using Robinson's bound in Lemma A2. It follows from (70) that $\phi(\cdot)$ is locally AC on $[0, +\infty)$ if $Q(\cdot)F(\cdot)u(\cdot)$ is locally AC on $[0, +\infty)$. To prove that $\phi(\cdot)$ is RCLBV on $[0, +\infty)$, the existence of a positive Radon measure $\mu$ on $[0, T]$ satisfying $d_H(\phi(s), \phi(t)) \le \mu((s, t])$ for all $0 \le s \le t \le T$, for all $T > 0$, must be ensured. Indeed, for each $T > 0$, since $Q(\cdot)F(\cdot)u(\cdot)$ is RCBV on $[0, T]$, there exists a vector measure (or known as differential measure) $d_{QFu}$ on $[0, T]$ associated with $Q(\cdot)F(\cdot)u(\cdot)$ such that

$$Q(t)F(t)u(t) - Q(s)F(s)u(s) = d_{QFu}((s, t]), \quad \forall 0 \le s \le t \le T, \qquad (71)$$

see [34, p.140], [36, 37]. Let $\tilde{\mu} := |d_{QFu}|$ be the variation measure of $d_{QFu}$, then $\tilde{\mu}$ is a positive Radon measure on $[0, T]$, see [38, p.383], [33]. From (71), it is inferred that:

$$\|Q(t)F(t)u(t) - Q(s)F(s)u(s)\| = \|d_{QFu}((s, t))\| \le \tilde{\mu}((s, t]), \quad \forall 0 \le s \le t \le T, \qquad (72)$$

where the inequality is obtained from the definition of variation measures [39, p.2]. Let $\mu := \gamma_K \tilde{\mu}$, it can be deduced from (70) and (72) that $\phi(\cdot)$ is RCBV on $[0, T]$, is thus RCLBV on $[0, +\infty)$. $\square$

As mentioned in the proof of Proposition 3, if $Q(\cdot)F(\cdot)u(\cdot)$ is RCBV on $[0, T]$, then there exists a positive Radon measure $\mu$ such that $d_H(\phi(s), \phi(t)) \le \mu((s, t])$, for all $0 \le s \le t \le T$. Moreover, there always exists a nondecreasing RCBV function $\tilde{u} : [0, T] \to [0, +\infty)$ such that $\mu((s, t]) = \tilde{u}(t) - \tilde{u}(s)$ for all $0 \le s \le t \le T$. Indeed, let $\tilde{u}(t) := \mu((0, t])$ for all $t \in [0, T]$, then $\tilde{u}(\cdot)$ is nondecreasing and right-continuous on $[0, T]$. Since $\mu = \gamma_K |d_{QFu}|$ and $Q(\cdot)F(\cdot)u(\cdot)$ is BV on $[0, T]$,

$$\mu((0, T]) = \gamma_K \sup_{0=t_0<t_1<\dots<t_k=T} \sum_{i=0}^{k-1} \|d_{QFu}((t_i, t_{i+1}))\| = \gamma_K V_{[0,T]}(QFu) < +\infty.$$

Hence, $V_{[0,T]}(\tilde{u}) = \tilde{u}(T) - \tilde{u}(0) = \mu((0, T]) < +\infty$, that is, $\tilde{u}(\cdot)$ is BV on $[0, T]$. Therefore, it is implied that [4, Assumption (A3)] holds, which is an important assumption when studying FOSwPs. The next proposition allows for the relaxation of item 2 in Proposition 3, at the price of assuming bounded sets $\phi(t)$. It can be observed that computing $\phi(t)$ is straightforward, but finding a mapping $Q(\cdot)$ that satisfies item 2 in Proposition 3 is generally not easy, or even impossible. The following result is useful in these cases when $\phi(t)$ is a bounded set for all $t \ge 0$ (it is a moving polytope).

**Proposition 4.** Consider the system (3) and its equivalent (68). Suppose that Assumptions 2, 4 and the following hold:

1. There is $Q : \mathbb{R}_+ \to \mathbb{R}^{j \times m}$ such that $K^*(t) = \{\xi \in \mathbb{R}^m : Q(t)\xi \ge 0\}$, for all $t \ge 0$ and some $j$.

2. $\phi(t)$ is bounded for all $t \ge 0$.

3. $F(t)u(t) \in \text{rge}(C(t)) + K^*(t)$, for all $t \ge 0$.

Then, $\phi(\cdot)$ in (69) is locally AC (resp. RCLBV) on $[0, +\infty)$ if $Q(\cdot)C(\cdot)R(\cdot)^{-1}$ and $Q(\cdot)F(\cdot)u(\cdot)$ are locally AC (resp. RCLBV) on $[0, +\infty)$.

*Proof.* Clearly $\phi(t) = \{x \in \mathbb{R}^n : Q(t)C(t)R(t)^{-1}x + Q(t)F(t)u(t) \in \mathbb{R}_+^j\}$, which is nonempty for all $t \ge 0$ due to item 3. Since $\phi(t)$ is bounded for all $t \ge 0$ (item 2), there exists $M > 0$ such that:

$$\|x\| \le M, \quad \forall x \in \phi(t), \forall t \ge 0.$$

To simplify the writing, set $\tilde{A}(t) := Q(t)C(t)R(t)^{-1} \in \mathbb{R}^{j \times n}$, $b(t) := Q(t)F(t)u(t) \in \mathbb{R}^j$ for all $t \ge 0$. Let $s, t$ such that $0 \le s \le t$. Using Corollary A1, $\text{dist}(x, \phi(s)) \le M(\|\tilde{A}(t) - \tilde{A}(s)\| + \|b(t) - b(s)\|)$,
is proposed in [34] and used in [10, 4], see Definition A1. Under this definition, the well-posedness of (58) has been shown in [10, 4], which is recalled here:

**Theorem 3.** [10, Theorem 3.5] Assume that $u \in L_1([0, +\infty), \mathbb{R}^u)$, $A(\cdot)$ and $E(\cdot)$ are measurable and uniformly bounded on $[0, T]$ for any $T > 0$, and the set-valued mapping $\phi(\cdot)$ is locally AC (resp. RCLBV) with non-empty values on $[0, +\infty)$. Then the FOSwP (58) has one and only one locally AC (resp. RCLBV) solution $z(\cdot)$ on $[0, +\infty)$, for every initial condition $z(0) = z_0 \in \phi(0)$.

The well-posedness of (3) is ensured by combining Propositions 3, 4 and Theorem 3. Intuitively, this may seem unrelated to Theorem 2 or Corollary 2, but in fact, they are very close. Indeed, (3) is equivalent to (58) under Assumptions 2, 4. Assume that $K^*(t) := \{\xi \in \mathbb{R}^m : Q(t)\xi \ge 0\}$ for some matrix $Q(t) \in \mathbb{R}^{j \times m}$, for all $t \ge 0$ and for some $j \in \mathbb{N}$ (i.e., item 1 in Proposition 3 holds). Then $\phi(t) = \{x \in \mathbb{R}^n : Q(t)C(t)R(t)^{-1}x + Q(t)F(t)u(t) \ge 0\}$ and (58) is rewritten as the following TVLCCS using (6), where $z(t) \in \mathbb{R}^n$ and $\hat{\lambda}(t) \in \mathbb{R}^j$.

$$ \left\{ \begin{array}{l} \dot{z}(t) = (R(t)A(t)R(t)^{-1} + \dot{R}(t)R(t)^{-1})z(t) + (Q(t)C(t)R(t)^{-1})^{\top} \hat{\lambda}(t) + R(t)E(t)u(t), \\ \hat{w}(t, z) = Q(t)C(t)R(t)^{-1}z(t) + Q(t)F(t)u(t), \\ 0 \le \hat{\lambda}(t) \perp \hat{w}(t, z) \ge 0. \end{array} \right. \qquad (69) $$

Here, we can easily see that the conditions to guarantee Proposition 3 and Theorem 3 being true are very close to the conditions ensuring Theorem 2 is satisfied on the system (69).

**Remark 4.** So far, to obtain the existence and uniqueness of an AC solution to (3) when $D(\cdot) \equiv 0$, either a combination of Proposition 3 (resp. Proposition 4) and Theorem 3, or Corollary 2 can be used. If all matrices in (3) are constant and $K(t) = \mathbb{R}_+^m$ for all $t \ge 0$, then these approaches (except Proposition 4) become equivalent. However, in the general case, they are different and can be used selectively based on each specific situation. This will be illustrated by examples in the next section.

### 2.4.2 Examples

Let us consider the circuit in Figure 3. Denote by $\lambda$ the current through the diode, and $w$ the voltage across the diode, $x(t) := [x_1(t) \ x_2(t)]^\top$, where $x_1, x_2$ denote the charge in the capacitor and the current through the inductor, respectively. The dynamics of this circuit is given by:

$$ \left\{ \begin{array}{l} \dot{x}(t) = \begin{bmatrix} \frac{1}{\mathbf{R}(t)\mathbf{C}(t)} & -1 \\ \frac{1}{\mathbf{L}(t)\mathbf{C}(t)} & 0 \end{bmatrix} x(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \lambda(t) + \begin{bmatrix} -\frac{1}{\mathbf{R}(t)} \\ \frac{1}{\mathbf{L}(t)} \end{bmatrix} u(t), \\ w(t, x) = \begin{bmatrix} 1 \\ \mathbf{C}(t) \end{bmatrix} x(t) + u(t), \\ 0 \le \lambda(t) \perp w(t, x) \ge 0. \end{array} \right. $$

Let $\mathbf{C}(t) = \frac{t^2+2}{t^2+1}, u(t) = 1+\sin(t)$ and $\mathbf{R}(t), \mathbf{L}(t) > \rho$, for all $t \ge 0$, for some $\rho > 0$ such that $\mathbf{R}(\cdot), \mathbf{L}(\cdot)$ are measurable functions. It can be checked that Assumptions 4, 3, 2 hold with $P(t) = \text{diag}(\frac{1}{C(t)}, 1)$ is a solution of (56), and items 1, 2, 3, 5 in Corollary 2 hold. However, this system does not satisfy
for all $x \in \phi(t)$. This implies $\sup_{x \in \phi(t)} \text{dist}(x, \phi(s)) \le M(\|\tilde{A}(t) - \tilde{A}(s)\| + \|b(t) - b(s)\|)$. By a symmetric argument, the right-hand side of this inequality is also a bound of $\sup_{x \in \phi(s)} \text{dist}(x, \phi(t))$. Hence, the following holds:

$$d_H(\phi(s), \phi(t)) \le M(\|\tilde{A}(t) - \tilde{A}(s)\| + \|b(t) - b(s)\|). \quad (73)$$

For each $h \ge 0$, for each $i \in \{1, ..., n\}$, let $a_i(h) \in \mathbb{R}^j$ denote the columns of $\tilde{A}(h)$. Recall that the norm being used is the Frobenius norm. Using [29, Fact 9.8.12-(vii)], we obtain:

$$\begin{aligned} \|\tilde{A}(t) - \tilde{A}(s)\| + \|b(t) - b(s)\| &\le \sqrt{n} \max_{1 \le i \le n} (\|a_i(t) - a_i(s)\|_1) + \|b(t) - b(s)\|_1 \\ &\le \sqrt{n} \left( \sum_{i=1}^n \|a_i(t) - a_i(s)\|_1 \right) + \|b(t) - b(s)\|_1, \end{aligned} \quad (74)$$

where $\|\cdot\|_1$ is the maximum absolute column sum norm. Suppose that $\tilde{A}(\cdot)$ and $b(\cdot)$ are locally AC, then for all $i \in \{1, ..., n\}$ there exist derivatives of $a_i(\cdot)$ and $b(\cdot)$, denoted as $\dot{a}_i$ and $\dot{b}$ respectively, such that $a_i(t) - a_i(s) = \int_s^t \dot{a}_i(\tau) d\tau$, and $b(t) - b(s) = \int_s^t \dot{b}(\tau) d\tau$. It follows that:

$$\sqrt{n} \left( \sum_{i=1}^{n} \|a_i(t) - a_i(s)\|_1 \right) + \|b(t) - b(s)\|_1 \le \int_{s}^{t} \left[ \sqrt{n} \left( \sum_{i=1}^{n} \|\dot{a}_i(\tau)\|_1 \right) + \|\dot{b}(\tau)\|_1 \right] d\tau. \quad (75)$$

It follows from (73), (74), and (75) that $d_H(\phi(s), \phi(t)) \le v(t) - v(s)$, where $v(\cdot)$ is a nondecreasing locally AC function and defined by:

$$v(\cdot) : [0, +\infty) \to \mathbb{R}_+, \quad \hat{t} \mapsto v(\hat{t}) = \int_0^{\hat{t}} M \left[ \sqrt{n} \left( \sum_{i=1}^n \|\dot{a}_i(\tau)\|_1 \right) + \|\dot{b}(\tau)\|_1 \right] d\tau.$$

Thus $\phi(\cdot)$ is locally AC. Let us now prove that $\phi(\cdot)$ is RCLBV if $\tilde{A}(\cdot)$ and $b(\cdot)$ are RCLBV. Indeed, for each $T > 0$, proceeding as the proof of Proposition 3, there exist positive Radon measures $\tilde{\mu}_i$ on $[0, T]$ for $i = 1, ..., n + 1$ such that:

$$\begin{aligned} M\sqrt{n}\left(\sum_{i=1}^{n}\|a_i(t)-a_i(s)\|_1\right)+M\|b(t)-b(s)\|_1 &\le M\sqrt{n}\left(\sum_{i=1}^{n}\tilde{\mu}_i((s,t])\right)+M\tilde{\mu}_{n+1}((s,t]) \\ &=\mu((s,t]), \quad \forall 0 \le s \le t \le T, \end{aligned} \quad (76)$$

where $\mu := M\sqrt{n}\left(\sum_{i=1}^{n}\tilde{\mu}_i\right) + M\tilde{\mu}_{n+1}$. Using [40, Theorem 7.1.7], $\mu$ is a positive Radon measure on $[0, T]$. It follows from (73), (74), (76) that $\phi(\cdot)$ is RCBV on $[0, T]$, is thus RCLBV on $[0, +\infty)$. $\square$

An appropriate concept for the solutions of the FOSwP (68) when the moving set $\phi(\cdot)$ is RCLBV, is proposed in [34] and used in [10, 4], see Definition A1. Under this definition, the well-posedness of (68) has been shown in [10, 4], which is recalled here:

**Theorem 3.** [10, Theorem 3.5] Assume that $u \in L_1([0, +\infty), \mathbb{R}^u)$, $A(\cdot)$ and $E(\cdot)$ are measurable and uniformly bounded on $[0, T]$ for any $T > 0$, and the set-valued mapping $\phi(\cdot)$ is locally AC (resp. RCLBV) with non-empty values on $[0, +\infty)$. Then the FOSwP (68) has one and only one locally AC (resp. RCLBV) solution $z(\cdot)$ on $[0, +\infty)$, for every initial condition $z(0) = z_0 \in \phi(0)$.
item 4 in Corollary 2. Indeed, let $s = 0$ and $t = 1$, we obtain:

$$ (C(s)R(s)^{-1})^{-1}(\mathbb{R}_+ - u(t)) = \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -\sqrt{2}(1 + \sin(1))\} \\ \notin (C(t)R(t)^{-1})^{-1}(\mathbb{R}_+ - u(t)) = \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -\sqrt{\frac{3}{2}}(1 + \sin(1))\}. $$

Therefore, the existence of an AC solution to this system is not guaranteed by Corollary 2. On the other hand, this system satisfies items 1, 2, and 3 in Proposition 3 with $Q(t) = \sqrt{C(t)}$, for all $t \ge 0$. Since $Q(\cdot)u(\cdot) = (1 + \sin(\cdot))\sqrt{\frac{(\cdot)^2+2}{(\cdot)^2+1}}$ is a locally AC function on $[0, +\infty)$, $\phi(\cdot)$ is locally AC using Proposition 3. Using Theorem 3, this system has a unique locally AC solution on $[0, +\infty)$ for every $x(0) \in \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -2\}$.

<figure><img src="image_4.png" /><figcaption>Figure 3: Time-varying RLC circuit with an ideal diode.</figcaption></figure>

The next example illustrates a TVLCCS system that possesses a unique RCLBV solution. The following system is considered where $x(t) \in \mathbb{R}$ and $\lambda(t) \in \mathbb{R}^2$.

$$ \begin{cases} \dot{x}(t) = x(t) + B(t)\lambda(t), \\ 0 \le \lambda(t) \perp w(t, x) = C(t)x(t) + u(t) \ge 0, \end{cases} \qquad (70) $$

in which, $B(t) = [1 \ -1]$, $C(t) = \begin{bmatrix} t+1 \\ -(t+1) \end{bmatrix}$, and $u(t) = \begin{cases} [0 \ \exp(2t)]^\top, & \text{if } t < 2 \\ [1 \ 0]^\top, & \text{if } t \ge 2 \end{cases}$, for all $t \ge 0$.

The system (70) is a TVLCCS and satisfies all assumptions in Proposition 3. Indeed,

1. Obviously, Assumptions 4, 2 hold where $P(t) = t + 1$ is a solution of (56).

2. We have $K^*(t) = \{\xi \in \mathbb{R}^2 : Q(t)\xi \ge 0\}$, where $Q(t) = \begin{bmatrix} \frac{1}{\sqrt{t+1}} & 0 \\ 0 & \frac{1}{\sqrt{t+1}} \end{bmatrix} \in \mathbb{R}^{2 \times 2}$. Moreover, $Q(t)C(t)R(t)^{-1} = [1 \ -1]^\top$ is a constant matrix for all $t \ge 0$. Hence, items 1, 2 in Proposition 3 are validated.

3. Clearly $K^*(t) - u(t) = \begin{cases} \mathbb{R}_+^2 - [0 \ \exp(2t)]^\top, & \text{if } t < 2 \\ \mathbb{R}_+^2 - [1 \ 0]^\top, & \text{if } t \ge 2 \end{cases}$, and $\text{rg}(C(t)) = \begin{cases} [1] & \xi, \forall \xi \in \mathbb{R} \\ [-1] & \xi, \forall \xi \in \mathbb{R} \end{cases}$ is a constant linear space for all $t \ge 0$. Therefore, $\text{rg}(C(t)) \cap (K^*(t) - u(t)) \ne \emptyset$ for all $t \ge 0$. Using Remark 3, then item 3 in Proposition 3 holds.
**Remark 4.** In Propositions 3, 4, item 3 plays a role in ensuring the non-emptiness of $\phi(t)$ for all $t \ge 0$. Additionally, an equivalent form of item 3 is obtained as follows:

$$ \mathrm{rge}(C(t)) \cap (K^*(t) - F(t)u(t)) \neq \emptyset, \quad \forall t \geq 0. \qquad (77) $$

When $D(\cdot) \equiv 0$, then (77) is a relaxed form of Assumption 3. There are cases where Assumption 3 is false but (77) is true. For instance, $C(t) = [1 - 1]^{\top}$, $F(t)u(t) = 0$ and $K^*(t) = \mathbb{R}_+^2$, for all $t \ge 0$. In addition, all items in Corollary 2 are also relaxed by Propositions 3, 4, and Theorem 3.

