# Stability characterization of impulsive linear switched systems

Yacine Chitour, Jamal Daafouz, Ihab Haidar, Paolo Mason, Mario Sigalotti

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Yacine Chitour, Jamal Daafouz, Ihab Haidar, Paolo Mason, Mario Sigalotti. Stability characterization of impulsive linear switched systems. 2025. hal-05137462

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# Stability characterization of impulsive linear switched systems

Yacine Chitour* Jamal Daafouz† Ihab Haidar‡ Paolo Mason§ Mario Sigalotti¶

June 30, 2025

## Abstract

This paper addresses a class of impulsive systems defined by a mix of continuous-time and discrete-time switched linear dynamics. We first analyze a related class of weighted discrete-time switched systems for which we establish two main stability results: a converse Lyapunov theorem and a Berger-Wang-type formula. These results are used to characterize the exponential stability of the considered class of systems via spectral and Lyapunov-based approaches, extending existing results in hybrid and switched systems theory.

## 1 Introduction

In the modeling and analysis of many real-world processes, continuous dynamics are often interrupted by sudden events or abrupt changes. Such phenomena are naturally described by *impulsive systems*, in which the state of the system undergoes discontinuous jumps at specific time instants. These systems are widely encountered in domains such as control engineering, robotics, and communication networks [20, 13]. The stability of impulsive systems has been extensively investigated in various frameworks, including Lyapunov-based approaches [2, 28, 15], Input-to-State Stability (ISS) [14, 10], and more recently, practical exponential stability under positivity constraints with applications to consensus in multi-agent systems [17].

Complementary to this, *switched systems* constitute another class of hybrid dynamical systems that consist of multiple subsystems and a rule that orchestrates the switching among them. These systems have been widely studied due to their relevance in control systems, power electronics, and fault-tolerant design. A thorough overview of stability and stabilizability results for switched linear systems can be found in [25, 21, 7].

In many practical applications, systems exhibit both impulsive and switching behaviors. This combination leads to the study of *switched impulsive systems*, which integrate the challenges of

<footnote>* Yacine Chitour is with Laboratoire des Signaux et Systèmes (L2S), Université Paris-Saclay, CNRS, Centrale-Supélec, Gif-sur-Yvette, France, yacine.chitour@12s.centralesupelec.fr.</footnote>

<footnote>† Jamal Daafouz is with CNRS, CRAN, Université de Loraine, Institut Universitaire de France (IUF), Vandoeuvre-lès-Nancy, France, Jamal.Daafouz@univ-lorraine.fr.</footnote>

<footnote>‡ Ihab Haidar is with Quartz, ENSEA, Cergy-Pontoise, France, and with Laboratoire Jacques-Louis Lions (LJLL), Inria, Sorbonne Université, Paris, France, ihab.haidar@ensea.fr.</footnote>

<footnote>§ Paolo Mason is with Laboratoire des Signaux et Systèmes (L2S), Université Paris-Saclay, CNRS, CentraleSupélec, Gif-sur-Yvette, France, paolo.mason@12s.centralesupelec.fr.</footnote>

<footnote>¶ Mario Sigalotti is with Laboratoire Jacques-Louis Lions (LJLL), Inria, Sorbonne Université, Université de Paris, CNRS, Paris, France, mario sigalotti@inria.fr.</footnote>
both impulsive and switching dynamics. The coexistence of impulses and mode transitions complicates the analysis, as both mechanisms can independently or jointly affect system stability. A general framework for analyzing such systems has been established using the concept of hybrid time domains, which unifies continuous flows and discrete events [12]. Furthermore, the ISS property for systems subject to switching and impulses has been studied in [1, 22, 19], contributing to the understanding of robustness in such hybrid contexts.

In the present paper we focus on characterizing the stability of impulsive linear switched systems described by

$$ \left\{ \begin{array}{ll} \dot{x}(t) = Z_1(t_k)x(t), & t \in [t_k, t_{k+1}), \\ x(t_{k+1}) = Z_2(t_k)x(t_{k+1}^-), & k \ge 0, \end{array} \right. \qquad (1) $$

where $t \mapsto (Z_1(t), Z_2(t))$ is a piecewise-constant function taking values in a bounded set $\mathcal{Z} \subset M_d(\mathbb{R}) \times M_d(\mathbb{R})$, and $(t_k)_{k \ge 0}$ is a strictly increasing sequence of switching times going to $+\infty$. The problem is motivated, in particular, by questions addressed in [5], where stability criteria for hybrid linear systems subject to singular perturbations are obtained thanks to the stability analysis of auxiliary systems of the form (1).

