hill. Energy functions, transient stability and voltage behaviour in power systems with nonlinear loads. *IEEE Transactions on Power Systems*, 4(4):1525–1533, 1989.

[52] F. Küsters and S. Trenn. Duality of switched DAEs. *Math. Control Signals Syst.*, 28, 2016. Article no 25.

[53] B.K. Le. On a class of Lur'e dynamical systems with state-dependent set-valued feedback. *Set-Valued Variational Anal.*, 28(3):537–557, 2020.

[54] B.K. Le. Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions. *Optimization*, 69(6):1187–1217, 2020.

[55] D. Liberzon and S. Trenn. Switched nonlinear differential algebraic equations: Solution theory, Lyapunov functions, and stability. *Automatica*, 48(5):954–963, 2012.

[56] M.D.P. Monteiro Marques. *Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction*, volume 9 of *Progress in Nonlinear Differential Equations and Their Applications*. Birkhäuser Verlag, Basel, 1993.

[57] M.D.P Monteiro Marques. Regularization and graph approximation of a discontinuous evolution problem. *Journal of Differential Equations*, 67(2):145–164, 1987.

[58] J.J. Moreau. Evolution problem associated with a moving convex set in a Hilbert space. *Journal of Differential Equations*, 26(3):347–374, 1977.

[59] J.J. Moreau. Liaisons unilatérales sans frottement et chocs inélastiques. *C.R. Acad. Sc. Paris Série II*, 296:1473–1476, May 1983.

[60] J.J. Moreau. *Nonsmooth Mechanics and Applications*, volume 302 of *International Centre for Mechanical Sciences*, chapter Unilateral contact and dry friction in finite freedom dynamics, pages 1–82. Springer, Vienna, 1988.

[61] A. Oliveri, M. Lodi, and M. Storace. Nonlinear models of power inductors: A survey. *International Journal of Circuit Theory and Applications*, 50(1):2–34, 2022.

[62] Q.H. Pham and B. Brogliato. Correction to "Analysis of the implicit Euler time-discretization of a class of descriptor-variable linear cone complementarity systems",
J. Convex Analysis 29/2 (2022) 481–517. *Journal of Convex Analysis*, 31(3):1035–1037, 2024.

[63] Q.H. Pham, B. Brogliato, and F. Miranda-Villatoro. Well-posedness of passive time-varying linear cone complementarity systems. *Journal of Convex Analysis*, 32(3):681–722, 2025.

[64] P.J. Rabier and W.C. Rheinboldt. On impasse points of quasilinear differential-algebraic equations. *J. Math. Analysis Applic.*, 181:429–454, 1994.

[65] P.J. Rabier and W.C. Rheinboldt. On the computation of impasse points of quasi-linear differential-algebraic equations. *Math. Comp.*, 62:133–154, 1994.

[66] P.J. Rabier and W.C. Rheinboldt. Discontinuous solutions of semilinear differential-algebraic-equations–I Distribution solutions. *Nonlinear Analysis. Theory, Methods and Applications*, 27(11):1241–1256, 1996.

[67] P.J. Rabier and W.C. Rheinboldt. Discontinuous solutions of semilinear differential-algebraic-equations–II P-consistency. *Nonlinear Analysis. Theory, Methods and Applications*, 27(11):1257–1280, 1996.

[68] P.J. Rabier and W.C. Rheinboldt. Time-dependent linear DAEs with discontinuous inputs. *Linear Algebra and its Applications*, 247:1–29, 1996.

[69] P.J. Rabier and W.C. Rheinboldt. Theoretical and numerical analysis of differential-algebraic equations. In *Solution of Equations in Rn (Part 4)*, *Techniques of Scientific Computing (Part 4)*, *Numerical Methods for Fluids (Part 2)*, volume 8 of *Handbook of Numerical Analysis*, pages 183–540. Elsevier, 2002.

[70] G. Reissig. Differential-algebraic equations and impasse points. *IEEE Trans. Circ. and Systems-II: Fundamental Theory*, 43(2):122–133, 1996.

[71] A. Rocca, V. Acary, and B. Brogliato. Index-2 hybrid DAE: a case study with well-posedness and numerical analysis. In *Proc. IFAC World Congress*, IFAC PaperOnLine, pages 1888–1893, Berlin, 2020.

[72] R.T. Rockafellar. *Convex Analysis*. Princeton Landmarks in Mathematics. Princeton University Press, 1970.

[73] R.T. Rockafellar and R.J.B. Wets. *Variational Analysis*. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, Heidelberg, 1998. 3rd printing.

[74] S.S. Sastry and C.A. Desoer. Jump behavior in circuits and systems. *IEEE Trans. Circ. Syst.*, 28:1109–1124, 1981.

[75] Y. Song, D.J. Hill, T. Liu, and X. Zhang. Impasse surface of differential–algebraic power system models: An interpretation based on admittance matrices. *IEEE Transactions on Automatic Control*, 68(10):6224–6231, 2023.

[76] P. Stechlinski, M. Patrascu, and P.I. Barton. *Applications of Differential-Algebraic Equations: Examples and Benchmarks*, chapter Nonsmooth DAEs with Applications in Modeling Phase Changes, pages 243–275. Differential-Algebraic Equations Forum. Springer, Cham, 2018.

[77] P. Stechlinski, M. Patrascu, and P.I. Barton. Nonsmooth differential-algebraic equations in chemical engineering. *Computers and Chemical Engineering*, 114:52–68, 2018.
[78] P.G. Stechlinski and P.I. Barton. Dependence of solutions of nonsmooth differential-algebraic equations on parameters. *Journal of Differential Equations*, 262(3):2254-2285, 2017.

[79] A. Tanwani, B. Brogliato, and C. Prieur. Well-posedness and output regulation for implicit time-varying evolution variational inequalities. *SIAM Journal on Control and Optimization*, 56(2):751-781, 2018.

[80] A. Tanwani and S. Trenn. Detectability and observer design for switched differential algebraic equations. *Automatica*, 99:289-300, 2019.

[81] L. Thibault. Moreau sweeping process with bounded truncated retraction. *Journal of Convex Analysis*, 23(4):1051-1098, 2016.

[82] S. Trenn. Regularity of distributional differential algebraic equations. *Math. Control Signals Syst.*, 21:229-264, 2009.

[83] S. Trenn. *Dynamics and Control of Switched Electronic Systems. Advanced Perspectives for Modeling, Simulation and Control of Power Converters*, chapter Switched Differential Algebraic Equations, pages 189-216. Advances in Industrial Control. Springer, London, 2012.

[84] P.T. Vuong. A second order dynamical system and its discretization for strongly pseudo-monotone variational inequalities. *SIAM Journal on Control and Optimization*, 59(4):2875-2897, 2021.

[85] G. Wachsmuth. From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability. *Journal of Nonsmooth Analysis and Optimization*, 3, January 2022.