**Remark 5.** In [10], the matrices $C(t), R(t)^{-1}, F(t)$ are constant and $K^*(t) = K(t) = \mathbb{R}_+^m$, for all $t \ge 0$. Therefore, to ensure the non-emptiness of $\phi(t)$, the constraint qualification (CQ): $\mathrm{rge}(C) - \mathbb{R}_+^m = \mathbb{R}^m$ is used in [10, p.8] (see its statement in Appendix D). Moreover, this CQ is indeed derived from:

$$ -Fu(t) - f \in \mathrm{rge}(C) - \mathbb{R}_+^m, \qquad (78) $$

where $f$ is a vector in the complementarity condition. Because in [10], they aim to avoid imposing assumptions on the vector $f$ (i.e., treating $f$ as an arbitrary vector in $\mathbb{R}^m$), so the CQ as previously is implicitly understood as a sufficient condition for (78) holding for every $f \in \mathbb{R}^m$. Thus, item 3 in Propositions 3, 4 is an extension of (78) when $f = 0$.

The well-posedness of (3) is ensured by combining Propositions 3, 4 and Theorem 3. Intuitively, this may seem unrelated to Theorem 2 or Corollary 2, but in fact, they are very close. Indeed, (3) is equivalent to (68) under Assumptions 2, 4. Assume that $K^*(t) := \{\xi \in \mathbb{R}^m : Q(t)\xi \ge 0\}$ for some matrix $Q(t) \in \mathbb{R}^{j \times m}$, for all $t \ge 0$ and for some $j \in \mathbb{N}$ (i.e., item 1 in Proposition 3 holds). Then $\phi(t) = \{x \in \mathbb{R}^n : Q(t)C(t)R(t)^{-1}x + Q(t)F(t)u(t) \ge 0\}$ and (68) is rewritten as the following TVLCCS using (6), where $z(t) \in \mathbb{R}^n$ and $\hat{\lambda}(t) \in \mathbb{R}^j$.

$$ \left\{ \begin{array}{l} \dot{z}(t) = (R(t)A(t)R(t)^{-1} + \dot{R}(t)R(t)^{-1})z(t) + (Q(t)C(t)R(t)^{-1})^{\top} \hat{\lambda}(t) + R(t)E(t)u(t), \\ \hat{w}(t, z) = Q(t)C(t)R(t)^{-1}z(t) + Q(t)F(t)u(t), \\ 0 \le \hat{\lambda}(t) \perp \hat{w}(t, z) \ge 0. \end{array} \right. \qquad (79) $$

Here, we can easily see that the conditions to guarantee Proposition 3 and Theorem 3 being true are very close to the conditions ensuring Theorem 2 is satisfied on the system (79).

**Remark 6.** When $z(\cdot)$ is an RCLBV solution of (68), it may possess jumps. Inspired by the state-jump model provided in [41, p.167], [42, p.91], [5, p.62], at time of jumps $t$, (68) can be rewritten as:

$$ z(t^{+}) - z(t^{-}) \in -\mathcal{N}_{\phi(t^{+})} \left( \frac{z(t^{+}) + ez(t^{-})}{1 + e} \right), \qquad (80) $$

where $e \in [0, 1]$ is the restitution coefficient (see e.g., [10, Remark 3.7]). Suppose that items 1, 3 in Proposition 3 (resp. Proposition 4) hold. Using (4), then (80) is equivalent to $z(t^{+}) = (1 + e)\mathrm{proj}_{\phi(t^{+})}(z(t^{-})) - ez(t^{-})$, and $z(t^{+})$ exists and is unique since $\phi(t^{+})$ (given in (69)) is a nonempty closed convex set. However, in many cases, it is not easy to compute this projection. Therefore, another way to calculate $z(t^{+})$ is based on the equivalence of (80) with the following:

$$ \left\{ \begin{array}{l} z(t^{+}) - z(t^{-}) = (Q(t^{+})C(t^{+})R(t)^{-1})^{\top} \hat{\lambda}, \\ 0 \le \hat{\lambda} \perp \hat{w} = Q(t^{+})C(t^{+})R(t)^{-1} \left( \frac{z(t^{+}) + ez(t^{-})}{1 + e} \right) + Q(t^{+})F(t^{+})u(t^{+}) \ge 0. \end{array} \right. \qquad (81) $$
$$4. Q(t)u(t) = \begin{cases} \left[ 0 \frac{\exp(2t)}{\sqrt{t+1}} \right]^{\top}, & \text{if } t < 2 \\ \left[ \frac{1}{\sqrt{t+1}} 0 \right]^{\top}, & \text{if } t \ge 2 \end{cases}, \text{ which yields } Q(\cdot)u(\cdot) \text{ is a RCLBV function on } [0, +\infty).$$

Thus, $\phi(\cdot)$ defined in (59) is RCLBV on $[0, +\infty)$. Using Theorem 3, the system (70) has a unique RCLBV solution on $[0, +\infty)$, for every $x(0) \in [0, 1]$. In this case, the moving set $\phi(t)$ and the Hausdorff distance of $\phi(\cdot)$ at two arbitrary times are stated as follows:

$$\phi(t) = \begin{cases} \left[ 0, \frac{\exp(2t)}{\sqrt{t+1}} \right], & \text{if } t < 2 \\ \left[ \frac{-1}{\sqrt{t+1}}, 0 \right], & \text{if } t \ge 2 \end{cases}, \quad d_H(\phi(t), \phi(s)) = \begin{cases} \frac{\exp(2t)}{\sqrt{t+1}} - \frac{\exp(2s)}{\sqrt{s+1}}, & \text{if } s \le t < 2 \\ \max \left\{ \frac{\exp(2s)}{\sqrt{s+1}}, \frac{1}{\sqrt{t+1}} \right\}, & \text{if } s < 2 \le t \\ \frac{1}{\sqrt{s+1}} - \frac{1}{\sqrt{t+1}}, & \text{if } 2 \le s \le t \end{cases}.$$

This implies that the Robinson's bound (61) in Proposition 3 holds with $\gamma_K = 1$.

# 3 Further examples and future works

## 3.1 Simple Examples

Let us illustrate some peculiarities of TVLCCS (3) compared to LCS (1), especially concerning state jumps (LCS possess state jumps at the initial time, or may jump at times when $Fu(t)$ is discontinuous [9, 10]). To this aim, several scalar examples are considered, to which the foregoing theoretical developments do not apply, and may thus motivate further investigations.

### 3.1.1 Example 1: State jumps in TVLCCS with constant $K(t)$

The following scalar system is considered where $x(t) \in \mathbb{R}$ and $\lambda(t) \in \mathbb{R}$.

$$\begin{cases} (a) \dot{x}(t) = 0.05x(t) + \frac{t^2+1}{t^2+2}\sin(t+1)\lambda(t), \\ (b) 0 \le \lambda(t) \perp w(t, x(t)) = \sin(t+1)x(t) \ge 0, x(0) = 1. \end{cases} \quad (71)$$

The system (71) is a special case of (3) when $A(t) = 0.05$, $B(t) = \frac{t^2+1}{t^2+2}\sin(t+1)$, $C(t) = \sin(t+1)$, $D(t) = E(t) = F(t) = 0$ and $K(t) = K^*(t) = \mathbb{R}_+$, for all $t \ge 0$. Let us analyze solutions of (71).

**Necessity of state jumps:** From (71)(b), we get $\lambda(0) = 0$ since $w(0, x(0)) = \sin(1) > 0$. It is deduced from (71)(a) that $\dot{x}(0) = 0.05x(0)$. We conclude that for each $t \in [0, \pi - 1)$ then $\lambda(t) = 0$ and $\dot{x}(t) = 0.05x(t) > 0$, so a solution is given by $x(t) = \exp(0.05t)$ on $[0, \pi - 1)$. Therefore, $x((\pi - 1)^-) > 1$. Obviously, $w((\pi - 1)^+, x((\pi - 1)^+)) \ge 0$ if and only if $x((\pi - 1)^+) \le 0$. Hence, for the function $w(t, x(t))$ not to violate the unilateral constraint, the time-function $x(\cdot)$ must jump from a positive value greater than one to a non-positive value at the time $t = \pi - 1$. Let us try $x((\pi - 1)^+) = 0$. It can be checked that for each $t \ge \pi - 1$, $x(t) = 0$ and $\lambda(t) = 0$ satisfy (71) and thus $x(t) = 0$ is a global solution on $[\pi - 1, +\infty)$. This short analysis shows that the trajectory of (71) cannot be continuous (i.e., a state jump has to occur) at $t = \pi - 1$.

**FOSwP interpretation:** Embedding (71) into a FOSwP helps to understand the underlying mechanisms. In order to clarify the above choice (i.e., $x((\pi - 1)^+) = 0$), let us use the transformation
$$ \Leftrightarrow \left\{ \begin{array}{l} \hat{w} = \underbrace{\frac{1}{1+e} Q(t^+) C(t^+) P(t)^{-1} C(t^+)^\top Q(t^+)^\top \hat{\lambda}}_{K} + \underbrace{Q(t^+) C(t^+) R(t)^{-1} z(t^-) + Q(t^+) F(t^+) u(t^+)}_q, \\ 0 \le \hat{\lambda} \perp \hat{w} \ge 0. \end{array} \right. \qquad (82) $$

by using (6). Clearly LCP($q, K$) is solvable, where the symmetric matrix $K$ is positive semi-definite. Therefore, it can be solved using programming languages such as Python or Matlab to get the value of $\hat{\lambda}$, and then $z(t^+)$. Note that when $D(\cdot) \equiv 0$, under Assumptions 2, 4 then (3) is equivalent to (68), and thus at time of jumps, (3) is equivalent to (82) where $z(t^-) = R(t)x(t^-)$.

**Remark 7.** So far, to obtain the existence and uniqueness of an AC solution to (3) when $D(\cdot) \equiv 0$, either a combination of Proposition 3 (resp. Proposition 4) and Theorem 3, or Corollary 2 can be used. If all matrices in (3) are constant and $K(t) = \mathbb{R}_+^m$ for all $t \ge 0$, then these approaches (except Proposition 4) become equivalent. However, in the general case, they are different and can be used selectively based on each specific situation. This will be illustrated by examples in the next section.

## 2.4.2 Examples

Let us consider the circuit in Figure 3. Denote by $\lambda$ the current through the diode, and $w$ the voltage across the diode, $x(t) := [x_1(t) \ x_2(t)]^\top$, where $x_1, x_2$ denote the charge in the capacitor and the current through the inductor, respectively. The dynamics of this circuit is given by:

$$ \left\{ \begin{array}{l} \dot{x}(t) = \begin{bmatrix} \frac{1}{\mathbf{R}(t)\mathbf{C}(t)} & -1 \\ \frac{1}{\mathbf{L}(t)\mathbf{C}(t)} & 0 \end{bmatrix} x(t) + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \lambda(t) + \begin{bmatrix} -\frac{1}{\mathbf{R}(t)} \\ \frac{1}{\mathbf{L}(t)} \end{bmatrix} u(t), \\ w(t, x) = \begin{bmatrix} 1 \\ \mathbf{C}(t) \end{bmatrix} x(t) + u(t), \\ 0 \le \lambda(t) \perp w(t, x) \ge 0. \end{array} \right. $$

Let $\mathbf{C}(t) = \frac{t^2+2}{t^2+1}, u(t) = 1+\sin(t)$ and $\mathbf{R}(t), \mathbf{L}(t) > \rho$, for all $t \ge 0$, for some $\rho > 0$ such that $\mathbf{R}(\cdot), \mathbf{L}(\cdot)$ are measurable functions. It can be checked that Assumptions 4, 3, 2 hold with $P(t) = \text{diag}(\frac{1}{C(t)}, 1)$ is a solution of (66), and items 1, 2, 3, 5 in Corollary 2 hold. However, this system does not satisfy item 4 in Corollary 2. Indeed, let $s = 0$ and $t = 1$, we obtain:

$$ \begin{aligned} (C(s)R(s)^{-1})^{-1}(\mathbb{R}_+ - u(t)) &= \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -\sqrt{2}(1 + \sin(1))\} \\ &\not\subset \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -\sqrt{\frac{3}{2}}(1 + \sin(1))\} \\ &= (C(t)R(t)^{-1})^{-1}(\mathbb{R}_+ - u(t)). \end{aligned} $$

Therefore, the existence of an AC solution to this system is not guaranteed by Corollary 2. On the other hand, this system satisfies items 1, 2, and 3 in Proposition 3 with $Q(t) = \sqrt{C(t)}$, for all $t \ge 0$. Since $Q(\cdot)u(\cdot) = (1 + \sin(\cdot))\sqrt{\frac{(\cdot)^2+2}{(\cdot)^2+1}}$ is a locally AC function on $[0, +\infty)$, $\phi(\cdot)$ is locally AC using Proposition 3. Using Theorem 3, this system has a unique locally AC solution on $[0, +\infty)$ for every $x(0) \in \{[\xi_1 \ \xi_2]^\top \in \mathbb{R}^2 : \xi_1 \ge -2\}$.
proposed in [13] [10, Section 3.1] to rewrite (71) equivalently as the FOSwP:

$$ \dot{z}(t) - H(t)z(t) \in -\mathcal{N}_{\phi(t)}(z(t)), \qquad (72) $$

where $z(t) = R(t)x(t)$, $H(t) = (\dot{R}(t)R(t)^{-1} + 0.05)$ with $R(t) = \sqrt{\frac{t^2+2}{t^2+1}}$, $\dot{R}$ is the derivative of $R$, and $\phi(t) = \{x \in \mathbb{R} : C(t)R(t)^{-1}x \ge 0\}$. Let us now analyze solutions of (72), then solutions of (71). Because $\phi(0) = \mathbb{R}_+$ and $z(0) = \sqrt{2} > 0$, then $\dot{z}(0) = H(0)z(0) > 0$ is obtained by (72). It can be checked that $\dot{z}(t) = H(t)z(t)$ on $[0, \pi - 1)$, so a solution is given by $z(t) = R(t) \exp(0.05t)$. Since $\phi((\pi - 1)^-) = \mathbb{R}_+$ and $\phi((\pi - 1)^+) = \mathbb{R}_-$, the set $\phi(t)$ jumps at the time $t = \pi - 1$. This explains why the state jump occurs. Using the rule of state jump which is proposed in [16] and used in [10, 4, 5], then at time $t = \pi - 1$, the state jump is represented by:

$$ z(t^+) - z(t^-) \in -\mathcal{N}_{\phi(t^+)}(z(t^+)). \qquad (73) $$

This implies that $z(t^+) = \text{proj}_{\phi(t^+)}(z(t^-))$ using (4). Since $\phi((\pi - 1)^+) = \mathbb{R}_-$ and $z((\pi - 1)^-) > 0$, the unique post-jump solution is $z((\pi - 1)^+) = 0$. On $[\pi - 1, +\infty)$, it can be verified that $z(t) = 0$ is the unique solution of (72). This leads to the same result as the analysis above that the global solution is given by $x(t) = \exp(0.05t)$ if $t \in [0, \pi - 1)$, and $x(t) = 0$ if otherwise.

**A more general state-jump rule with nonzero restitution:** Following [40], let us now consider a second model of state jump defined by:

$$ z(t^+) - z(t^-) \in -\mathcal{N}_{\phi(t^+)} \left( \frac{z(t^+) + ez(t^-)}{1 + e} \right), \qquad (74) $$

where $e \in [0, 1]$ plays the role of a restitution coefficient, familiar in Contact Mechanics [41]. It follows that $z(t^+) = (1+e) \text{proj}_{\phi(t^+)}(z(t^-)) - ez(t^-)$ using (4). This model becomes (73) if $e = 0$. Let us analyze solutions of (72) in the case $e = 1$. Similarly as above, $z(t) = R(t) \exp(0.05t)$ is a solution on $[0, \pi - 1)$. It follows from (74) that at time $t = \pi - 1$, $z((\pi - 1)^+) = -z((\pi - 1)^-) < 0$. Then, it can be checked that $z(t) = -R(t) \exp(0.05t)$ is a solution on $[\pi - 1, 2\pi - 1)$. Repeating this process, the state jump occurs at times $t = k\pi - 1$, for any $k \in \mathbb{N} \setminus \{0\}$, and it is concluded that $z(t) = R(t) \exp(0.05t)$ if $t \in [k\pi - 1, (k+1)\pi - 1)$ for some $k \in \mathbb{N}$, $k$ is even, and $z(t) = -R(t) \exp(0.05t)$ if otherwise. Thus, the global solution is defined by $x(t) = \exp(0.05t)$ if $t \in [k\pi - 1, (k+1)\pi - 1)$ for some $k \in \mathbb{N}$, $k$ is even, and $x(t) = -\exp(0.05t)$ if otherwise. In general, let us consider (71) where $A(t) = a$ is a constant. Proceeding as above, a global solution is given by:

$$ x(t) = \begin{cases} e^k \exp(at), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, k \text{ is even,} \\ -e^k \exp(at), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, k \text{ is odd,} \end{cases} $$

which is RCLBV on $[0, +\infty)$ for all $a \in \mathbb{R}$ and $e \in [0, 1]$, even though the set-valued mapping $\phi(\cdot)$ is not RCLBV.

**Existence of finite accumulations of state jumps:** Let $C(t) = \sin(\frac{1}{t-1})$ in (3) (which is not BV on $[0, 1]$). Intuitively, there should exist a finite accumulation of state jumps at $t = 1$, because such $C(t)$ makes the admissible domain $C(t)x \ge 0$ jump with accumulation. Let us investigate this
<figure><img src="image_4.png" /><figcaption>Figure 3: Time-varying RLC circuit with an ideal diode.</figcaption></figure>

The next example illustrates a TVLCCS system that possesses a unique RCLBV solution. The following system is considered where $x(t) \in \mathbb{R}$ and $\lambda(t) \in \mathbb{R}^2$.

$$ \begin{cases} \dot{x}(t) = x(t) + B(t)\lambda(t), \\ 0 \le \lambda(t) \perp w(t, x) = C(t)x(t) + u(t) \ge 0, \end{cases} \qquad (83) $$

in which, $B(t) = [1 \ -1]$, $C(t) = \begin{bmatrix} t+1 \\ -(t+1) \end{bmatrix}$, and $u(t) = \begin{cases} [0 \ \exp(2t)]^\top, & \text{if } t < 2 \\ [1 \ 0]^\top, & \text{if } t \ge 2 \end{cases}$, for all $t \ge 0$.