In order to characterize the stability of systems of the form (1), we first examine a general class of discrete-time switched systems, referred to as *weighted discrete-time switched systems*. More precisely, given a family $\mathcal{N} \subset M_d(\mathbb{R}) \times \mathbb{R}_{\ge 0}$, we consider the following class of discrete-time systems

$$ x(k + 1) = N(k)x(k), \quad (N(k), \tau(k)) \in \mathcal{N}, \quad k \ge 0, \qquad (2) $$

where the transition from $x(k)$ to $x(k + 1)$ takes a time duration $\tau(k)$. A standard discrete-time switched system can be seen as a special case of system (2), in which each mode has a unit weight. The notion of exponential stability of weighted discrete-time switched systems is defined in a manner analogous to the classical (unit-weight) case. It is important to note that the stability and instability of systems such as (2) are independent of the weights (cf. [6]). However, the exponential growth rate, which is formally defined and studied in [6], depends fundamentally on the associated weights. For this class of systems, we develop here two main results. The first result is a converse Lyapunov theorem characterizing the stability of (2) through the existence of a smooth Lyapunov function with suitable conditions. The second result is a Berger-Wang-type formula [3] establishing the equality, under a suitable irreducibility condition, between two measures of asymptotic stability associated with system (2): the first one based on the operator norm and the other on the spectral radius. This result plays a key role in the stability analysis of system (2). Similar types of results have been studied in [27] in the context of discrete inclusions, for linear switched dynamical systems on graphs in [8], and have been extended in [9, 18] for Markovian systems.

Returning to system (1), we consider its associated weighted discrete-time switched system with modes from

$$ \mathcal{N} = \{(Z_2 e^{tZ_1}, t) \mid t \ge 0, (Z_1, Z_2) \in \mathcal{Z}\}. $$

Using the Berger-Wang formula developed for general weighted discrete-time switched systems and extending it to the reducible case in this setting, we give a characterization of the stability of (1) in terms of the sign of its maximal Lyapunov exponent, in the sense that system (1) is exponentially stable if and only if its maximal Lyapunov exponent is negative and exponentially unstable if and only if its maximal Lyapunov exponent is positive. Building on this characterization, along with the converse Lyapunov theorem developed for general weighted discrete-time switched systems and the correspondence between the stability of (1) and its associated weighted system, we
derive a converse Lyapunov theorem for system (1). Similar converse Lyapunov results have been previously developed in the literature (see, e.g., [4]) in the general framework of hybrid systems. However, in our case, we go further by providing additional structural properties of the Lyapunov function. Let us also mention [24], where similar results have been obtained for linear switched differential-algebraic equations. In contrast to our setting, the flow and jump matrices in that framework necessarily commute, which considerably simplifies the analysis.

The paper is organized as follows. Section 2 presents the problem statement and outlines the main results. In Section 3 we establish the main results for general weighted discrete-time switched system. Section 4 is dedicated to system (1), where we apply and extend the previous results.

## 1.1 Notation

By $\mathbb{R}$ we denote the set of real numbers and by $\mathbb{R}_{\ge \tau}$ the set of real numbers greater than $\tau \ge 0$. We use $\mathbb{N}$ for the set of positive integers. We use $M_{n,m}(\mathbb{R})$ to denote the set of $n \times m$ real matrices and simply $M_n(\mathbb{R})$ if $n = m$. The $n \times n$ identity matrix is denoted by $I_n$. The spectral radius of a square matrix $M$ (i.e., the maximal modulus of its eigenvalues) is denoted by $\rho(M)$ and its spectral abscissa (i.e., the maximal real part of its eigenvalues) by $\alpha(M)$.

The Euclidean norm of a vector $x \in \mathbb{R}^n$ is denoted by $|x|$, while $\| \cdot \|$ denotes the induced norm on $M_n(\mathbb{R})$, that is, $\|M\| = \max_{x \in \mathbb{R}^n \setminus \{0\}} \frac{|Mx|}{|x|}$ for $M \in M_n(\mathbb{R})$. The vector space generated by a set of vectors $S$ is denoted by $\text{span}(S)$, and $\ker(A)$ denotes the kernel of a matrix $A$.

Given a set $\mathcal{Z}$, we denote by $S_{\mathcal{Z}}$ the set of right-continuous piecewise-constant functions from $\mathbb{R}_{\ge 0}$ to $\mathcal{Z}$, that is, those functions $Z : \mathbb{R}_{\ge 0} \to \mathcal{Z}$ such that there exists an increasing sequence $(t_k = t_k(Z))_{k \in \Theta^*(Z)}$ of switching times in $(0, +\infty)$ which is locally finite (i.e., has no finite density point) and for which $Z|_{[t_k, t_{k+1})}$ is constant for $k, k+1 \in \Theta^*(Z)$ (with $Z|_{[0, t_1)}$ and $Z|_{(\sup_{k \in \Theta^*(Z)} t_k, +\infty)}$ also constant). Here $\Theta^*(Z) = \emptyset$, $\Theta^*(Z) = \{1, \dots, N\}$, or $\Theta^*(Z) = \mathbb{N}$, depending on whether $Z$ has no, $N \in \mathbb{N}$, or infinitely many switchings, respectively. Set $t_0 = 0$ and, when $\Theta^*(Z)$ is finite with cardinality $N$, $t_{N+1} = +\infty$.

Notice that it is allowed that the value of $Z$ is the same on two subsequent intervals between switching times.

Given $\tau \ge 0$, we denote by $S_{\mathcal{Z},\tau} \subset S_{\mathcal{Z},0} = S_{\mathcal{Z}}$ the set of piecewise-constant signals with dwell time $\tau \ge 0$ (i.e., such that $t_{k+1} \ge t_k + \tau$ for $k \in \Theta(Z) := \{0\} \cup \Theta^*(Z)$).

The Hausdorff distance between two nonempty subsets $X$ and $Y$ of $\mathbb{R}^n$ is the quantity defined by

$$d_H(X, Y) = \max \left\{ \sup_{x \in X} d(x, Y), \sup_{y \in Y} d(y, X) \right\},$$

where $d(x, Y) = \inf_{y \in Y} |x - y|$ and $d(y, X) = \inf_{x \in X} |x - y|$.

# 2 Problem statement and main results

Let $d \in \mathbb{N}$ and $\mathcal{Z}$ be a bounded subset of $M_d(\mathbb{R}) \times M_d(\mathbb{R})$. Consider the linear switched systems with state jumps

$$\Sigma_{\mathcal{Z},\tau} : \begin{cases} \dot{x}(t) = Z_1(t_k)x(t), & t \in [t_k, t_{k+1}), k \in \Theta(Z), \\ x(t_k) = Z_2(t_{k-1}) \lim_{t \uparrow t_k} x(t), & k \in \Theta^*(Z), \end{cases} \quad (3)$$
for $Z \in S_{Z,\tau}$, where $\Theta(Z)$ and $\Theta^*(Z)$, introduced in Section 1.1, are used to parameterize the switching instants of the signal $Z$. We denote by $\Phi_Z(t, 0)$ the flow from time 0 to time $t$ of $\Sigma_{Z,\tau}$ associated with the switching signal $Z$, i.e., the matrix such that $x_0 \mapsto \Phi_Z(t, 0)x_0$ maps the initial condition $x(0) = x_0$ to the evolution at time $t$ of the corresponding solution of $\Sigma_{Z,\tau}$.

**Definition 1.** System $\Sigma_{Z,\tau}$ is said to be

1. exponentially stable (ES, for short) if there exist $c, \delta > 0$ such that

$$ \|\Phi_Z(t, 0)\| \le ce^{-\delta t}, \quad \forall t \ge 0, \forall Z \in S_{Z,\tau}; \quad (4) $$

2. exponentially unstable (EU, for short) if there exist $c, \delta > 0$, $Z \in S_{Z,\tau}$, and $x_0 \in \mathbb{R}^d \setminus \{0\}$ such that

$$ |\Phi_Z(t, 0)x_0| \ge ce^{\delta t}|x_0|, \quad \forall t \ge 0. $$

The maximal Lyapunov exponent of $\Sigma_{Z,\tau}$ is defined as

$$ \lambda(\Sigma_{Z,\tau}) = \limsup_{t \to +\infty} \frac{1}{t} \sup_{Z \in S_{Z,\tau}} \log(\|\Phi_Z(t, 0)\|), $$

with the convention that $\log(0) = -\infty$. We define also the quantity $\mu(\Sigma_{Z,\tau})$ given by

$$ \mu(\Sigma_{Z,\tau}) = \sup_{Z \in S_{Z,\tau}, k \in \Theta^*(Z)} \frac{\log(\rho(\Phi_Z(t_k, 0)))}{t_k}. $$

We prove the following results.