The system (83) is a TVLCCS and satisfies all assumptions in Proposition 3. Indeed,

1. Obviously, Assumptions 4, 2 hold where $P(t) = t+1$ is a solution of (66).

2. We have $K^*(t) = \{\xi \in \mathbb{R}^2: Q(t)\xi \ge 0\}$, where $Q(t) = \text{diag}(\frac{1}{\sqrt{t+1}}, \frac{1}{\sqrt{t+1}}) \in \mathbb{R}^{2 \times 2}$. Moreover, $Q(t)C(t)R(t)^{-1} = [1 \ -1]^\top$ is a constant matrix for all $t \ge 0$. Hence, items 1, 2 in Proposition 3 are validated.

3. Clearly $K^*(t) - u(t) = \begin{cases} \mathbb{R}_+^2 - [0 \ \exp(2t)]^\top, & \text{if } t < 2 \\ \mathbb{R}_+^2 - [1 \ 0]^\top, & \text{if } t \ge 2 \end{cases}$, and $\text{rge}(C(t)) = \begin{cases} [1] \\ [-1] \end{cases}$, $\xi, \forall \xi \in \mathbb{R}$ is a constant linear space for all $t \ge 0$. Therefore, $\text{rge}(C(t)) \cap (K^*(t) - u(t)) \ne \emptyset$ for all $t \ge 0$. Using Remark 4, then item 3 in Proposition 3 holds.

4. $Q(t)u(t) = \begin{cases} [0 \ \frac{\exp(2t)}{\sqrt{t+1}}]^\top, & \text{if } t < 2 \\ [\frac{1}{\sqrt{t+1}} \ 0]^\top, & \text{if } t \ge 2 \end{cases}$, which yields $Q(\cdot)u(\cdot)$ is a RCLBV function on $[0, +\infty)$. Indeed, first, $Q(\cdot)u(\cdot)$ is obviously right-continuous at $t = 2$. Let any $T > 0$ and $(Qu)(t) := Q(t)u(t)$ for all $t \in [0, T]$, then the variation of $Q(\cdot)u(\cdot)$ over $[0, T]$ is given by:

$$ \begin{aligned} V_{[0,T]}(Qu) &= \sup_{\sigma \in \mathcal{P}} \sum_{i=0}^{N_{\sigma}-1} \|(Qu)(t_{i+1}) - (Qu)(t_i)\| \le \sup_{\sigma \in \mathcal{P}} \left( \sum_{i=0}^{i^*-1} \|(Qu)(t_{i+1}) - (Qu)(t_i)\| \right. \\ & \qquad \left. + \|(Qu)(t_{i^*+1}) - (Qu)(t_{i^*})\| + \sum_{i=i^*+1}^{N_{\sigma}-1} \|(Qu)(t_{i+1}) - (Qu)(t_i)\| \right) \\ &= \sup_{\sigma \in \mathcal{P}} \left( \frac{\exp(2t_{i^*})}{\sqrt{t_{i^*}+1}} - \frac{\exp(2t_0)}{\sqrt{t_0+1}} + \sqrt{\frac{1}{t_{i^*+1}+1} + \frac{\exp(4t_{i^*})}{t_{i^*}+1}} + \frac{1}{\sqrt{t_{i^*+1}+1}} - \frac{1}{\sqrt{t_{N_{\sigma}}+1}} \right), \end{aligned} $$
fact for the following TVLCCS:

$$ \begin{cases} \dot{x}(t) = 0.05x(t) + \frac{t^2+1}{t^2+2} \sin\left(\frac{1}{t-1}\right)\lambda(t), \\ 0 \le \lambda(t) \perp w(t, x(t)) = \sin\left(\frac{1}{t-1}\right)x(t) \ge 0, \quad x(0) = -1. \end{cases} \tag{75} $$

In the FOSwP format, the system (75) can be rewritten equivalently as:

$$ \dot{z}(t) - H(t)z(t) \in -\mathcal{N}_{\phi(t)}(z(t)), \tag{76} $$

where $z(t) = R(t)x(t)$, $H(t) = (\dot{R}(t)R(t)^{-1} + 0.05)$, and $\phi(t) = \{x \in \mathbb{R} : C(t)R(t)^{-1}x \ge 0\}$ with $R(t) = \sqrt{\frac{t^2+2}{t^2+1}}$. After few manipulations, $\phi(t)$ is calculated as:

$$ \phi(t) = \begin{cases} \mathbb{R}_+, & \text{if } t \in \left(1 + \frac{1}{k\pi}, 1 + \frac{1}{(k-1)\pi}\right) \cup \left(1 + \frac{1}{\pi}, +\infty\right), & \text{for some } k \in \mathbb{Z}\setminus\{0, 1\} \text{ is odd,} \\ \mathbb{R}, & \text{if } t = 1 + \frac{1}{k\pi}, & \text{for some } k \in \mathbb{Z}\setminus\{0\}, \\ \mathbb{R}_-, & \text{otherwise.} \end{cases} $$

Clearly $\phi(t)$ will switch an infinite number of times in the interval $[0, 2]$. In this example, to indicate finite accumulation, we consider the model of state jumps with restitution $e \in [0, 1)$ as in (74). At time $t = 1 + \frac{1}{k\pi}$ for some $k \in \mathbb{Z}\setminus\{0\}$, (76) can be rewritten as $z(t^+) = (1+e)\text{proj}_{\phi(t^+)}(z(t^-)) - ez(t^-)$, in view of (74). The unique solution to (75) is defined by:

$$ x(t) = \begin{cases} -\exp(0.05t), & \text{if } t \in [0, 1 - \frac{1}{\pi}), \\ (-1)^{k+1}e^k \exp(0.05t), & \text{if } t \in [1 - \frac{1}{k\pi}, 1 - \frac{1}{(k+1)\pi}), \text{ for some } k \in \mathbb{N}\setminus\{0\}, \\ 0, & \text{if } t \in [1, +\infty). \end{cases} $$

Moreover, it can be proven that the solution $x(\cdot)$ defined above is RCBV on $[0, +\infty)$, even though $\phi(\cdot)$ is not RCLBV. This conclusion holds for any $e \in [0, 1)$. The crucial property in the above example is the existence of an accumulation of jumps which implies that the state vanishes at the “singular” time $t = 1$s. The merit of this example is to prove that some TVLCCS are equivalent to an FOSwP with varying sets that are not RCLBV, but which have a unique RCBV solution with an accumulation of state jumps. When $e = 1$, it is difficult to give a meaning to the solution at $t = 1$s. The solution is BV on intervals $[0, 1 - \epsilon]$ and $[1 + \epsilon, \epsilon']$ with $\epsilon' > 1 + \epsilon, \epsilon > 0$. But the variation of $x(\cdot)$ on such interval tends to infinity when $\epsilon$ tends to zero. Also, it is unclear how to “cross” $t = 1$s.

### 3.1.2 Example 2: State jumps in TVLCCS with time-varying $K(t)$

Let us consider the following two-dimensional TVLCCS:

$$ \begin{cases} \dot{x}(t) = A(t)x(t) + B(t)\lambda(t), \\ K(t) \ni \lambda(t) \perp w(t, x(t)) = C(t)x(t) \in K^*(t), \quad x(0) = [1 \ 2]^\top, \end{cases} \tag{77} $$

where $x(t), \lambda(t) \in \mathbb{R}^2$, $A(t) = \text{diag}(a_1, a_2)$ for some $a_1, a_2 \in \mathbb{R}$, and $K(t) = [\mathbb{R}_+ (t_0 - t)\mathbb{R}_+]^\top$, $B(t) = C(t) = \text{diag}(b_1(t), b_2(t))$ such that on $[0, t_0)$, $b_1(t) > 0, b_2(t) = 0$, on $(t_0, +\infty)$, $b_1(t) = 0, b_2(t) > 0$, and $b_1(t_0) = b_2(t_0) = 0$ for some $t_0 \in \mathbb{R}_+$. Let us analyze solutions of (77). Firstly, we can rewrite (77) in the FOSwP format as:

$$ \dot{x}(t) - A(t)x(t) \in -\mathcal{N}_{\phi(t)}(x(t)), \tag{78} $$
where $\mathcal{P}$ is the set of all partition of $[0, T]$ and $i^*$ be an index such that $t_{i^*} < 2$ and $t_{i^*+1} \ge 2$. It follows that $V_{[0,T]}(Qu) < (\exp(4) - 1) + (\sqrt{1 + \exp(8)}) + (1 - \frac{1}{\sqrt{T+1}}) < +\infty$. This implies that $Q(\cdot)u(\cdot)$ is BV on $[0, T]$ for all $T > 0$, is hence LBV on $[0, +\infty)$.

Thus, $\phi(\cdot)$ defined in (69) is RCLBV on $[0, +\infty)$. Using Theorem 3, the system (83) has a unique RCLBV solution on $[0, +\infty)$, for every $x(0) \in [0, 1]$. In this case, the moving set $\phi(t)$ and the Hausdorff distance of $\phi(\cdot)$ at two arbitrary times are stated as follows:

$$ \phi(t) = \begin{cases} \begin{bmatrix} 0, \frac{\exp(2t)}{\sqrt{t+1}} \\ \frac{-1}{\sqrt{t+1}}, 0 \end{bmatrix}, & \text{if } t < 2 \\ \begin{bmatrix} 0, \frac{\exp(2t)}{\sqrt{t+1}} \\ \frac{-1}{\sqrt{t+1}}, 0 \end{bmatrix}, & \text{if } t \ge 2 \end{cases}, \quad d_H(\phi(t), \phi(s)) = \begin{cases} \frac{\exp(2t)}{\sqrt{t+1}} - \frac{\exp(2s)}{\sqrt{s+1}}, & \text{if } s \le t < 2 \\ \max\left\{\frac{\exp(2s)}{\sqrt{s+1}}, \frac{1}{\sqrt{t+1}}\right\}, & \text{if } s < 2 \le t \\ \frac{1}{\sqrt{s+1}} - \frac{1}{\sqrt{t+1}}, & \text{if } 2 \le s \le t \end{cases} $$

This implies that the Robinson's bound (70) in Proposition 3 holds with $\gamma_K = 1$. Using Remark 6, at jump time $t = 2$ then (83) is equivalent to the following:

$$ \begin{cases} x(t^+) - x(t^-) = \begin{bmatrix} \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{3}} \end{bmatrix} \hat{\lambda}, \\ \hat{w} = \frac{1}{1+e} \begin{bmatrix} \sqrt{3} & -\sqrt{3} \end{bmatrix}^\top (x(t^+) + ex(t^-)) + \begin{bmatrix} \frac{1}{\sqrt{3}} & 0 \end{bmatrix}^\top, \\ 0 \le \hat{\lambda} \perp \hat{w} \ge 0. \end{cases} $$

It is rewritten as: $0 \le \hat{\lambda} \perp \hat{w} = \frac{1}{1+e} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \hat{\lambda} + \begin{bmatrix} \sqrt{3} \\ -\sqrt{3} \end{bmatrix} x(t^-) + \begin{bmatrix} \frac{1}{\sqrt{3}} \\ 0 \end{bmatrix} \ge 0$, where $e \in [0, 1]$ is the restitution coefficient. To obtain a numerical simulation of (83), let us time-discretize this system. The following scheme is interpreted as utilizing a state jump model with $e = 0$.

### A Direct Backward (Implicit) Euler Scheme

Let $[0, T]$, $T > 0$ be the interval that we need to obtain the solution. Imitating the ideas for discretizing the LCS systems proposed in [5], the time-discretization of (83) is:

$$ \begin{cases} x_{k+1} - x_k = h x_{k+1} + h B_{k+1} \lambda_{k+1}, \\ 0 \le \lambda_{k+1} \perp w_{k+1} = C_{k+1} x_{k+1} + u_{k+1} \ge 0, \end{cases} \quad (84) $$

where $(t_0, t_1, ..., t_n)$ is a partition of $[0, T]$ with $t_{k+1} - t_k = h > 0$ the time step, $n = \frac{T}{h}$, and $x_k := x(t_k)$, $\lambda_k := \lambda(t_k)$, $w_k := w(t_k, x_k)$, $B_{k+1} := B(t_{k+1})$, $C_{k+1} := C(t_{k+1})$, $u_{k+1} := u(t_{k+1})$, for all $k = 0, n-1$. With $h$ small enough, after some algebraic manipulations, (84) is equivalent to:

$$ \begin{cases} x_{k+1} = \frac{1}{1-h} (x_k + h B_{k+1} \lambda_{k+1}), \\ 0 \le \lambda_{k+1} \perp (h C_{k+1} B_{k+1}) \lambda_{k+1} + C_{k+1} x_k + (1-h) u_{k+1} \ge 0, \end{cases} \quad (85) $$

where for each $k \in \{0, 1, ..., n-1\}$, the second line of (85) represents the LCP($q(t_{k+1})$, $M$) defined by:

$$ 0 \le \lambda_{k+1} \perp \underbrace{\begin{bmatrix} h & 1 \\ -1 & 1 \end{bmatrix}}_{M(t_{k+1})=M \ge 0} \lambda_{k+1} + \underbrace{\begin{bmatrix} 1 \\ -1 \end{bmatrix} x_k + (1-h) \frac{u_{k+1}}{t_{k+1}+1}}_{q(t_{k+1})} \ge 0. $$
where $\phi(t) = \{x \in \mathbb{R}^2 : C(t)x(t) \in \mathcal{K}^*(t)\}$. It follows that $\phi(t) = [\mathbb{R}_+ \mathbb{R}]^\top$ if $t < t_0$, $\phi(t) = \mathbb{R}^2$ if $t = t_0$, and $\phi(t) = [\mathbb{R} \mathbb{R}_-]^\top$ if $t > t_0$. Obviously, $\phi(t)$ jumps at time $t_0$, and the Hausdorff distance $d_H(\phi(t_0^-), \phi(t_0^+)) = +\infty$. On $[0, t_0)$, since $x(t) \in \text{int}(\phi(t))$, (78) can be rewritten as $\dot{x}(t) = A(t)x(t)$, which implies that $x(t) = [\exp(a_1t) \ 2\exp(a_2t)]^\top > 0$. Since $x(t_0^-) > 0$ and $\phi(\cdot)$ switches at time $t = t_0$, the second component of the state $x(\cdot)$ jumps at $t = t_0$. Using the model of state jump with restitution coefficient $e \in [0, 1]$ as in (74), we obtain:

$$x(t_0^+) - x(t_0^-) \in -\mathcal{N}_{\phi(t_0^+)} \left( \frac{x(t_0^+) + ex(t_0^-)}{1+e} \right).$$

Therefore, $x(t_0^+) = (1+e)\text{proj}_{\phi(t_0^+)}(x(t_0^-)) - ex(t_0^-) = [\exp(a_1t_0) - 2e \exp(a_2t_0)]^\top$. On $[t_0, +\infty)$, proceeding as above, $x(t)$ has the form $x(t) = [\exp(a_1t) - 2e \exp(a_2t)]^\top$. From all of the above, a global solution of the system (77) is given by $x(t) = [\exp(a_1t) \ 2\exp(a_2t)]^\top$ if $t < t_0$, and $x(t) = [\exp(a_1t) - 2e \exp(a_2t)]^\top$ if $t \ge t_0$. In can be concluded that (77) always has a RCLBV solution for all $a_1, a_2 \in \mathbb{R}$ and for all $e \in [0, 1]$.

### 3.1.3 Recapitulation

These examples have shown important differences concerning the ability to transform TVLCCS into FOSwP, as well as the jump times of the state in TVLCCS, compared to LCS, which have been studied in [10, 4, 9, 8, 42]. Firstly, it is possible to transform from TVLCCS to FOSwP and vice versa, even when $\phi(\cdot)$ (i.e., time-varying set-valued mapping in FOSwP) is not LBV, which violates a usual condition in the analysis of FOSwP used in Theorem 3, and in [4, Section 5.2, (A3)], [43, 10]. This is impossible for LCS. For instance, the FOSwP: $-\dot{x}(t) \in \mathcal{N}_{\phi(t)}(x(t))$, $x(0) = x_0 \in \phi(0)$, $\phi(t) = \mathbb{R}_+$ if $t \in [0, t_1)$, $\phi(t) = \mathbb{R}_-$ if $t \in [t_1, +\infty)$, cannot be shown to be equivalent to any LCS (1) with $w(t, x) = Cx + Fu(t)$. The key to this issue lies in the time variation of $C(\cdot)$. The matrix $C(t)$ in (3) plays a significant role in the study of TVLCCS. Secondly, when LCS (1) is autonomous (i.e., $E = F = 0$), results in [8, 42] have shown that state jumps occur only at the initial time due to inconsistent initialization, i.e., $Cx(0) \notin (\text{SOL}(0, D))^*$. When LCS (1) has time-varying external inputs, the results in [9] pointed out that state jumps occur at arbitrary time $t$ where $Fu(t)$ is discontinuous (i.e., $\{t : x(t^+) \neq x(t^-)\} \subseteq \{t : Fu(t^+) \neq Fu(t^-)\}$) or at the initial time if $Cx(0) + Fu(0) \notin (\text{SOL}(0, D))^*$. In other words, if $Fu(\cdot)$ is continuous then $x(\cdot)$ is continuous (excepted possibly at the initial time). However, in the examples above, even though $F(\cdot)u(\cdot) = 0$ is a continuous function, the state $x(\cdot)$ is discontinuous at $t = \pi - 1$ in (71), and at $t = t_0$ in (77) even if $C(\cdot)$ is a continuous function and at $t = 1 - \frac{1}{\pi}$ in (75). This is a new finding when studying TVLCCS.

## 3.2 Analysis of the causes of state jumps in Section 3.1

In this section, we will use the theoretical results presented in Section 2.4 to clarify the reason why state jumps appear in examples in Section 3.1. Firstly, let us come back to the system (71). Note that state jumps occur when $T > \pi - 1$. It can be checked that (71) does not satisfy Assumption 3 and items 2, 4 in Corollary 2, where $P(t) = \frac{t^2+2}{t^2+1}$ is a solution of (56). Indeed, we have:

$$\text{rgc}(C(t)) = \begin{cases} 0, & \text{if } t = k\pi - 1 \text{ for some } k \in \mathbb{N} \setminus \{0\}, \\ \mathbb{R}, & \text{otherwise.} \end{cases}$$

Thus, item 2 fails. Let $s = \frac{\pi-1}{2}$, $t = \frac{3\pi-3}{2}$, then $s < t$ and $(C(s)R(s)^{-1})^{-1}(\mathbb{R}_+) = \mathbb{R}_+ \not\subseteq \mathbb{R}_- = (C(t)R(t)^{-1})^{-1}(\mathbb{R}_+)$. So item 4 fails. Finally, Assumption 3 fails due to the following:
Since item 3 holds, $q(t_{k+1}) \in \text{rg}(C_{k+1}) + \mathbb{R}_+^2 = (\text{SOL}(0, M))^*$ for all $k = 0, n-1$. Clearly $M = M^\top \succcurlyeq 0$, is thus copositive-plus [22, Exercise 3.12.1]. Therefore, for each $k \in \{0, 1, ..., n-1\}$, the LCP($q(t_{k+1})$, $M$) has a solution, using [22, Theorem 3.8.6]. Moreover, the LCP($q(t_{k+1})$, $M$) is completely equivalent to the following quadratic program (QP) [22, p.5]:

$$ \begin{array}{ll} \text{minimize} & f(\lambda) = \frac{1}{2}\lambda^\top M\lambda + q(t_{k+1})^\top \lambda \\ \text{subject to} & \lambda \ge 0. \end{array} $$

This QP is solved by CVXPY (a domain-specific language for convex optimization) in Python. Take $h = 5 \times 10^{-5}s$ and $x(0) = 0.5$, then the numerical simulation of (83) is depicted in Figure 4.