**Theorem 2.** System $\Sigma_{Z,\tau}$ is ES if and only if $\sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1) < 0$ and there exist $c, \gamma > 0$ and $V: \mathbb{R}^d \to \mathbb{R}_+$ 1-homogeneous and Lipschitz continuous such that, for every $x \in \mathbb{R}^d$, $(Z_1, Z_2) \in \mathcal{Z}$ and $t \in \mathbb{R}_{\ge \tau}$, we have

$$ |x| \le V(x) \le c|x|, \quad (5) $$

$$ V(Z_2 e^{tZ_1} x) \le e^{-\gamma t} V(x), \quad (6) $$

**Theorem 3.** Let $\lambda(\Sigma_{Z,\tau}) < +\infty$. Then

$$ \lambda(\Sigma_{Z,\tau}) = \max \left( \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \mu(\Sigma_{Z,\tau}) \right). $$

**Theorem 4.** Let $\lambda(\Sigma_{Z,\tau}) < +\infty$. Then System $\Sigma_{Z,\tau}$ is ES if and only if $\lambda(\Sigma_{Z,\tau}) < 0$ and EU if and only if $\lambda(\Sigma_{Z,\tau}) > 0$.

# 3 Stability of weighted discrete-time switched systems

Let $\mathcal{N}$ be a subset of $M_d(\mathbb{R}) \times [0, +\infty)$. We denote by $\Omega = \Omega_{\mathcal{N}}$ the set of all sequences $\omega = ((A_n, \tau_n))_{n \in \mathbb{N}}$ in $\mathcal{N}$ such that $\sum_{k \in \mathbb{N}} \tau_k = +\infty$. For every $\omega \in \Omega$ and $k \in \mathbb{N}$, let $\omega_k$ be the finite sequence made of the first $k$ elements of $\omega$ and, using the notation for $\omega$ introduced above, associate with $\omega_k$ the weight

$$ |\omega_k| = \tau_1 + \cdots + \tau_k $$
and the matrix product

$$ \Pi_{\omega_k} = A_k \cdots A_1. $$

For $k_1 \le k_2$, we define also the matrix product

$$ \Pi_{\omega_{k_1 \to k_2}} = A_{k_2} \cdots A_{k_1+1}, $$

with the convention that $\Pi_{\omega_{k_1 \to k_2}} = I_d$ if $k_1 = k_2$.

We associate with $\mathcal{N}$ a system $\Xi = \Xi_{\mathcal{N}}$ whose trajectories are the sequences $(x(k))_{k \in \mathbb{N}}$ in $\mathbb{R}^d$ such that there exists $\omega \in \Omega$ for which

$$ x(k) = \Pi_{\omega_k} x(0), \quad k \in \mathbb{N}. \tag{7} $$

We say that $\Xi$ is a weighted discrete-time system.

**Remark 5.** As we will see in Section 4, we can associate with a switched system with jumps a natural weighted discrete-time system. Weighted discrete-time systems can be used to study also more general classes of systems: for instance, we could consider the case where the dwell time depends on the mode and has also an upper bound, in the sense that there exist $\tau_-$, $\tau_+$: $\mathbb{Z} \to \mathbb{R}_{\ge 0} \cup \{+\infty\}$ with $\tau_- \le \tau_+$ such that $t_k + \tau_+(Z(t_k)) \ge t_{k+1} \ge t_k + \tau_-(Z(t_k))$ for every $k \in \Theta^*(Z)$. (See, for instance, [11, 23]).

**Definition 6.** We say that $\Xi$ is

1. exponentially stable (ES, for short) if there exist $c, \delta > 0$ such that

$$ \|\Pi_{\omega_k}\| \le c e^{-\delta|\omega_k|}, \quad \forall \omega \in \Omega, \forall k \in \mathbb{N}; \tag{8} $$

2. exponentially unstable (EU, for short) if there exist $c, \delta > 0$, $x_0 \in \mathbb{R}^d \setminus \{0\}$, and $\omega \in \Omega$ such that

$$ |\Pi_{\omega_k} x_0| \ge c e^{\delta|\omega_k|} |x_0|, \quad \forall k \in \mathbb{N}. $$

## 3.1 Converse Lyapunov theorem for weighted discrete-time switched systems

The following result is a converse Lyapunov theorem stated in the context of weighted discrete-time switched systems.