<figure><img src="image_5.png" /><figcaption>Figure 4: Simulation of (83) with Backward Euler algorithm, time step $h = 5 \times 10^{-5}s$, the initial condition $x(0) = 0.5$.</figcaption></figure>
$$ \text{rgc}(C(t)) \cap \text{rint}(\mathbb{R}_+) = \begin{cases} \emptyset, & \text{if } t = k\pi - 1 \text{ for some } k \in \mathbb{N} \setminus \{0\}, \\ \mathbb{R}_+ \setminus \{0\}, & \text{otherwise.} \end{cases} $$

Therefore, the existence of an AC solution to (71) is not guaranteed by Corollary 2. In addition, it can also be verified that there is no mapping $Q(\cdot)$ that satisfies assumptions in Proposition 3. Proposition 4 can not be used either since $\phi(t)$ in the FOSwP (72), which is equivalent to (71), is not bounded. Hence, we may expect that state jumps will be necessary, and they will be analyzed by certain conditions beyond the results in this article. Secondly, let us consider the system (75) (resp. (77)) and note that state jumps occur when $T > 1 - \frac{1}{\pi}$ (resp. $T > t_0$). Similar to the above analysis, it is verified that (75) does not satisfy Assumption 3, items 2, 4, and 5 in Corollary 2, since $C(\cdot)$ is not right-continuous at $t = 1s$ (resp. (77) does not satisfy Assumption 3, items 1, 2, 4, and 5 in Corollary 2, since $C(\cdot)$ is not right-continuous and $K^*(\cdot)$ is not inner semicontinuous from right at $t_0$). In addition, these examples also do not satisfy assumptions in Propositions 3, 4. Thus state jumps (as described in Section 3.1) can be expected for these system.

### 3.3 Further comments

Some assumptions which are made in the foregoing theoretical results deserve to be relaxed. It would be interesting to relax items 1 (iii) and 2 in Theorem 2. As the examples in Section 3.1 demonstrate, the role played by $C(\cdot)$ is prominent in TVLCCS. It yields equivalent FOSwP with switching unbounded sets for which the Hausdorff distance is unsuitable, and with Zeno behavior (infinitely many switches in a finite time). In this article, TVLCCS are studied through the analysis of Lur'e DIs as in (10). As pointed out after the proof of Proposition 1, one can study Lur'e DIs which do not stem from TVLCCS. In this setting, extending the results in [19, 18] to varying matrices and/or cones which depend also on the state, certainly is of interest. The above examples show that modifying the state jump rule can significantly influence the solution's behavior. Extensions to higher dimensions could be tackled.

## 4 Conclusions

This article deals with the well-posedness of finite-dimensional time-varying linear cone complementarity systems. All the system's data are allowed to depend on time, including the cones defining the complementarity constraints. The passivity of a suitable operator is a key assumption. It allows us to transform the dynamics into a suitable form known as Lur'e differential inclusion. Various cases are tackled depending on the system's data, with absolutely continuous or right-continuous of local bounded variation solutions. Examples from electrical circuits illustrate the theoretical developments.

# Appendices

## A Some useful definitions and results

**Theorem A1.** [26, Proposition C.3.8] Consider the following nonlinear differential equation:

$$ \dot{x}(t) = f(t, x(t)), \quad t \ge 0; \quad x(0) = x_0, \qquad (1) $$

where $x(t) \in \mathbb{R}^n$ and $f: [0, +\infty) \times \mathbb{R}^n \to \mathbb{R}^n$. Assume that the following conditions hold:
# 3 Further examples and future works

## 3.1 Simple examples

Let us illustrate some peculiarities of TVLCCS (3) compared to LCS (1), especially concerning state jumps (LCS possess state jumps at the initial time, or may jump at times when $Fu(t)$ is discontinuous [9, 10]). To this aim, several scalar examples are considered, to which the foregoing theoretical developments do not apply, and may thus motivate further investigations.

### 3.1.1 Example 1: State jumps in TVLCCS with constant $K(t)$

The following scalar system is considered where $x(t) \in \mathbb{R}$ and $\lambda(t) \in \mathbb{R}$.

$$ \begin{cases} (a) \dot{x}(t) = 0.05x(t) + \frac{t^2+1}{t^2+2} \sin(t+1)\lambda(t), \\ (b) 0 \le \lambda(t) \perp w(t, x(t)) = \sin(t+1)x(t) \ge 0, x(0) = 1. \end{cases} \tag{86} $$

The system (86) is a special case of (3) when $A(t) = 0.05$, $B(t) = \frac{t^2+1}{t^2+2} \sin(t+1)$, $C(t) = \sin(t+1)$, $D(t) = E(t) = F(t) = 0$ and $K(t) = K^*(t) = \mathbb{R}_+$, for all $t \ge 0$. Let us analyze solutions of (86).

**Necessity of state jumps:** From (86)(b), we get $\lambda(0) = 0$ since $w(0, x(0)) = \sin(1) > 0$. It is deduced from (86)(a) that $\dot{x}(0) = 0.05x(0)$. We conclude that for each $t \in [0, \pi - 1)$ then $\lambda(t) = 0$ and $\dot{x}(t) = 0.05x(t) > 0$, so a solution is given by $x(t) = \exp(0.05t)$ on $[0, \pi - 1)$. Therefore, $x((\pi - 1)^-) > 1$. Obviously, $w((\pi - 1)^+, x((\pi - 1)^+)) \ge 0$ if and only if $x((\pi - 1)^+) \le 0$. Hence, for the function $w(t, x(t))$ not to violate the unilateral constraint, the time-function $x(\cdot)$ must jump from a positive value greater than one to a non-positive value at the time $t = \pi - 1$. Let us try $x((\pi - 1)^+) = 0$. It can be checked that for each $t \ge \pi - 1$, $x(t) = 0$ and $\lambda(t) = 0$ satisfy (86) and thus $x(t) = 0$ is a global solution on $[\pi - 1, +\infty)$. This short analysis shows that the trajectory of (86) cannot be continuous (i.e., a state jump has to occur) at $t = \pi - 1$.

**FOSwP interpretation, and time-discretization:** Embedding (86) into a FOSwP helps to understand the underlying mechanisms. In order to clarify the above choice (i.e., $x((\pi - 1)^+) = 0$), let us use the transformation proposed in [13] [10, Section 3.1] to rewrite (86) equivalently as the FOSwP:

$$ \dot{z}(t) - H(t)z(t) \in -\mathcal{N}_{\phi(t)}(z(t)), \tag{87} $$

where $z(t) = R(t)x(t)$, $H(t) = (\dot{R}(t)R(t)^{-1} + 0.05)$ with $R(t) = \sqrt{\frac{t^2+2}{t^2+1}}$, $\dot{R}$ is the derivative of $R$, and $\phi(t) = \{x \in \mathbb{R} : C(t)R(t)^{-1}x \ge 0\}$. Let us propose a time-discretization of (87). Moreau's catching-up algorithm [16] (which can be viewed as an extension of the backward (implicit) Euler algorithm) for (87) reads as: $z_{k+1} \in z_k + hH_{k+1}z_{k+1} - \mathcal{N}_{\phi_{k+1}}(z_{k+1})$, where $z_k := z(t_k)$, $H_{k+1} := H(t_{k+1})$, $\phi_{k+1} := \phi(t_{k+1})$ and $h$ is a sufficiently small time step. Set $F(z_{k+1}) := (1 - hH_{k+1})z_{k+1} - z_k$, then the previous expression is equivalent to $0 \in F(z_{k+1}) + \mathcal{N}_{\phi_{k+1}}(z_{k+1})$. Since $(1 - hH_{k+1})$ is positive for sufficiently small $h > 0$, $F(\cdot)$ is strongly monotone for any given $z_k$. In addition, since $\phi_{k+1}$ is a closed convex set for all $k \ge 0$, the generalized equation $0 \in F(z_{k+1}) + \mathcal{N}_{\phi_{k+1}}(z_{k+1})$ has a unique solution (see e.g., [5, Theorem 2.35]). However, we could not use the expression $0 \in F(z_{k+1}) + \mathcal{N}_{\phi_{k+1}}(z_{k+1})$ to numerically simulate (86), but must rewrite it in the form $z_{k+1} = \text{proj}_{\phi_{k+1}}((1 - hH_{k+1})^{-1}z_k)$ using (4). A numerical simulation of (86) is shown in Figure 5 and it demonstrates the existence of a jump at $t = \pi - 1$, as predicted.
1. $f(\cdot, x) : [0, +\infty) \to \mathbb{R}^n$ is measurable for each fixed $x$.

2. $f(t, \cdot) : \mathbb{R}^n \to \mathbb{R}^n$ is continuous for each fixed $t$.

3. There is a locally integrable function $\alpha : [0, +\infty) \to \mathbb{R}_+$ such that for each $t \ge 0$:

$$ \|f(t, x) - f(t, y)\| \le \alpha(t)\|x - y\|, \quad \forall x, y \in \mathbb{R}^n. $$

4. For each fixed $x_0 \in \mathbb{R}^n$, there is a locally integrable function $\beta : [0, +\infty) \to \mathbb{R}_+$ such that:

$$ \|f(t, x_0)\| \le \beta(t), \quad \text{for almost all } t. $$

Then, there exists a unique locally AC solution to (i1) on $[0, +\infty)$, for any $x_0 \in \mathbb{R}^n$.

**Lemma A1. (Hoffman's bound)** [44, Lemma 3.2.3] For any matrices $C \in \mathbb{R}^{l \times n}$ and $A \in \mathbb{R}^{m \times n}$, there exists a constant $c > 0$ (depending only on $C$ and $A$) such that for all vectors $d \in \mathbb{R}^l$ and $b \in \mathbb{R}^m$ for which $P := \{x \in \mathbb{R}^n : Cx = d, Ax \le b\}$ is nonempty,

$$ \mathrm{dist}(x, P) \le c(\|Cx - d\| + \|\max(0, Ax - b)\|), \quad \forall x \in \mathbb{R}^n. $$

Denote by $P(A, b) := \{x \in \mathbb{R}^n : Ax \le b\}$ a convex polyhedral set, for some $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$.

**Corollary A1.** [44, Corollary 3.2.5] Let the matrix $A \in \mathbb{R}^{m \times n}$ be given. Then, for every bounded subset $S$ of $\mathbb{R}^n$, there exists a constant $L > 0$ such that for every vector $b \in \mathbb{R}^m$ for which $P(A, b)$ is nonempty and for every $x \in S \cap P(A', b')$,

$$ \mathrm{dist}(x, P(A, b)) \le L (\|A - A'\| + \|b - b'\|). $$

Note that $L$ is a bound of $S$.

**Proposition A1. (Chain rule)** [29, 21, 4] Let $f : \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ be a proper convex lower semicontinuous function, and $A : \mathbb{R}^n \to \mathbb{R}^m$ be a linear operator. Assume that either the function $f$ is polyhedral or for some $x_0$ with $Ax_0 \in \mathrm{dom}(f)$: $\mathrm{rge}(A) - \mathbb{R}_+(\mathrm{dom}(f) - Ax_0)$ is a vector subspace of $\mathbb{R}^m$, where $\mathrm{dom}(f) := \{y \in \mathbb{R}^m : f(y) < +\infty\}$. Then the subdifferential in the sense of convex analysis of the composite functional $f \circ A : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is given by:

$$ \partial(f \circ A)(x) = A^{\top} \partial f(Ax), \quad \forall x \in \mathbb{R}^n. $$

**Lemma A2. (Robinson's bound)** [45, p.760] Let $\mathcal{K} := \{y \in \mathbb{R}^m : Dy \ge 0\}$ be a closed convex polyhedral cone, for some matrix $D$. Let $S := \{x \in \mathbb{R}^n : Ax - b \in \mathcal{K}\}$, and $S' := \{x \in \mathbb{R}^n : Ax - b' \in \mathcal{K}\}$, where $A \in \mathbb{R}^{m \times n}$ is a matrix, and $b, b' \in \mathbb{R}^m$ are vectors. Suppose that $S$ and $S'$ are nonempty, then there exist a constant $\gamma_{A, \mathcal{K}}$ (depending only on $A$ and $\mathcal{K}$) such that:

$$ d_H(S, S') \le \gamma_{A, \mathcal{K}} \|D\| \|b - b'\|. $$

**Definition A1.** [10, 34, 4] Consider the following FOSwP:

$$ \dot{z}(t) \in -\mathcal{N}_{\phi(t)}(z(t)) + f(t, z(t)), \quad z(0) = z_0 \in \phi(0). \quad (i2) $$

Assume that $\phi(\cdot)$ is RCLBV on $[0, +\infty)$, that is, for each $T > 0$, there exists a positive Radon measure $\mu$ on $[0, T]$ such that $d_H(\phi(s), \phi(t)) \le \mu((s, t])$, for all $0 \le s \le t \le T$. Assume further that
Let us now analyze solutions of (87), then solutions of (86). Because $\phi(0) = \mathbb{R}_+$ and $z(0) = \sqrt{2} > 0$, then $\dot{z}(0) = H(0)z(0) > 0$ is obtained by (87). It can be checked that $\dot{z}(t) = H(t)z(t)$ on $[0, \pi - 1)$, so a solution is given by $z(t) = c \exp \left( \int_0^t H(v) dv \right)$ for some $c \in \mathbb{R}$. Since $z(0) = \sqrt{2}$, this implies that $c = \sqrt{2}$, and $z(t) = R(t) \exp(0.05t)$. Since $\phi((\pi - 1)^+) = \mathbb{R}_+$ and $\phi((\pi - 1)^+) = \mathbb{R}_-$, the set $\phi(t)$ jumps at the time $t = \pi - 1$. This explains why the state jump occurs. Using the rule of state jump which is proposed in [16] and used in [10], [5, Section 2.4.1], and [4, Section 5.2], then at time $t = \pi - 1$, the state jump is represented by:

$$z(t^+) - z(t^-) \in -\mathcal{N}_{\phi(t^+)}(z(t^+)). \qquad (88)$$

This implies that $z(t^+) = \text{proj}_{\phi(t^+)}(z(t^-))$ using (4). Since $\phi((\pi - 1)^+) = \mathbb{R}_-$ and $z((\pi - 1)^-) > 0$, the unique post-jump solution is $z((\pi - 1)^+) = 0$. On $[\pi - 1, +\infty)$, it can be verified that $z(t) = 0$ is the unique solution of (87). This leads to the same result as the analysis above that the global solution is given by $x(t) = \exp(0.05t)$ if $t \in [0, \pi - 1)$, and $x(t) = 0$ if otherwise, is unique in the sense that if there is another global solution $\bar{x}(t)$ then $x(t) = \bar{x}(t)$ almost everywhere on $[0, +\infty)$.

<figure><img src="image_6.png" /><figcaption>Figure 5: Simulation of (86) using the catching-up algorithm with time step $h = 10^{-4}s$.</figcaption></figure>

**A more general state-jump rule with nonzero restitution:** Following [41], let us now consider a second model of state jump defined by:

$$z(t^+) - z(t^-) \in -\mathcal{N}_{\phi(t^+)} \left( \frac{z(t^+) + ez(t^-)}{1 + e} \right), \qquad (89)$$

where $e \in [0, 1]$ plays the role of a restitution coefficient, familiar in Contact Mechanics [43]. It follows that $z(t^+) = (1 + e) \text{proj}_{\phi(t^+)}(z(t^-)) - ez(t^-)$ using (4). This model becomes (88) if $e = 0$. Let us analyze solutions of (87) in the cases $e = 1$ and $e = 0.7$. When $e = 1$, similarly as above, $z(t) = R(t) \exp(0.05t)$ is a solution on $[0, \pi - 1)$. It follows from (89) that at time $t = \pi - 1$, $z((\pi - 1)^+) = -z((\pi - 1)^-) < 0$. Then, it can be checked that $z(t) = -R(t) \exp(0.05t)$ is a solution on $[\pi - 1, 2\pi - 1)$. Repeating this process, the state jump occurs at times $t = k\pi - 1$, for any $k \in \mathbb{N} \setminus \{0\}$, and it is concluded that $z(t) = R(t) \exp(0.05t)$ if $t \in [k\pi - 1, (k + 1)\pi - 1)$ for some
there exists a locally integrable function $\beta(\cdot) : [0, +\infty) \to \mathbb{R}_+$ such that $\|f(t, z)\| \le \beta(t)(1 + \|z\|)$ for all $z \in \mathbb{R}^n$, for all $t \ge 0$. A mapping $z : [0, +\infty) \to \mathbb{R}^n$ is a solution of (i2) if:

(i) $z(\cdot)$ is RCLBV on $[0, +\infty)$ and satisfies $z(0) = z_0$, $z(t) \in \phi(t)$ for all $t \ge 0$;

(ii) there exists a positive Radon measure $\nu$ AC equivalent to the measure $\mu + \lambda$ and with respect to which the differential measure $dz$ of $z(\cdot)$ is AC with the density $\frac{dz}{d\nu}$ is a locally integrable function from $[0, +\infty)$ to $\mathbb{R}^n$ with respect to the measure $\nu$, and the following holds:

$$ \frac{dz}{d\nu}(t) \in -\mathcal{N}_{S(t)}(z(t)) + f(t, z(t)) \frac{d\lambda}{d\nu}(t), \quad \nu \text{ - almost everywhere on } [0, +\infty), $$

where $\lambda$ is a one-dimensional Lebesgue measure, and $\frac{d\lambda}{d\nu}(\cdot)$ is the density relative to $\nu$ of $\lambda$.