**Theorem 7.** Let $\Xi = \Xi_{\mathcal{N}}$ be a weighted discrete-time switched system. System $\Xi$ is ES if and only if there exist $c, \gamma > 0$ and a 1-homogeneous and Lipschitz continuous function $V: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ such that

$$ |x| \le V(x) \le c|x|, \quad \forall x \in \mathbb{R}^d, \tag{9} $$

$$ V(Ax) \le e^{-\gamma\tau} V(x), \quad \forall x \in \mathbb{R}^d, \forall (A, \tau) \in \mathcal{N}. \tag{10} $$

*Proof.* We prove first the sufficiency part. Let $x \in \mathbb{R}^d$, $\omega \in \Omega$, and $k \ge 1$. Using inequalities (9)-(10), we have

$$ |\Pi_{\omega_k} x| \le V(\Pi_{\omega_k} x) \le e^{-\gamma|\omega_k|} V(x) \le c e^{-\gamma|\omega_k|} |x|. $$

By arbitrariness of $x \in \mathbb{R}^d$, we have $\|\Pi_{\omega_k}\| \le c e^{-\gamma|\omega_k|}$ and $\Xi$ is ES.
Now, suppose that $\Xi$ is ES. Let $\gamma$ be any positive scalar smaller than or equal to the constant $\delta$ appearing in Definition 6. Consider the function $V: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ defined by

$$V(x) = \sup_{\omega \in \Omega} \sup_{k \ge 0} |\Pi_{\omega_k} x| e^{\gamma|\omega_k|}, \quad x \in \mathbb{R}^d, \quad (11)$$

where we set $\Pi_{\omega_0} = I_d$ and $|\omega_0| = 0$ for every $\omega \in \Omega$.

The left-hand side of (9) is obtained by considering $k = 0$ while the right-hand side of inequality (9) follows from the definition of ES of $\Xi$.

Let now $(A, \tau) \in \mathcal{N}$, and consider $\omega \in \Omega$ such that $\omega_1 = (A, \tau)$. We have

$$ \begin{aligned} V(x) &\ge \sup_{k \ge 0} |\Pi_{\omega_k} x| e^{\gamma(\tau_k + \dots + \tau_1)} \\ &\ge \sup\{ |Ax| e^{\gamma\tau}, |A_2Ax| e^{\gamma(\tau_2+\tau)}, \dots, \\ &\qquad |A_k \dots A_2Ax| e^{\gamma(\tau_k + \dots + \tau_2+\tau)}, \dots \} \\ &= e^{\gamma\tau} \sup\{ |Ax|, |A_2Ax| e^{\gamma\tau_2}, \dots, \\ &\qquad |A_k \dots A_2Ax| e^{\gamma(\tau_k + \dots + \tau_2)}, \dots \}. \end{aligned} $$

Taking the sup over $\{\omega \in \Omega | \omega_1 = (A, \tau)\}$, we get the inequality

$$V(x) \ge e^{\gamma\tau} \sup_{\tilde{\omega} \in \Omega} \sup_{k \ge 0} |\Pi_{\tilde{\omega}_k} Ax| e^{\gamma|\tilde{\omega}_k|} = e^{\gamma\tau} V(Ax),$$

from which we obtain (10).

Concerning the Lipschitz continuity of $V$, let $x, y \in \mathbb{R}^d$. Since $\Xi$ is ES, it follows that $(\omega, k) \mapsto \|\Pi_{\omega_k} e^{\gamma|\omega_k|}\|$ is upper-bounded by some positive constant $L$ over $\Omega \times \mathbb{N}$. By consequence, we have

$$ \begin{aligned} V(x) - V(y) &= \sup_{\omega \in \Omega} \sup_{k \ge 0} |\Pi_{\omega_k} x| e^{\gamma|\omega_k|} - \sup_{\omega \in \Omega} \sup_{k \ge 0} |\Pi_{\omega_k} y| e^{\gamma|\omega_k|} \\ &\le \sup_{\omega \in \Omega} \sup_{k \ge 0} \left( |\Pi_{\omega_k} x| e^{\gamma|\omega_k|} - |\Pi_{\omega_k} y| e^{\gamma|\omega_k|} \right) \\ &\le \sup_{\omega \in \Omega} \sup_{k \ge 0} \|\Pi_{\omega_k} e^{\gamma|\omega_k|}\| |x - y| \\ &\le L|x - y|, \end{aligned} $$

from which we get that $|V(x) - V(y)| \le L|x - y|$ for every $x, y \in \mathbb{R}^d$. $\square$