## B The proof of Remark 1 when $D(t) = D(t)^\top \succcurlyeq 0$

Firstly, for each $t \in [0, T]$, since $\mathrm{rge}(D(t)) = (\ker D(t)^\top)^\perp$ [30, (2.4.14)-p.95], using the symmetry of $D(t)$ and [21, Example 6.23], we get $\mathrm{rge}(D(t)) = (\ker D(t))^*$. Therefore, $\mathcal{K}^*(t) + \mathrm{rge}(D(t)) = (\mathcal{K}(t) \cap \ker D(t))^*$ [21, Corollary 11.25]. So now, let us prove that $\mathcal{H}(t) = (\mathcal{K}(t) \cap \ker D(t))^*$, for all $t \in [0, T]$. Indeed, let any $t \in [0, T]$.

Let $[\subseteq]$ Let any $y \in \mathcal{H}(t)$, then $\hat{Q}(t)Q(t)y \ge 0$, that is, $Q(t)y \in (\mathrm{SOL}(0, Q(t)D(t)Q(t)^\top))^*$. This implies that

$$ \langle Q(t)y, z \rangle \ge 0, \quad \forall z \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top). \qquad (ii1) $$

Take any $\alpha \in \mathcal{K}(t) \cap \ker D(t)$, then there exists $\xi \in \mathbb{R}_+^j$ such that $\alpha = Q(t)^\top \xi$ and $D(t)\alpha = 0$. Therefore, we obtain $Q(t)D(t)Q(t)^\top \xi = 0$, and thus $\xi \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top)$. It follows from (ii1) that $\langle y, \alpha \rangle = \langle y, Q(t)^\top \xi \rangle = \langle Q(t)y, \xi \rangle \ge 0$. Because $\alpha$ is chosen arbitrarily, $\langle y, \alpha \rangle \ge 0$ for all $\alpha \in \mathcal{K}(t) \cap \ker D(t)$. This means $y \in (\mathcal{K}(t) \cap \ker D(t))^*$.

Let $[\supseteq]$ Let any $h \in (\mathcal{K}(t) \cap \ker D(t))^*$, then $\langle h, z \rangle \ge 0$ for all $z \in \mathcal{K}(t) \cap \ker D(t)$. To obtain $h \in \mathcal{H}(t)$, we shall prove that $Q(t)h \in (\mathrm{SOL}(0, Q(t)D(t)Q(t)^\top))^*$. Indeed, take any $\alpha \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top)$, then we have $\alpha \in \mathbb{R}_+^j$ and $\langle \alpha, Q(t)D(t)Q(t)^\top \alpha \rangle = 0$. Since $D(t) = D(t)^\top \succcurlyeq 0$, there exists $\tilde{D}(t) = \tilde{D}(t)^\top \succcurlyeq 0$ such that $\tilde{D}(t)^2 = D(t)$. Moreover, it follows from the symmetry of $\tilde{D}(t)$ that $\ker D(t) = \ker \tilde{D}(t)$. Therefore, we get $0 = \langle \alpha, Q(t)D(t)Q(t)^\top \alpha \rangle = \langle \tilde{D}(t)Q(t)^\top \alpha, \tilde{D}(t)Q(t)^\top \alpha \rangle = \|\tilde{D}(t)Q(t)^\top \alpha\|^2$. Hence, we get $Q(t)^\top \alpha \in \ker \tilde{D}(t) = \ker D(t)$. Alternatively, since $\alpha \in \mathbb{R}_+^j$, $Q(t)^\top \alpha \in \mathcal{K}(t)$. Therefore, $Q(t)^\top \alpha \in \mathcal{K}(t) \cap \ker D(t)$ and thus, $\langle h, Q(t)^\top \alpha \rangle \ge 0$, i.e., $\langle Q(t)h, \alpha \rangle \ge 0$.

## C The proof of Theorem 1 in Section 2.2.3

The proof is the same as the proof of [20, Theorem 23], with some small changes in steps 1 and 2 as shown below. In step 1: At the beginning of the proof, using (A1) and the Lipschitzness of $f(t, \cdot)$, then for all $t \in [0, T]$ we also obtain:

$$ \begin{aligned} \langle \dot{x}_{l+1}(t) - \dot{x}_l(t), x_{l+1}(t) - x_l(t) \rangle &\le \langle f(t, x_l(t)) - f(t, x_{l-1}(t)), x_{l+1}(t) - x_l(t) \rangle \\ &\le L_f \|x_l(t) - x_{l-1}(t)\| \|x_{l+1}(t) - x_l(t)\| \\ &\le L_f \|x_l(t) - x_{l-1}(t)\| \|x_{l+1}(t) - x_l(t)\|. \end{aligned} $$

where $L_f(t)$ is a (bounded) Lipschitz constant and $L_f := \max_{t \in [0, T]} (L_f(t))$. In step 2: At the beginning of the case where $F$ satisfies (LG) (see [20, p.1047]), let $\varphi_l(t) := \int_0^t \|f(\tau, x_{l-1}(\tau))\| + \|\bar{u}(\tau)\| d\tau$ for
$k \in \mathbb{N}$, $k$ is even, and $z(t) = -R(t) \exp(0.05t)$ if otherwise. Thus, the global solution is defined by $x(t) = \exp(0.05t)$ if $t \in [k\pi - 1, (k+1)\pi - 1)$ for some $k \in \mathbb{N}$, $k$ is even, and $x(t) = -\exp(0.05t)$ if otherwise. When $e = 0.7$, doing the same analysis as above then a global solution is given by:

$$x(t) = \begin{cases} (0.7)^k \exp(0.05t), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, k \text{ is even,} \\ -(0.7)^k \exp(0.05t), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, k \text{ is odd.} \end{cases}$$

To numerically simulate (86) in the cases $e = 1$ or $0.7$, we shall discretize the following measure differential inclusion (MDI) [16, 5, 42]:

$$dz - H(t)z(t)dt \in -\mathcal{N}_{\phi(t^+)} \left( \frac{z(t^+) + ez(t^-)}{1+e} \right), \quad (90)$$

where $dz$ is the differential measure associated with the function $z(\cdot)$, which is expected to be LBV on $[0, +\infty)$ based on the mentioned analysis. Clearly (90) is a combination of (87) and (89). If the state is continuous at $t$ then $z(t^+) = z(t^-) = z(t)$, $\frac{dz}{dt} = \dot{z}(t)$, and (90) becomes (87). Otherwise, that is, the state jump occurs at $t$, then (90) is equivalent to the MDI:

$$dz(\{t\}) \in -\mathcal{N}_{\phi(t^+)} \left( \frac{z(t^+) + ez(t^-)}{1+e} \right). \quad (91)$$

Since $dz(\{t\}) = z(t^+) - z(t^-)$, (90), (91) become (89). Let us consider (90) on the interval $(t_k, t_{k+1}]$:

$$dz((t_k, t_{k+1}]) - \int_{t_k}^{t_{k+1}} H(t)z(t)dt \in -\mathcal{N}_{\phi(t_{k+1})} \left( \frac{z(t_{k+1}^+) + ez(t_{k+1}^-)}{1+e} \right), \quad (92)$$

which is approximated by:

$$z(t_{k+1}) - z(t_k) - hH(t_{k+1})z(t_{k+1}) \in -\mathcal{N}_{\phi(t_{k+1})} \left( \frac{z(t_{k+1}) + ez(t_k)}{1+e} \right), \quad (93)$$

where $h := t_{k+1} - t_k$ is the time step. Set $z(t_k) := z_k$, $z(t_{k+1}) := z_{k+1}$, $\phi(t_{k+1}) := \phi_{k+1}$, and $H(t_{k+1}) := H_{k+1}$, then (93) can be rewritten as: $(1 - hH_{k+1})z_{k+1} - z_k \in -\mathcal{N}_{\phi_{k+1}} \left( \frac{z_{k+1} + ez_k}{1+e} \right)$.

This is equivalent to $z_{k+1} = (1 + e)\text{proj}_{\phi_{k+1}} \left( \frac{(1 - hH_{k+1})e + 1}{(1 + e)(1 - hH_{k+1})} z_k \right) - ez_k$ using (4). A numerical simulation of (86) when using the model (89) with $e = 1$ or $0.7$ is presented in Figures 6.

**Calculation of the sets and solutions of local and global variations:** Let us consider the variation of the moving set $\phi(t)$ and state $x(t)$ of the system (86) on $[0, +\infty)$. Denote by $\mathcal{P}$ the set of all partitions of $[0, T]$ for some $T > 0$.

(a) The moving set $\phi(\cdot)$ is not LBV, hence not BV on $[0, +\infty)$.

Let us choose a partition $\sigma = (t_0, ..., t_{N_\sigma}) \in \mathcal{P}$ such that $0 = t_0 < t_1 < ... < t_k = \pi - 1 < ... < t_{N_\sigma} = T$ for some $T > \pi - 1$. Then we have

$$V_{[0,T]}(\phi) \geq \sum_{i=0}^{N_\sigma-1} d_H(\phi(t_{i+1}), \phi(t_i)) \geq d_H(\mathbb{R}_+, \mathbb{R}) \geq \sup_{a \in \mathbb{R}} \text{dist}(a, \mathbb{R}_+) = +\infty.$$

This implies that $\phi(t)$ is not BV on $[0, T]$, and therefore is not LBV on $[0, +\infty)$.
all $t \in [0, T]$ and $l \ge 1$. By assumption, $\|f(t, z)\| \le M(T) + N(T)\|z\|$ for all $z \in \text{dom } f(t, \cdot)$ and for all $t \in [0, T]$. Therefore, for all $0 \le \underline{\tau} < \bar{\tau} \le T$, the following holds:

$$ \varphi_l(\bar{\tau}) - \varphi_l(\underline{\tau}) \le (N(T)C_1 + M(T))(\bar{\tau} - \underline{\tau}) + \int_{\underline{\tau}}^{\bar{\tau}} \|\bar{u}(\tau)\| d\tau, $$

where $C_1$ is a bound of $x_l(\cdot)$ on $[0, T]$ for all $l \ge 1$, $N(T)$, $M(T)$ coincide with $L_f$ and $\|f(t, 0)\|$ in [20, Theorem 23], respectively. The rest of the proof is the same as the proof of [20, Theorem 23].

## References

[1] F. Vasca, L. Iannelli, and K. Camlibel. A new perspective in power converters modelling: complementarity systems. In *2007 IEEE Power Electronics Specialists Conference*, pages 1817–1823, 2007.

[2] F. Vasca, L. Iannelli, M.K. Camlibel, and R. Frasca. A new perspective for modeling power electronics converters: Complementarity framework. *IEEE Transactions on Power Electronics*, 24(2):456–468, 2009.

[3] R. Frasca, M.K. Camlibel, I.C. Goknar, L. Iannelli, and F. Vasca. Linear passive networks with ideal switches: Consistent initial conditions and state discontinuities. *IEEE Transactions on Circuits and Systems I: Regular Papers*, 57(12):3138–3151, 2010.

[4] 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.

[5] V. Acary, O. Bonnefon, and B. Brogliato. *Nonsmooth Modeling and Simulation for Switched Circuits*, volume 69 of *Lecture Notes in Electrical Engineering*. Springer Verlag, October 2010. Erratum at https://inria.hal.science/hal-01311078v2/document.

[6] W.P.M.H. Heemels, J.M. Schumacher, and S. Weiland. The rational complementarity problem. *Linear Algebra and its Applications*, 294(1):93–135, 1999.

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

[8] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. The nature of solutions to linear passive complementarity systems. In *Proc. of the 38th IEEE Conference on Decision and Control*, pages 3043–3048, 1999.

[9] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. On linear passive complementarity systems. *European Journal of Control*, 8:220–237, 12 2002.

[10] 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.

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

[12] B. Brogliato and D. Goeleven. Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems. *Nonlinear Analysis: Theory, Methods Applications*, 74(1):195–212, 2011.
<figure><img src="image_7.png" /><figcaption>Figure 6: Simulation of (86) using the state jump model (89) with time step $h = 5 \times 10^{-4}$s.</figcaption></figure>

(b) The solution $x(\cdot)$ without restitution is BV on $[0, +\infty)$ (see Figure 5).

For each $T > 0$ and partition $\sigma = (t_0, ..., t_{N_\sigma}) \in \mathcal{P}$, denote by $S_\sigma^{[0,\pi-1)}$, $e_{max}^\sigma$ the set of all index $i$ where $t_i \in [0, \pi - 1)$, the largest index in the set $S_\sigma^{[0,\pi-1)}$, respectively. Then we have

$$ 
\begin{aligned}
V_{[0, +\infty)}(x) &= \lim_{T \to \infty} V_{[0, T]}(x) = \lim_{T \to \infty} \left( \sup_{\sigma \in \mathcal{P}} \sum_{i=0}^{N_\sigma - 1} |x(t_{i+1}) - x(t_i)| \right) = \sup_{\sigma \in \mathcal{P}} \sum_{i \in S_\sigma^{[0, \pi-1)}} |x(t_{i+1}) - x(t_i)| \\
&= \sup_{\sigma \in \mathcal{P}} (2x(t_{e_{max}^\sigma}) - x(t_0)) < \exp(0.05(\pi - 1)) - 1 < +\infty.
\end{aligned}
$$

(c) The solution $x(\cdot)$ with restitution $e = 1$ is not BV, but LBV on $[0, +\infty)$ (see Figure 6).

Firstly, $x(\cdot)$ is not BV on $[0, +\infty)$ since it is unbounded on $[0, +\infty)$. Now, let us fix $T > 0$ and take $h = \lfloor \frac{T+1}{\pi} \rfloor$. Then, the variation of $x(\cdot)$ on $[0, T]$ is given by:

$$ V_{[0,T]}(x) = \sup_{\sigma \in \mathcal{P}} \left( \sum_{j=0}^{h-1} \sum_{i \neq e_{\text{max}}^{\sigma,j+1}}^{i \in S_{\sigma}^{j+1}} |x(t_{i+1}) - x(t_i)| + \sum_{j=1}^{h-1} |x(t_{e_{\text{min}}^{\sigma,j+1}}) - x(t_{e_{\text{max}}^{\sigma,j}})| \right), $$

with respect to a partition $\sigma \in \mathcal{P}$, $S_\sigma^{j+1} := S_\sigma^{[j\pi-1,(j+1)\pi-1)}$ is the set of all indices $i$ such that $t_i \in [j\pi - 1, (j+1)\pi - 1)$, the values $e_{\text{min}}^{\sigma,j+1}, e_{\text{max}}^{\sigma,j+1}$ are the smallest and largest indices of $S_\sigma^{j+1}$ for all $j = 0, h-1$, respectively, and $N_\sigma$ is the number of elements of $\sigma$. By some basic calculations, we obtain:

$$ 
\begin{aligned}
V_{[0,T]}(x) &= \sup_{\sigma \in \mathcal{P}} \left( \sum_{j=1}^{h} (-1)^{j+1} (x(t_{e_{\text{max}}^{\sigma,j}}) - x(t_{e_{\text{min}}^{\sigma,j}})) + \sum_{j=1}^{h-1} (-1)^{j} (x(t_{e_{\text{min}}^{\sigma,j+1}}) - x(t_{e_{\text{max}}^{\sigma,j}})) \right) \\
&= \sup_{\sigma \in \mathcal{P}} \left( \left( 2 \sum_{j=1}^{h-1} (-1)^{j+1} x(t_{e_{\text{max}}^{\sigma,j}}) \right) + (-1)^{h+1} x(t_{e_{\text{max}}^{\sigma,h}}) - x(t_{e_{\text{min}}^{\sigma,1}}) \right) \\
&\le \sup_{\sigma \in \mathcal{P}} \left( \left( 2 \sum_{j=1}^{h} (-1)^{j+1} x(t_{e_{\text{max}}^{\sigma,j}}) \right) - x(t_{e_{\text{min}}^{\sigma,1}}) \right).
\end{aligned}
$$
[13] B. Brogliato. Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings. *Systems and Control Letters*, 51(5):343–353, 2004.

[14] M.K. Camlibel, L. Iannelli, and F. Vasca. Passivity and complementarity. *Math. Program.*, 145(1-2):531–563, 2014.

[15] 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.

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

[17] M.K. Camlibel and Hans Schumacher. Linear passive systems and maximal monotone mappings. *Mathematical Programming*, 157(2):397–420, June 2016.

[18] B.K. Le. On a class of Lur'e dynamical systems with state-dependent set-valued feedback. *Set-Valued and Variational Analysis*, 2020.

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

[20] M.K. Camlibel, L. Iannelli, and A. Tanwani. Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators. *Mathematical Programming, Series A*, July 2021.

[21] R.T. Rockafellar and R.J.B. Wets. *Variational Analysis*, volume 317 of *Grundlehren der mathematischen Wissenschaften*. Springer, 1998.

[22] R.W. Cottle, J.S. Pang, and R.E. Stone. *The Linear Complementarity Problem*. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1992.

[23] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. *Dissipative Systems Analysis and Control*. Communication and Control Engineering. Springer International Publishing, 3rd edition, 2020.

[24] J.F. Edmond and L. Thibault. Relaxation of an optimal control problem involving a perturbed sweeping process. *Mathematical Programming*, 104:347–373, 11 2005.

[25] C. Castaing and M.D.P. Monteiro Marques. Evolution problems associated with nonconvex closed moving sets with bounded variation. *Portugaliae Mathematica*, 53, 01 1996.

[26] E.D. Sontag. *Mathematical Control Theory: Deterministic Finite Dimensional Systems*. Texts in Applied Mathematics. Springer New York, 2013.

[27] V.I. Bogachev. *Measure Theory*, volume II. Springer Berlin Heidelberg, 2006.

[28] J.B. Conway. *A Course in Functional Analysis*, volume 96 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 2nd ed edition, 1990.

[29] H.H. Bauschke and P.L. Combettes. *Convex Analysis and Monotone Operator Theory in Hilbert Spaces*. CMS Books in Mathematics. Springer, 2011.
In this case, we have $|x(t_{e_{max}^{\sigma,j}})| \le \exp(0.05(j\pi - 1))$ and $x(t_{e_{min}^{\sigma,1}}) = x(t_0) = 1$, for all $j = \overline{1,h}, \sigma \in \mathcal{P}$.
Hence the boundedness of $V_{[0,T]}(x)$ is obtained:

$$V_{[0,T]}(x) \le 2 \sum_{j=1}^{h} \exp(0.05(j\pi - 1)) - 1 = 2 \exp(0.05(\pi - 1)) \frac{1 - \exp(0.05h\pi)}{1 - \exp(0.05\pi)} - 1 < +\infty.$$

Therefore, the solution $x(\cdot)$ is BV on $[0, T]$ for any $T > 0$ and thus, it is LBV on $[0, +\infty)$.

(d) The solution $x(\cdot)$ with restitution $e = 0.7$ is BV on $[0, +\infty)$ (see Figure 6).

Using the same idea as above, the variation of $x(\cdot)$ on $[0, T]$ is evaluated as:

$$V_{[0,T]}(x) \le \sup_{\sigma \in \mathcal{P}} \left( \left( 2 \sum_{j=1}^{h} (-1)^{j+1} x(t_{e_{max}^{\sigma,j}}) \right) - x(t_{e_{min}^{\sigma,1}}) \right),$$

where $h = \lceil \frac{T+1}{\pi} \rceil$. In this case, $|x(t_{e_{max}^{\sigma,j}})| \le (0.7)^{j-1} \exp(0.05(j\pi - 1))$ for all $j = \overline{1,h}, \sigma \in \mathcal{P}$.
Therefore, the previous expression implies that:

$$\begin{aligned}
V_{[0, +\infty)}(x) &= \lim_{T \to \infty} V_{[0, T]}(x) \le \lim_{h \to \infty} \left( 2 \sum_{j=1}^{h} (0.7)^{j-1} \exp(0.05(j\pi - 1)) \right) - 1 \\
&= \frac{2 \exp(0.05(\pi - 1))}{1 - 0.7 \exp(0.05\pi)} - 1 < +\infty.
\end{aligned}$$

In general, let us consider the system (86) where $A(t) = a$ is a constant. We point out now the relation between the coefficient $a$ and the restitution coefficient $e$ such that the solution $x(\cdot)$ is BV or LBV. Clearly, for each value $a$ and $e$ we get a unique solution $x(\cdot)$ of the form:

$$x(t) = \begin{cases} e^k \exp(at), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, \ k \text{ is even,} \\ -e^k \exp(at), & \text{if } t \in [k\pi - 1, (k+1)\pi - 1) \text{ with } k \in \mathbb{N}, \ k \text{ is odd.} \end{cases}$$

From the above analysis, only three types of solutions exist, as depicted in Figures 5, 6. Therefore, the solution $x(\cdot)$ is BV on $[0, +\infty)$ if and only if the following constraint holds:

$$e^k \exp(a((k+1)\pi - 1)) > e^{k+2} \exp(a((k+3)\pi - 1)), \forall k \in \mathbb{N}. \quad (94)$$

Simplifying these constraints, then (94) is equivalent to: $e \exp(a\pi) < 1$. It follows that:

* If $a < 0$ then $x(\cdot)$ is BV on $[0, +\infty)$ for all $e \in [0, 1]$.

* If $a > 0$ and $e \ge \frac{1}{\exp(a\pi)}$ then $x(\cdot)$ is LBV, but it is not BV on $[0, +\infty)$.

* If $a > 0$ and $e < \frac{1}{\exp(a\pi)}$ then $x(\cdot)$ is BV on $[0, +\infty)$.

* If $a = 0$ and $e = 1$ then $x(\cdot)$ is LBV, but it is not BV on $[0, +\infty)$.

* If $a = 0$ and $e \in [0, 1)$ then $x(\cdot)$ is BV on $[0, +\infty)$.

It is concluded that in the framework of the system (86) where the coefficient $A(t)$ is a constant, the solution $x(\cdot)$ is always RCLBV on $[0, +\infty)$, even though the set-valued mapping $\phi(\cdot)$ is not RCLBV.
[30] D.S. Bernstein. *Matrix Mathematics: Theory, Facts, and Formulas*. Princeton University Press, second edition, 2011.

[31] P.R. Halmos. *Finite-Dimensional Vector Spaces: Second Edition*. Dover Books on Mathematics. Dover Publications, 2017.

[32] A.A. Vladimirov. Nonstationary dissipative evolution equations in a Hilbert space. *Nonlinear Analysis: Theory, Method Applications*, 17(6):499–518, 1991.

[33] M. Kunze and M. D. P. Monteiro Marques. BV solutions to evolution problems with time-dependent domains. *Set-Valued Analysis*, 5(1):57–72, 1997.

[34] J.F. Edmond and L. Thibault. BV solutions of nonconvex sweeping process differential inclusion with perturbation. *Journal of Differential Equations*, 226(1):135–179, January 2006.

[35] N. Dinculeanu, I.N. Sneddon, and M. Stark. *Vector Measures*. ISSN. Elsevier Science, 2014.

[36] J.J. Moreau. Sur les mesures différentielles des fonctions vectorielles à variation locale-ment bornée. *Séminaire d'Analyse Convexe*, 5(17), 1975. http://dml.mathdoc.fr/item/hal-01878584.

[37] C. Castaing and M.D.P. Monteiro Marques. BV periodic solutions of an evolution problem associated with continuous moving convex sets. *Set-Valued Analysis*, 3:381–399, 1995.

[38] C. Castaing, Tráng Xuân Dúc Hā, and M. Valadier. Evolution equations governed by the sweeping process. *Set-Valued Analysis*, 1:109–139, 1993.

[39] J. Diestel and J.J. Uhl. *Vector Measures*. Mathematical surveys and monographs. American Mathematical Society, 1977.

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

[41] B. Brogliato. *Nonsmooth Mechanics. Models, Dynamics and Control*. Communications and Control Engineering. Springer International Publishing Switzerland, 3rd edition, 2016.

[42] W.P.M.H. Heemels, M.K. Camlibel, and J.M. Schumacher. On the dynamic analysis of piecewise-linear networks. *IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, 49(3):315–327, 2002.

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

[44] F. Facchinei and J.S. Pang. *Finite-Dimensional Variational Inequalities and Complementarity Problems*. Springer Series in Operations Research and Financial Engineering. Springer New York, 2003.

[45] S.M. Robinson. Stability theory for systems of inequalities. part I: Linear systems. *SIAM Journal on Numerical Analysis*, 12(5):754–769, 1975.
**Existence of finite accumulations of state jumps:** Let $C(t) = \sin\left(\frac{1}{t-1}\right)$ in (3) (which is not BV on $[0, 1]$). Intuitively, there should exist a finite accumulation of state jumps at $t = 1$, because such $C(t)$ makes the admissible domain $C(t)x \ge 0$ jump with accumulation. Let us investigate this fact for the following TVLCCS:

$$ \begin{cases} \dot{x}(t) = 0.05x(t) + \frac{t^2+1}{t^2+2} \sin\left(\frac{1}{t-1}\right) \lambda(t), \\ 0 \le \lambda(t) \perp w(t, x(t)) = \sin\left(\frac{1}{t-1}\right) x(t) \ge 0, \quad x(0) = -1. \end{cases} \quad (95) $$

In the FOSwP format, the system (95) can be rewritten equivalently as:

$$ \dot{z}(t) - H(t)z(t) \in -\mathcal{N}_{\phi(t)}(z(t)), \quad (96) $$

where $z(t) = R(t)x(t)$, $H(t) = (\dot{R}(t)R(t)^{-1} + 0.05)$, and $\phi(t) = \{x \in \mathbb{R} : C(t)R(t)^{-1}x \ge 0\}$ with $R(t) = \sqrt{\frac{t^2+2}{t^2+1}}$. After few manipulations, $\phi(t)$ is calculated as:

$$ \phi(t) = \begin{cases} \mathbb{R}_+, & \text{if } t \in \left(1 + \frac{1}{k\pi}, 1 + \frac{1}{(k-1)\pi}\right) \cup \left(1 + \frac{1}{\pi}, +\infty\right), & \text{for some } k \in \mathbb{Z}\setminus\{0, 1\} \text{ is odd,} \\ \mathbb{R}, & \text{if } t = 1 + \frac{1}{k\pi}, & \text{for some } k \in \mathbb{Z}\setminus\{0\}, \\ \mathbb{R}_-, & \text{otherwise.} \end{cases} $$

Clearly $\phi(t)$ will switch an infinite number of times in the interval $[0, 2]$. In this example, to indicate finite accumulation, we consider the model of state jumps with restitution $e \in [0, 1)$ as in (89). At time $t = 1 + \frac{1}{k\pi}$ for some $k \in \mathbb{Z}\setminus\{0\}$, (96) can be rewritten as $z(t^+) = (1+e) \text{proj}_{\phi(t^+)}(z(t^-)) - ez(t^-)$, in view of (89). The unique solution to (95) is defined by:

$$ x(t) = \begin{cases} -\exp(0.05t), & \text{if } t \in [0, 1 - \frac{1}{\pi}), \\ (-1)^{k+1}e^k \exp(0.05t), & \text{if } t \in \left[1 - \frac{1}{k\pi}, 1 - \frac{1}{(k+1)\pi}\right), \text{ for some } k \in \mathbb{N}\setminus\{0\}, \\ 0, & \text{if } t \in [1, +\infty). \end{cases} \quad (97) $$

The numerical simulations of (95) when $e = 0.98$, time steps $h = 3 \times 10^{-4}s$, $3 \times 10^{-5}s$, $3 \times 10^{-6}s$ are depicted in Figures 8, 9. If we decrease time step $h$, then the discrete-time solution approaches the analytical solution as in (97) with an accumulation time of jumps at $t = 1s$. The numerical simulations of (95) in the interesting cases $e = 0$ and $1$ with time step $h = 3 \times 10^{-4}s$ are also shown in Figures 7. If we consider (95) with the model of state jump when $e \in [0, 1)$, there always exists an RCBV solution on $[0, +\infty)$. Indeed, first, the solution $x(\cdot)$ in (97) is evidently right-continuous. Let $T > 0$ and we consider the variation of $x(\cdot)$:

$$ V_{[0,T]}(x) = \sup_{\sigma \in \mathcal{P}} \sum_{i=0}^{N_{\sigma}-1} |x(t_{i+1}) - x(t_i)|. $$

For each partition $\sigma \in \mathcal{P}$, let $S_{\sigma}^{-1}$, $S_{\sigma}^{0}$, $S_{\sigma}^{k}$, be the sets of all indices $i$ such that $t_i \ge 1$, $t_i \in [0, 1 - \frac{1}{\pi})$, and $t_i \in [1 - \frac{1}{k\pi}; 1 - \frac{1}{(k+1)\pi})$, the values $e_{min}^{\sigma,k}$, $e_{max}^{\sigma,k}$ are the smallest and largest indices of $S_{\sigma}^{k}$ for all $k \in \{-1, 0, 1, ...\}$, and $e^{\sigma,\infty}$ is the index such that $t_{e^{\sigma,\infty}} < 1$ and $t_{e^{\sigma,\infty}+1} \ge 1$, respectively. Hence,
we obtain the following assessment:

$$
\begin{align}
V_{[0,T]}(x) &= \sup_{\sigma \in \mathcal{P}} \left( \sum_{k=-1}^{\infty} \sum_{i \neq e_{\sigma,k}^{\sigma,k}}^{i \in S_{\sigma}^{k}} |x(t_{i+1}) - x(t_i)| + \sum_{k=0}^{\infty} |x(t_{e_{\sigma,k+1}}) - x(t_{e_{\sigma,k}})| + |x(t_{e_{\sigma,-1}}) - x(t_{e^{\sigma,\infty}})| \right) \nonumber \\
&= \sup_{\sigma \in \mathcal{P}} \left( \sum_{k=0}^{\infty} \sum_{i \neq e_{\sigma,k}^{\sigma,k}}^{i \in S_{\sigma}^{k}} |x(t_{i+1}) - x(t_i)| + \sum_{k=0}^{\infty} |x(t_{e_{\sigma,k+1}}) - x(t_{e_{\sigma,k}})| + |x(t_{e^{\sigma,\infty}})| \right) \nonumber \\
&= \sup_{\sigma \in \mathcal{P}} \left( \underbrace{\sum_{k=0}^{\infty} |x(t_{e_{\sigma,k}}) - x(t_{e_{\sigma,k}})|}_{P_1} + \underbrace{\sum_{k=0}^{\infty} |x(t_{e_{\sigma,k+1}}) - x(t_{e_{\sigma,k}})|}_{P_2} + \underbrace{|x(t_{e^{\sigma,\infty}})|}_{P_3} \right). \tag{98}
\end{align}
$$

Since $t_{e^{\sigma,\infty}} < 1$, there exists $k \in \mathbb{N}$ such that $t_{e^{\sigma,\infty}} \in S_{\sigma}^{k}$, then $P_3 = e^k \exp(0.05t_{e^{\sigma,\infty}}) < \exp(0.05)$.
Moreover, we also have:

$$
P_1 \le \sum_{k=0}^{\infty} \left| e^k \left( \exp(0.05t_{e_{\min}^{\sigma,k}}) - \exp(0.05t_{e_{\max}^{\sigma,k}}) \right) \right| \le 2 \exp(0.05) \sum_{k=0}^{\infty} e^k = \frac{2 \exp(0.05)}{1-e}.
$$

And similarly, the following is verified.

$$
P_2 = \sum_{k=0}^{\infty} \left| e^{k+1} \exp(0.05t_{e_{\min}^{\sigma,k+1}}) - e^k \exp(0.05t_{e_{\max}^{\sigma,k}}) \right| \le \exp(0.05) \sum_{k=0}^{\infty} (e^{k+1} + e^k) = \frac{(e+1) \exp(0.05)}{1-e}.
$$

From bounds of $P_1$, $P_2$, and $P_3$, it implies that $V_{[0, +\infty)}(x) = \lim_{T \to \infty} V_{[0,T]}(x) < \frac{4 \exp(0.05)}{1-e} < +\infty$.
That is, the solution $x(\cdot)$ is BV on $[0, +\infty)$.

**Remark 8.** This conclusion holds for any $e \in [0,1)$. The crucial property in the above example is the existence of an accumulation of jumps which implies that the state vanishes at the “singular” time $t = 1$s. In general TVLCCS the existence of such accumulation cannot be shown. The merit of this example is to prove that some TVLCCS are equivalent to an FOSwP with varying sets that are not RCLBV, but which have a unique RCBV solution with an accumulation of state jumps.

**Remark 9.** When $e = 1$, it is difficult to give a meaning to the solution at $t = 1$. The solution is BV on interval $[0, 1 - \epsilon]$ and $[1 + \epsilon, \epsilon']$ with $\epsilon' > 1 + \epsilon, \epsilon > 0$. But the variation of $x(\cdot)$ on such interval tends to infinity when $\epsilon$ tends to zero. Also, it is unclear how to “cross” $t = 1$. Figure 7(b) illustrates this fact. The catching-up algorithm goes through the “singular time” $t = 1$, if there is no time step $k$ such that $t_k = 1$.

**Remark 10.** In Figures 8, we observe that $\lambda(t)$ and $w(t, x)$ may take negative values when $t \ge 1$ which violates the linear complementarity conditions stated in the second line of (95). This is because the obtained solution is an approximate solution due to the discretization of the system, and therefore it and complementarity variables (i.e., $\lambda, w$) become inaccurate when the time step $h$ is large. For instance, at the jump times $t$ then $w(t^+, x(t^+)) = \sin(\frac{1}{t^+-1}) x(t^+)$ is inaccurate because the value $x(t^+)$ is incorrect when $t \ge 1$. Here, as observed, when we decrease the time step from $h = 3 \times 10^{-4}$s (or $h = 3 \times 10^{-5}$s) to $h = 3 \times 10^{-6}$s, this issue is improved. Therefore, if we consider a sufficiently small time step $h$, this issue will disappear. A similar issue can be found in [5, p.19].
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7: Simulation of (95) using the state jump model (89) with time step $h = 3 \times 10^{-4}$s.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: Simulation of (95) using the state jump model (89) with the restitution coefficient $e = 0.98$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9: Simulation of (95) using the state jump model (89) with $e = 0.98$, time step $h = 3 \times 10^{-6}$s.</figcaption></figure>

### 3.1.2 Example 2: State jumps in TVLCCS with time-varying $K(t)$

Let us consider the following two-dimensional TVLCCS:

$$ \begin{cases} \dot{x}(t) = A(t)x(t) + B(t)\lambda(t), \\ K(t) \ni \lambda(t) \perp w(t, x(t)) = C(t)x(t) \in K^*(t), \\ x(0) = [1 \ 2]^T, \end{cases} \qquad (99) $$
where $x(t), \lambda(t) \in \mathbb{R}^2$, $A(t) = \text{diag}(a_1, a_2)$ for some $a_1, a_2 \in \mathbb{R}$, and $\mathcal{K}(t) = [\mathbb{R}_+(t_0-t)\mathbb{R}_+]^\top$, $B(t) = C(t) = \text{diag}(b_1(t), b_2(t))$ such that on $[0, t_0)$, $b_1(t) > 0, b_2(t) = 0$, on $(t_0, +\infty)$, $b_1(t) = 0$, $b_2(t) > 0$, and $b_1(t_0) = b_2(t_0) = 0$ for some $t_0 \in \mathbb{R}_+$. Let us analyze solutions of (99). Firstly, we can rewrite (99) in the FOSwP format as:

$$ \dot{x}(t) - A(t)x(t) \in -\mathcal{N}_{\phi(t)}(x(t)), \qquad (100) $$

where $\phi(t) = \{x \in \mathbb{R}^2 : C(t)x(t) \in \mathcal{K}^*(t)\}$. It follows that $\phi(t) = [\mathbb{R}_+ \mathbb{R}]^\top$ if $t < t_0$, $\phi(t) = \mathbb{R}^2$ if $t = t_0$, and $\phi(t) = [\mathbb{R} \mathbb{R}_-]^\top$ if $t > t_0$. Obviously, $\phi(t)$ jumps at time $t_0$, and the Hausdorff distance $d_H(\phi(t_0^+), \phi(t_0^-)) = +\infty$. On $[0, t_0)$, since $x(t) \in \text{int}(\phi(t))$, (100) can be rewritten as $\dot{x}(t) = A(t)x(t)$, which implies that $x(t) = [\exp(a_1t) \ 2\exp(a_2t)]^\top > 0$. Since $x(t_0^-) > 0$ and $\phi(\cdot)$ switches at time $t = t_0$, the second component of the state $x(\cdot)$ jumps at $t = t_0$. Using the model of state jump with restitution coefficient $e \in [0, 1]$ as in (89), we obtain:

$$ x(t_0^+) - x(t_0^-) \in -\mathcal{N}_{\phi(t_0^+)} \left( \frac{x(t_0^+) + ex(t_0^-)}{1+e} \right). $$

Therefore, $x(t_0^+) = (1+e)\text{proj}_{\phi(t_0^+)}(x(t_0^-)) - ex(t_0^-) = [\exp(a_1t_0) - 2e\exp(a_2t_0)]^\top$. On $[t_0, +\infty)$, proceeding as above, $x(t)$ has the form $x(t) = [\exp(a_1t) - 2e\exp(a_2t)]^\top$. From all of the above, a global solution of the system (99) is given by $x(t) = [\exp(a_1t) \ 2\exp(a_2t)]^\top$ if $t < t_0$, and $x(t) = [\exp(a_1t) - 2e\exp(a_2t)]^\top$ if $t \ge t_0$. Choosing $a_1 = a_2 = 0.05, t_0 = 4, e = 0$ and $b_1(\cdot), b_2(\cdot)$ to be any functions that satisfy conditions set out at the beginning, and the time step $h = 5 \times 10^{-4}s$. Then a numerical simulation of (99) is depicted in Figure 10. From the form of the solution $x(\cdot)$, clearly $x(\cdot)$ is always right-continuous, and in this case, it is not BV, but LBV on $[0, +\infty)$. Consider the general cases of $a_1, a_2$ and $e$, we conclude that:

* If $a_1, a_2 \in (-\infty, 0]$ then $x(t)$ is BV on $[0, +\infty)$ for all $e \in [0, 1]$.

* If $a_1 > 0$ or $a_2 > 0$ then $x(t)$ is not BV, but LBV on $[0, +\infty)$ for all $e \in [0, 1]$.

Thus the system (99) always has a RCLBV solution for all $a_1, a_2 \in \mathbb{R}$ and for all $e \in [0, 1]$.

<figure><img src="image_12.png" /><figcaption>Figure 10: Simulation of (99) using the state jump model (89) with $e = 0$, time step $h = 5 \times 10^{-4}s$, $a_1 = a_2 = 0.05$, and $t_0 = 4$.</figcaption></figure>

### 3.1.3 Recapitulation

These examples have shown important differences concerning the ability to transform TVLCCS into FOSwP, as well as the jump times of the state in TVLCCS, compared to LCS, which have been studied in [10, 4, 9, 8, 44]. Firstly, it is possible to transform from TVLCCS to FOSwP and vice versa, even when $\phi(\cdot)$ (i.e., time-varying set-valued mapping in FOSwP) is not LBV, which violates a usual condition in the analysis of FOSwP used in Theorem 3, and in [4, Section 5.2, (A3)], [45, 10]. This is impossible for LCS. Actually, using the same reasoning as in [10, Proposition 3.2],
it can be proved that a time-varying $C(t)$ is necessary to pass from the TVLCCS to a FOSwP with such “unbounded” set jump. Indeed a TVLCCS with constant $C$ (and zero $D$) and constant $\mathcal{K}$ yields for each $t$ a polyhedral set $\phi(t)$ where the time-dependency stems solely from $Fu(t)$ in (3b). Moreover $d_H(\phi(t), \phi(s)) \le \gamma \|Fu(t) - Fu(s)\|$ for some $\gamma > 0$ and all $s, t \in \mathbb{R}_+$ (using Lemma A2 which compiles previous results by Walkup and Wets, and by Robinson). Thus, provided $u(\cdot)$ is a bounded function, it is impossible to obtain an FOSwP as above if $C$ and $\mathcal{K} = \mathbb{R}_+^m$ are constant. For instance, the FOSwP: $-\dot{x}(t) \in \mathcal{N}_{\phi(t)}(x(t))$, $x(0) = x_0 \in \phi(0)$, $\phi(t) = \mathbb{R}_+$ if $t \in [0, t_1)$, $\phi(t) = \mathbb{R}_-$ if $t \in [t_1, +\infty)$, cannot be shown to be equivalent to any LCS (1) with $w(t, x) = Cx + Fu(t)$. The key to this issue lies in the time variation of $C(\cdot)$. The matrix $C(t)$ in (3) plays a significant role in the study of TVLCCS. Secondly, when LCS (1) is autonomous (i.e., $E = F = 0$), results in [8, 44] have shown that state jumps occur only at the initial time due to inconsistent initialization, i.e., $Cx(0) \notin (\text{SOL}(0, D))^*$. When LCS (1) has time-varying external inputs, the results in [9] pointed out that state jumps occur at arbitrary time $t$ where $Fu(t)$ is discontinuous (i.e., $\{t : x(t^+) \neq x(t^-)\} \subseteq \{t : Fu(t^+) \neq Fu(t^-)\}$) or at the initial time if $Cx(0) + Fu(0) \notin (\text{SOL}(0, D))^*$. In other words, if $Fu(\cdot)$ is continuous then $x(\cdot)$ is continuous (excepted possibly at the initial time). However, in the examples above, even though $F(\cdot)u(\cdot) = 0$ is a continuous function, the state $x(\cdot)$ is discontinuous at $t = \pi - 1$ in (86), and at $t = t_0$ in (99) even if $C(\cdot)$ is a continuous function and at $t = 1 - \frac{1}{\pi}$ in (95). This is a new finding when studying TVLCCS.

## 3.2 Analysis of the causes of state jumps in Section 3.1

In this section, we will use the theoretical results presented in Section 2.4 to clarify the reason why state jumps appear in examples in Section 3.1. Firstly, let us come back to the system (86). Note that state jumps occur when $T > \pi - 1$. It can be checked that (86) does not satisfy Assumption 3 and items 2, 4 in Corollary 2, where $P(t) = \frac{t^2+2}{t^2+1}$ is a solution of (66). Indeed, we have:

$$ \text{rgc}(C(t)) = \text{rgc}(R(t)^{-1}C(t)^{\top}) = \begin{cases} 0, & \text{if } t = k\pi - 1 \text{ for some } k \in \mathbb{N}\setminus\{0\}, \\ \mathbb{R}, & \text{otherwise.} \end{cases} $$

Thus, item 2 fails. Let $s = \frac{\pi-1}{2}$, $t = \frac{3\pi-3}{2}$, then $s < t$ and $(C(s)R(s)^{-1})^{-1}(\mathbb{R}_+) = \mathbb{R}_+ \not\subseteq \mathbb{R}_- = (C(t)R(t)^{-1})^{-1}(\mathbb{R}_+)$. So item 4 fails. Finally, Assumption 3 fails due to the following:

$$ \text{rgc}(C(t)) \cap \text{rint}(\mathbb{R}_+) = \begin{cases} \emptyset, & \text{if } t = k\pi - 1 \text{ for some } k \in \mathbb{N}\setminus\{0\}, \\ \mathbb{R}_+\setminus\{0\}, & \text{otherwise.} \end{cases} $$

Therefore, the existence of an AC solution to (86) is not guaranteed by Corollary 2. In addition, it can also be verified that there is no mapping $Q(\cdot)$ that satisfies assumptions in Proposition 3. Proposition 4 can not be used either since $\phi(t)$ in the FOSwP (87), which is equivalent to (86), is not bounded. Hence, we may expect that state jumps will be necessary, and they will be analyzed by certain conditions beyond the results in this article. Secondly, let us consider the system (95) (resp. (99)) and note that state jumps occur when $T > 1 - \frac{1}{\pi}$ (resp. $T > t_0$). Similar to the above analysis, it is verified that (95) does not satisfy Assumption 3, items 2, 4, and 5 in Corollary 2, since $C(\cdot)$ is not right-continuous at $t = 1s$ (resp. (99) does not satisfy Assumption 3, items 1, 2, 4, and 5 in Corollary 2, since $C(\cdot)$ is not right-continuous and $\mathcal{K}^*(\cdot)$ is not inner semicontinuous from right at $t_0$). In addition, these examples also do not satisfy assumptions in Propositions 3, 4. Thus state jumps (as described in Section 3.1) can be expected for these system.
## 3.3 Further comments

Some assumptions which are made in the foregoing theoretical results deserve to be relaxed. It would be interesting to relax items 1 (iii) and 2 in Theorem 2. As the examples in Section 3.1 demonstrate, the role played by $C(\cdot)$ is prominent in TVLCCS. It yields equivalent FOSwP with switching unbounded sets for which the Hausdorff distance is unsuitable, and with Zeno behavior (infinitely many switches in a finite time). In this article, TVLCCS are studied through the analysis of Lur'e DIs as in (10). As pointed out after the proof of Proposition 1, one can study Lur'e DIs which do not stem from TVLCCS. In this setting, extending the results in [19, 18] to varying matrices and/or cones which depend also on the state, certainly is of interest. The above examples show that modifying the state jump rule can significantly influence the solution's behavior. Extensions to higher dimensions could be tackled.

# 4 Conclusions

This article deals with the well-posedness of finite-dimensional time-varying linear cone complementarity systems. All the system's data are allowed to depend on time, including the cones defining the complementarity constraints. The passivity of a suitable operator is a key assumption. It allows us to transform the dynamics into a suitable form known as Lur'e differential inclusion. Various cases are tackled depending on the system's data, with absolutely continuous or right-continuous of local bounded variation solutions. Examples from electrical circuits illustrate the theoretical developments.

# Appendices

## A Some useful definitions and results

**Theorem A1.** [26, Proposition C.3.8] Consider the following nonlinear differential equation:

$$ \dot{x}(t) = f(t, x(t)), \quad t \ge 0; \quad x(0) = x_0, \qquad (i1) $$

where $x(t) \in \mathbb{R}^n$ and $f: [0, +\infty) \times \mathbb{R}^n \to \mathbb{R}^n$. Assume that the following conditions hold:

1. $f(\cdot, x): [0, +\infty) \to \mathbb{R}^n$ is measurable for each fixed $x$.

2. $f(t, \cdot): \mathbb{R}^n \to \mathbb{R}^n$ is continuous for each fixed $t$.

3. There is a locally integrable function $\alpha: [0, +\infty) \to \mathbb{R}_+$ such that for each $t \ge 0$:

$$ \|f(t, x) - f(t, y)\| \le \alpha(t)\|x - y\|, \quad \forall x, y \in \mathbb{R}^n. $$

4. For each fixed $x_0 \in \mathbb{R}^n$, there is a locally integrable function $\beta: [0, +\infty) \to \mathbb{R}_+$ such that:

$$ \|f(t, x_0)\| \le \beta(t), \quad \text{for almost all } t. $$

Then, there exists a unique locally AC solution to (i1) on $[0, +\infty)$, for any $x_0 \in \mathbb{R}^n$.
**Remark A1.** Denote by $\mathfrak{B}(\mathbb{R})$, $\mathfrak{B}([0, +\infty))$ the Borel $\sigma$-algebra of $\mathbb{R}$ and $[0, +\infty)$, respectively [40, Definition 1.2.10]. Since $\mathfrak{B}(\mathbb{R})$ contains open and closed sets in $\mathbb{R}$, using [40, Lemma 6.2.4] then $\mathfrak{B}([0, +\infty)) = \{A \in \mathfrak{B}(\mathbb{R}) : A \subset [0, +\infty)\}$. Clearly if $f^{-1}(B) \in \mathfrak{B}([0, +\infty))$, then $f^{-1}(B)$ is a Lebesgue measurable set. A sufficient condition for this is that $[0, +\infty) \setminus f^{-1}(B)$ is a closed set. Indeed, it is obtained by the following:

$$f^{-1}(B) = \underbrace{[0, +\infty)}_{\in \mathfrak{B}(\mathbb{R})} \cap \underbrace{\mathbb{R} \setminus ([0, +\infty) \setminus f^{-1}(B))}_{\in \mathfrak{B}(\mathbb{R})} \in \mathfrak{B}(\mathbb{R}).$$

**Lemma A1. (Hoffman's bound)** [46, Lemma 3.2.3] For any matrices $C \in \mathbb{R}^{l \times n}$ and $A \in \mathbb{R}^{m \times n}$, there exists a constant $c > 0$ (depending only on $C$ and $A$) such that for all vectors $d \in \mathbb{R}^l$ and $b \in \mathbb{R}^m$ for which $P := \{x \in \mathbb{R}^n : Cx = d, Ax \le b\}$ is nonempty,

$$\mathrm{dist}(x, P) \le c(\|Cx - d\| + \|\max(0, Ax - b)\|), \quad \forall x \in \mathbb{R}^n.$$

Denote by $P(A, b) := \{x \in \mathbb{R}^n : Ax \le b\}$ a convex polyhedral set, for some $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$.

**Corollary A1.** [46, Corollary 3.2.5] Let the matrix $A \in \mathbb{R}^{m \times n}$ be given. Then, for every bounded subset $S$ of $\mathbb{R}^n$, there exists a constant $L > 0$ such that for every vector $b \in \mathbb{R}^m$ for which $P(A, b)$ is nonempty and for every $x \in S \cap P(A', b')$,

$$\mathrm{dist}(x, P(A, b)) \le L (\|A - A'\| + \|b - b'\|).$$

Note that $L$ is a bound of $S$.

**Proposition A1. (Chain rule)** [28, 21, 4] Let $f : \mathbb{R}^m \to \mathbb{R} \cup \{+\infty\}$ be a proper convex lower semicontinuous function, and $A : \mathbb{R}^n \to \mathbb{R}^m$ be a linear operator. Assume that either the function $f(\cdot)$ is polyhedral or for some $x_0$ with $Ax_0 \in \mathrm{dom}(f)$: $\mathrm{rge}(A) - \mathbb{R}_+(\mathrm{dom}(f) - Ax_0)$ is a vector subspace of $\mathbb{R}^m$, where $\mathrm{dom}(f) := \{y \in \mathbb{R}^m : f(y) < +\infty\}$. Then the subdifferential in the sense of convex analysis of the composite functional $f \circ A : \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$ is given by:

$$\partial(f \circ A)(x) = A^{\top} \partial f(Ax), \quad \forall x \in \mathbb{R}^n.$$

**Lemma A2. (Robinson's bound)** [47, p.760] Let $\mathcal{K} := \{y \in \mathbb{R}^m : Dy \ge 0\}$ be a closed convex polyhedral cone, for some matrix $D$. Let $S := \{x \in \mathbb{R}^n : Ax - b \in \mathcal{K}\}$, and $S' := \{x \in \mathbb{R}^n : Ax - b' \in \mathcal{K}\}$, where $A \in \mathbb{R}^{m \times n}$ is a matrix, and $b, b' \in \mathbb{R}^m$ are vectors. Suppose that $S$ and $S'$ are nonempty, then there exist a constant $\gamma_{A,\mathcal{K}}$ (depending only on $A$ and $\mathcal{K}$) such that:

$$d_H(S, S') \le \gamma_{A,\mathcal{K}} \|D\| \|b - b'\|.$$

**Definition A1.** [10, 34, 4] Consider the following FOSwP:

$$\dot{z}(t) \in -\mathcal{N}_{\phi(t)}(z(t)) + f(t, z(t)), \quad z(0) = z_0 \in \phi(0). \qquad (i2)$$

Assume that $\phi(\cdot)$ is RCLBV on $[0, +\infty)$, that is, for each $T > 0$, there exists a positive Radon measure $\mu$ on $[0, T]$ such that $d_H(\phi(s), \phi(t)) \le \mu((s, t])$, for all $0 \le s \le t \le T$. Assume further that there exists a locally integrable function $\beta(\cdot) : [0, +\infty) \to \mathbb{R}_+$ such that $\|f(t, z)\| \le \beta(t)(1 + \|z\|)$ for all $z \in \mathbb{R}^n$, for all $t \ge 0$. A mapping $z : [0, +\infty) \to \mathbb{R}^n$ is a solution of (i2) if:
(i) $z(\cdot)$ is RCLBV on $[0, +\infty)$ and satisfies $z(0) = z_0$, $z(t) \in \phi(t)$ for all $t \ge 0$;

(ii) there exists a positive Radon measure $\nu$ AC equivalent to the measure $\mu + \lambda$ and with respect to which the differential measure $dz$ of $z(\cdot)$ is AC with the density $\frac{dz}{d\nu}$ is a locally integrable function from $[0, +\infty)$ to $\mathbb{R}^n$ with respect to the measure $\nu$, and the following holds:

$$ \frac{dz}{d\nu}(t) \in -\mathcal{N}_{S(t)}(z(t)) + f(t, z(t)) \frac{d\lambda}{d\nu}(t), \quad \nu \text{ almost everywhere on } [0, +\infty), $$

where $\lambda$ is a one-dimensional Lebesgue measure, and $\frac{d\lambda}{d\nu}(\cdot)$ is the density relative to $\nu$ of $\lambda$.

## B The proof of Remark 1 when $D(t) = D(t)^\top \succcurlyeq 0$

Firstly, for each $t \in [0, T]$, since $\mathrm{rge}(D(t)) = (\ker D(t)^\top)^\perp$ [29, (2.4.14)-p.95], using the symmetry of $D(t)$ and [21, Example 6.23], we get $\mathrm{rge}(D(t)) = (\ker D(t))^*$. Therefore, $\mathcal{K}^*(t) + \mathrm{rge}(D(t)) = (\mathcal{K}(t) \cap \ker D(t))^*$ [21, Corollary 11.25]. So now, let us prove that $\mathcal{H}(t) = (\mathcal{K}(t) \cap \ker D(t))^*$, for all $t \in [0, T]$. Indeed, let any $t \in [0, T]$.

Let $[\subseteq]$ Let any $y \in \mathcal{H}(t)$, then $\hat{Q}(t)Q(t)y \ge 0$, that is, $Q(t)y \in (\mathrm{SOL}(0, Q(t)D(t)Q(t)^\top))^*$. This implies that

$$ \langle Q(t)y, z \rangle \ge 0, \quad \forall z \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top). \qquad (ii1) $$

Take any $\alpha \in \mathcal{K}(t) \cap \ker D(t)$, then there exists $\xi \in \mathbb{R}_+^j$ such that $\alpha = Q(t)^\top \xi$ and $D(t)\alpha = 0$. Therefore, we obtain $Q(t)D(t)Q(t)^\top \xi = 0$, and thus $\xi \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top)$. It follows from (ii1) that $\langle y, \alpha \rangle = \langle y, Q(t)^\top \xi \rangle = \langle Q(t)y, \xi \rangle \ge 0$. Because $\alpha$ is chosen arbitrarily, $\langle y, \alpha \rangle \ge 0$ for all $\alpha \in \mathcal{K}(t) \cap \ker D(t)$. This means $y \in (\mathcal{K}(t) \cap \ker D(t))^*$.

Let $[\supseteq]$ Let any $h \in (\mathcal{K}(t) \cap \ker D(t))^*$, then $\langle h, z \rangle \ge 0$ for all $z \in \mathcal{K}(t) \cap \ker D(t)$. To obtain $h \in \mathcal{H}(t)$, we shall prove that $Q(t)h \in (\mathrm{SOL}(0, Q(t)D(t)Q(t)^\top))^*$. Indeed, take any $\alpha \in \mathrm{SOL}(0, Q(t)D(t)Q(t)^\top)$, then we have $\alpha \in \mathbb{R}_+^j$ and $\langle \alpha, Q(t)D(t)Q(t)^\top \alpha \rangle = 0$. Since $D(t) = D(t)^\top \succcurlyeq 0$, there exists $\tilde{D}(t) = \tilde{D}(t)^\top \succcurlyeq 0$ such that $\tilde{D}(t)^2 = D(t)$. Moreover, it follows from the symmetry of $\tilde{D}(t)$ that $\ker D(t) = \ker \tilde{D}(t)$. Therefore, we get $0 = \langle \alpha, Q(t)D(t)Q(t)^\top \alpha \rangle = \langle \tilde{D}(t)Q(t)^\top \alpha, \tilde{D}(t)Q(t)^\top \alpha \rangle = \|\tilde{D}(t)Q(t)^\top \alpha\|^2$. Hence, we get $Q(t)^\top \alpha \in \ker \tilde{D}(t) = \ker D(t)$. Alternatively, since $\alpha \in \mathbb{R}_+^j$, $Q(t)^\top \alpha \in \mathcal{K}(t)$. Therefore, $Q(t)^\top \alpha \in \mathcal{K}(t) \cap \ker D(t)$ and thus, $\langle h, Q(t)^\top \alpha \rangle \ge 0$, i.e., $\langle Q(t)h, \alpha \rangle \ge 0$.

## C The proof of Theorem 1 in Section 2.2.3

The proof is the same as the proof of [20, Theorem 23], with some small changes in steps 1 and 2 as shown below. In step 1: At the beginning of the proof, using (A1) and the Lipschitzness of $f(t, \cdot)$, then for all $t \in [0, T]$ we also obtain:

$$ \begin{aligned} \langle \dot{x}_{l+1}(t) - \dot{x}_l(t), x_{l+1}(t) - x_l(t) \rangle &\le \langle f(t, x_l(t)) - f(t, x_{l-1}(t)), x_{l+1}(t) - x_l(t) \rangle \\ &\le L_f \|x_l(t) - x_{l-1}(t)\| \|x_{l+1}(t) - x_l(t)\| \\ &\le L_f \|x_l(t) - x_{l-1}(t)\| \|x_{l+1}(t) - x_l(t)\|. \end{aligned} $$

where $L_f(t)$ is a (bounded) Lipschitz constant and $L_f := \max_{t \in [0, T]} (L_f(t))$. In step 2: At the beginning of the case where $F$ satisfies (LG) (see [20, p.1047]), let $\varphi_l(t) := \int_0^t \|f(\tau, x_{l-1}(\tau))\| + \|u(\tau)\| d\tau$ for all $t \in [0, T]$ and $l \ge 1$. By assumption, $\|f(t, z)\| \le M(T) + N(T)\|z\|$ for all $z \in \mathrm{dom}\,f(t, \cdot)$ and for all $t \in [0, T]$. Therefore, the following holds:
$$ \varphi_l(\bar{\tau}) - \varphi_l(\underline{\tau}) \leq (N(T)C_1 + M(T))(\bar{\tau} - \underline{\tau}) + \int_{\underline{\tau}}^{\bar{\tau}} \|u(\tau)\| d\tau, \quad \forall 0 \leq \underline{\tau} < \bar{\tau} \leq T, $$

where $C_1$ is a bound of $x_l(\cdot)$ on $[0, T]$ for all $l \geq 1$, $N(T)$, $M(T)$ coincide with $L_f$ and $\|f(t, 0)\|$ in [20, Theorem 23], respectively. The rest of the proof is the same as the proof of [20, Theorem 23].

## D Constraint qualification [10, Equation (22)]

For each $t \in [0, +\infty)$, let us consider the moving set $\phi(t) := \{x \in \mathbb{R}^n : CR^{-1}x + Fu(t) + f \geq 0\}$, where $C \in \mathbb{R}^{m \times n}$, $F \in \mathbb{R}^{m \times u}$, $R^{-1} \in \mathbb{R}^{n \times n}$, $f \in \mathbb{R}^m$ are the constant matrices and vector, $u(t) \in \mathbb{R}^u$. Then, $\phi(t)$ is nonempty for all $t \geq 0$ if the following (i.e., [10, Equation (22)]) holds:

$$ \mathrm{rge}(C) - \mathbb{R}_+^m = \mathbb{R}^m. $$

This means for all $y \in \mathbb{R}^m$, there exist $y_1 \in \mathrm{rge}(C)$ and $y_2 \in \mathbb{R}_+^m$ such that $y = y_1 - y_2$.

## E Summary of main results in [20]

The paper [20] has focused on investigating the existence of solutions to the following differential inclusion:

$$ \dot{x}(t) \in -F(t, x(t)), \quad \forall t \in [0, T]; \quad x(0) = x_0, \qquad (v1) $$

where $F(t, \cdot) : \mathbb{R}^n \to \mathbb{R}^n$ is maximal monotone for each $t \in [0, T]$. We say that $x$ is an absolutely continuous solution of (v1) if $x \in AC([0, T], \mathbb{R}^n)$, and $x(t) \in \mathrm{Dom} F(t, \cdot)$ and $x$ satisfies (v1) almost everywhere on $[0, T]$. The following theorem represents the primary result of the paper. Additionally, the paper presents several results concerning broader classes and corollaries when applied to systems consisting of differential equations, where certain variables are combined using a maximal monotone operator (LCS for instance). It is important to emphasize that the assumptions (A1), (A2), (A3), (LG), and (YB) in Section 2.2.3 will be put to use in the subsequent part.

**Theorem E1.** [20, Theorem 11] Consider (v1) and assume that (A1)-(A3) hold. If the linear growth assumption (LG) holds, then there exists a unique AC solution of (v1) for every $x_0 \in \mathrm{cl}(\mathrm{Dom} F(0, \cdot))$. In case the Yosida approximation bound assumption (YB) holds, then there exists a unique AC solution of (v1) for every $x_0 \in \mathrm{Dom} F(0, \cdot)$.

### E.1 Extensions of Theorem E1

The first extension is provided when considering a non-autonomous DI in the form of:

$$ \dot{x}(t) \in -F(t, x(t)) + u(t), \quad \forall t \in [0, T]; \quad x(0) = x_0, \qquad (v2) $$

where $F(t, \cdot) : \mathbb{R}^n \to \mathbb{R}^n$ is maximal monotone for all $t \geq 0$ and $u \in L_1([0, T], \mathbb{R}^n)$.

**Theorem E2.** [20, Theorem 22] Consider (v2) and suppose that $u \in L_1([0, T], \mathbb{R}^n)$ and $F(t, \cdot)$ satisfies assumptions (A1)-(A3). If (LG) holds, then for every $x_0 \in \mathrm{cl}(\mathrm{Dom} F(0, \cdot))$, there exists a unique solution $x \in AC([0, T], \mathbb{R}^n)$. If (YB) holds, then for every $x_0 \in \mathrm{Dom} F(0, \cdot)$, there exists a unique solution $x \in AC([0, T], \mathbb{R}^n)$.

The second extension is presented when a Lipschitz perturbation appears. Let us consider DIs of the form:

$$ \dot{x}(t) \in -F(t, x(t)) + f(x(t)) + u(t), \quad \forall t \in [0, T]; \quad x(0) = x_0, \qquad (v3) $$
where $F(t, \cdot) : \mathbb{R}^n \to \mathbb{R}^n$ is maximal monotone for all $t \in [0, T]$, $f : \mathbb{R}^n \to \mathbb{R}^n$ is a function and $u \in L_1([0, T], \mathbb{R}^n)$. Based on Theorem E2, the following result guarantees the existence and uniqueness of a solution to (v3).

**Theorem E3.** [20, Theorem 23] Consider (v3) and suppose that $u \in L_1([0, T], \mathbb{R}^n)$, $f : \mathbb{R}^n \to \mathbb{R}^n$ is a Lipschitz continuous function, and $F(t, \cdot)$ satisfies assumptions **(A1)-(A3)**. If **(LG)** holds, then for every $x_0 \in \text{cl}(\text{Dom } F(0, \cdot))$, there exists a unique solution $x \in AC([0, T], \mathbb{R}^n)$. If **(YB)** holds, then for every $x_0 \in \text{Dom } F(0, \cdot)$, there exists a unique solution $x \in AC([0, T], \mathbb{R}^n)$.

## E.2 A corollary of Theorem E2

A corollary of Theorem E2 is derived when we apply it to a class of systems created by linear passive systems and the maximal monotone relationships among certain variables in those systems. In particular, let us consider the following system:

$$ egin{cases} 
\dot{x}(t) = Ax(t) + Bz(t) + u(t), & (v4a) \\ 
w(t) = Cx(t) + Dz(t) + v(t), & (v4b) \\ 
w(t) \in \mathcal{M}(-z(t)), & (v4c) 
\end{cases} $$

where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^n$, $v \in \mathbb{R}^m$ are external inputs and $\mathcal{M} : \mathbb{R}^m \to \mathbb{R}^m$ is some set-valued map. From (v4b) and (v4c), we get $z(t) \in -(\mathcal{M} + D)^{-1}(Cx(t) + v(t))$ and substituting it into (v4a), then (v4) is equivalent to the following, where it takes the form of (v2).

$$ \dot{x}(t) \in -H(t, x(t)) + u(t), $$

where

$$ H(t, x(t)) = -Ax + B(\mathcal{M} + D)^{-1}(Cx + v(t)) \quad (v5) $$

and $\text{Dom } H(t, \cdot) = C^{-1}(\text{Im}(\mathcal{M} + D) - v(t))$. The next theorem specifies the conditions under which $H(\cdot, \cdot)$, defined in (v5), satisfies the assumptions **(A1)-(A3)** and **(LG)** in Theorem E2.

**Theorem E4.** [20, Theorem 24] Let $T > 0$. Suppose that

(i) The system $(A, B, C, D)$ is passive with the storage function $x \mapsto \frac{1}{2}x^\top x$.

(ii) $\mathcal{M}$ is maximal monotone.

(iii) For all $t \in [0, T]$, we have $\text{rge}(C) \cap \text{rint}(\text{Im}(\mathcal{M} + D) - v(t)) \neq \emptyset$.

(iv) $v$ is bounded on $[0, T]$.

(v) There exist an AC nondecreasing function $\theta : [0, T] \to \mathbb{R}$ such that for all $s, t$ with $0 \le s \le t \le T$:

$$ \sup_{w \in \text{rge}(C) \cap (\text{Im}(\mathcal{M}+D)-v(s))} \text{dist}(w, \text{rge}(C) \cap (\text{Im}(\mathcal{M}+D)-v(t))) \le \theta(t) - \theta(s). $$

(vi) There exists a positive number $\alpha$ such that for all $\eta \in \text{Dom}(\mathcal{M} + D)^{-1}$, then

$$ \|B((\mathcal{M} + D)^{-1})^0(\eta)\| \le \alpha(1 + \|\eta\|). $$

Then, $H(\cdot, \cdot)$ satisfies assumptions **(A1)**, **(A2)**, **(A3)**, and **(LG)**.

## E.3 A corollary of Theorem E4

In this section, the linear complementarity systems with time-dependent external inputs are considered as a special case of (v4) if $\mathcal{M} \equiv \mathcal{P} : \mathbb{R}^m \to \mathbb{R}^m$ is the maximal monotone set-valued mapping defined by:
$$ \mathcal{P}(\xi) := \{\eta \in \mathbb{R}^m : \eta \ge 0, \xi \le 0, \text{ and } \langle\eta, \xi\rangle = 0\}, \text{ for any } \xi \in \mathbb{R}^m. $$

The aforementioned LCS systems have the form:

$$ \left\{ \begin{array}{l} \dot{x}(t) = Ax(t) + Bz(t) + u(t), \\ w(t) = Cx(t) + Dz(t) + v(t), \\ 0 \le z(t) \perp w(t) \ge 0, \end{array} \right. \qquad (v6) $$

where $x \in \mathbb{R}^n$ is the state, $u \in \mathbb{R}^n$ and $v \in \mathbb{R}^m$ are external inputs. The following theorem is a corollary of Theorem E4, obtained by directly applying Theorem E4 to (v6).

**Theorem E5.** [20, Theorem 26] Let $T > 0$. Suppose that

(i) The system $(A, B, C, D)$ is passive with the storage function $x \mapsto \frac{1}{2}x^\top x$.

(ii) $\text{rg}(C) \cap \text{rint}(\text{Im}(P + D) - v(t)) \neq \emptyset$ for all $t \in [0, T]$.

(iii) $v \in AC([0, T], \mathbb{R}^m)$.

Then, $H(\cdot, \cdot)$ (defined by (v5) with $M \equiv P$) satisfies assumptions (A1), (A2), (A3), and (LG).

## References

[1] F. Vasca, L. Iannelli, and K. Camlibel. A new perspective in power converters modelling: complementarity systems. In *2007 IEEE Power Electronics Specialists Conference*, pages 1817–1823, 2007.

[2] F. Vasca, L. Iannelli, M.K. Camlibel, and R. Frasca. A new perspective for modeling power electronics converters: Complementarity framework. *IEEE Transactions on Power Electronics*, 24(2):456–468, 2009.

[3] R. Frasca, M.K. Camlibel, I.C. Goknar, L. Iannelli, and F. Vasca. Linear passive networks with ideal switches: Consistent initial conditions and state discontinuities. *IEEE Transactions on Circuits and Systems I: Regular Papers*, 57(12):3138–3151, 2010.

[4] 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.

[5] V. Acary, O. Bonnefon, and B. Brogliato. *Nonsmooth Modeling and Simulation for Switched Circuits*, volume 69 of *Lecture Notes in Electrical Engineering*. Springer Verlag, October 2010. Erratum at https://inria.hal.science/hal-01311078v2/document.

[6] W.P.M.H. Heemels, J.M. Schumacher, and S. Weiland. The rational complementarity problem. *Linear Algebra and its Applications*, 294(1):93–135, 1999.

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

[8] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. The nature of solutions to linear passive complementarity systems. In *Proc. of the 38th IEEE Conference on Decision and Control*, pages 3043–3048, 1999.

[9] M.K. Camlibel, W.P.M.H. Heemels, and J.M. Schumacher. On linear passive complementarity systems. *European Journal of Control*, 8:220–237, 12 2002.
[10] 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.

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

[12] B. Brogliato and D. Goeleven. Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems. *Nonlinear Analysis: Theory, Methods Applications*, 74(1):195–212, 2011.

[13] B. Brogliato. Absolute stability and the Lagrange–Dirichlet theorem with monotone multivalued mappings. *Systems and Control Letters*, 51(5):343–353, 2004.

[14] M.K. Camlibel, L. Iannelli, and F. Vasca. Passivity and complementarity. *Math. Program.*, 145(1-2):531–563, 2014.

[15] 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.

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

[17] M.K. Camlibel and Hans Schumacher. Linear passive systems and maximal monotone mappings. *Mathematical Programming*, 157(2):397–420, June 2016.

[18] B.K. Le. On a class of Lur'e dynamical systems with state-dependent set-valued feedback. *Set-Valued and Variational Analysis*, 2020.

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

[20] M.K. Camlibel, L. Iannelli, and A. Tanwani. Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators. *Mathematical Programming, Series A*, July 2021.

[21] R.T. Rockafellar and R.J.B. Wets. *Variational Analysis*, volume 317 of *Grundlehren der mathematischen Wissenschaften*. Springer, 1998.

[22] R.W. Cottle, J.S. Pang, and R.E. Stone. *The Linear Complementarity Problem*. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1992.

[23] B. Brogliato, R. Lozano, B. Maschke, and O. Egeland. *Dissipative Systems Analysis and Control*. Communication and Control Engineering. Springer International Publishing, 3rd edition, 2020.

[24] J.F. Edmond and L. Thibault. Relaxation of an optimal control problem involving a perturbed sweeping process. *Mathematical Programming*, 104:347–373, 11 2005.

[25] C. Castaing and M.D.P. Monteiro Marques. Evolution problems associated with nonconvex closed moving sets with bounded variation. *Portugaliae Mathematica*, 53, 01 1996.
[26] E.D. Sontag. *Mathematical Control Theory: Deterministic Finite Dimensional Systems*. Texts in Applied Mathematics. Springer New York, 2013.

[27] J.B. Conway. *A Course in Functional Analysis*, volume 96 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 2nd ed edition, 1990.

[28] H.H. Bauschke and P.L. Combettes. *Convex Analysis and Monotone Operator Theory in Hilbert Spaces*. CMS Books in Mathematics. Springer, 2011.

[29] D.S. Bernstein. *Matrix Mathematics: Theory, Facts, and Formulas*. Princeton University Press, second edition, 2011.

[30] P.R. Halmos. *Finite-Dimensional Vector Spaces: Second Edition*. Dover Books on Mathematics. Dover Publications, 2017.

[31] A.A. Vladimirov. Nonstationary dissipative evolution equations in a Hilbert space. *Nonlinear Analysis: Theory, Method Applications*, 17(6):499–518, 1991.

[32] M. Kunze and M. D. P. Monteiro Marques. BV solutions to evolution problems with time-dependent domains. *Set-Valued Analysis*, 5(1):57–72, 1997.

[33] C. Castaing, Trúng Xuân Dúc Hā, and M. Valadier. Evolution equations governed by the sweeping process. *Set-Valued Analysis*, 1:109–139, 1993.

[34] J.F. Edmond and L. Thibault. BV solutions of nonconvex sweeping process differential inclusion with perturbation. *Journal of Differential Equations*, 226(1):135–179, January 2006.

[35] A. Bouach, T. Haddad, and L. Thibault. Nonconvex integro-differential sweeping process with applications. *SIAM Journal on Control and Optimization*, 60(5):2971–2995, 2022.

[36] N. Dinculeanu, I.N. Sneddon, and M. Stark. *Vector Measures*. ISSN. Elsevier Science, 2014.

[37] J.J. Moreau. Sur les mesures différentielles des fonctions vectorielles à variation locale-ment bornée. *Séminaire d'Analyse Convexe*, 5(17), 1975. http://dml.mathdoc.fr/item/hal-01878584.

[38] C. Castaing and M.D.P. Monteiro Marques. BV periodic solutions of an evolution problem associated with continuous moving convex sets. *Set-Valued Analysis*, 3:381–399, 1995.

[39] J. Diestel and J.J. Uhl. *Vector Measures*. Mathematical surveys and monographs. American Mathematical Society, 1977.

[40] V.I. Bogachev. *Measure Theory*, volume II. Springer Berlin Heidelberg, 2006.

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

[42] V. Acary and B. Brogliato. *Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics*. Lecture Notes in Applied and Computational Mechanics. Springer Berlin Heidelberg, 2008.
[43] B. Brogliato. *Nonsmooth Mechanics. Models, Dynamics and Control*. Communications and Control Engineering. Springer International Publishing Switzerland, 3rd edition, 2016.

[44] W.P.M.H. Heemels, M.K. Camlibel, and J.M. Schumacher. On the dynamic analysis of piecewise-linear networks. *IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications*, 49(3):315–327, 2002.

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

[46] F. Facchinei and J.S. Pang. *Finite-Dimensional Variational Inequalities and Complementarity Problems*. Springer Series in Operations Research and Financial Engineering. Springer New York, 2003.

[47] S.M. Robinson. Stability theory for systems of inequalities. part I: Linear systems. *SIAM Journal on Numerical Analysis*, 12(5):754–769, 1975.