## 3.2 Maximal Lyapunov exponents for weighted discrete time switched systems

The maximal Lyapunov exponent of a weighted discrete-time switched system $\Xi = \Xi_N$ is defined as

$$\lambda(\Xi) = \limsup_{t \to +\infty} \sup_{\omega \in \Omega, k \in \mathbb{N}, |\omega_k| = t} \frac{\log(\|\Pi_{\omega_k}\|)}{|\omega_k|}, \quad (12)$$

with the convention that $\sup \emptyset = -\infty$ and $\log 0 = -\infty$.
We define also

$$ \hat{\lambda}(\Xi) = \limsup_{t \to +\infty} \sup_{\{A|(A,t) \in \mathcal{N}\}} \frac{\log(\|A\|)}{t}. \quad (13) $$

We now define $\mu(\Xi)$, which is the counterpart of $\lambda(\Xi)$ in which the norm of a product $\Pi_{\omega_k}$ is replaced by its spectral radius, that is,

$$ \mu(\Xi) = \limsup_{t \to +\infty} \sup_{\omega \in \Omega, k \in \mathbb{N}, |\omega_k| = t} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|}. \quad (14) $$

**Remark 8.** It holds

$$ \mu(\Xi) = \sup_{\omega \in \Omega, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|}. $$

Indeed, by definition of $\mu(\Xi)$, one immediately has $\mu(\Xi) \le \sup_{\omega \in \Omega, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|}$. Conversely, fix $\omega \in \Omega$ and $k \in \mathbb{N}$, and denote by $\omega_k^* \in \Omega$ the repetition of the finite sequence $\omega_k$ for infinitely many times. Then, for every $m \in \mathbb{N}$,

$$ \rho(\Pi_{(\omega_k^*)_{mk}}) = \rho(\Pi_{\omega_k}^m) = \rho(\Pi_{\omega_k})^m $$

and $|(\omega_k^*)_{mk}| = m|\omega_k|$, whence the inequality $\frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \le \mu(\Xi)$.

Let, for $\xi \in \mathbb{R}$, $\Omega^\xi$ be the set of sequences $((e^{\xi\tau_n}A_n, \tau_n))_{n \in \mathbb{N}}$ such that $(A_n, \tau_n) \in \mathcal{N}$ and denote by $\Xi^\xi$ the corresponding discrete-time weighted system.

**Lemma 9.** For every $\xi \in \mathbb{R}$ we have

$$ \lambda(\Xi^\xi) = \xi + \lambda(\Xi), \quad \hat{\lambda}(\Xi^\xi) = \xi + \hat{\lambda}(\Xi), \quad \mu(\Xi^\xi) = \xi + \mu(\Xi). \quad (15) $$

*Proof.* The proof is direct from the definition of $\lambda(\Xi)$, $\hat{\lambda}(\Xi)$ and $\mu(\Xi)$. $\square$

Let us recall the definition of irreducible set of matrices.

**Definition 10.** We say that a subspace $E$ of $\mathbb{R}^d$ is invariant with respect to a set $M \subset M_d(\mathbb{R})$ if and only if it is invariant with respect to every matrix $M \in M$, i.e., $Mx \in E$ for every $x \in E$. A set $M \subset M_d(\mathbb{R})$ is said to be irreducible if its only invariant subspaces are $\{0\}$ and $\mathbb{R}^d$. Otherwise it is said to be reducible.

We will make use of the following result.

**Lemma 11.** Let $M \subset M_d(\mathbb{R})$ be irreducible and consider a subset $M_1$ of $M$ containing at least one nonzero matrix. Then, for every $x \in \mathbb{R}^d \setminus \{0\}$ there exists a product $\Pi$ of matrices in $M$ containing at least one element of $M_1$ and such that $\Pi x \ne 0$.

*Proof.* In the following we denote as $M^k$ the set of all possible products of $k$ matrices of $M$ and by $M^kx$ the set of all possible evaluations of matrices of $M^k$ at $x \in \mathbb{R}^d$.

Let $x \in \mathbb{R}^d \setminus \{0\}$. If $x \notin \cap_{M \in M_1} \ker(M)$ then in particular $x \notin \ker(\bar{M})$ for some $\bar{M} \in M_1$ and we can take $\Pi = \bar{M}$.

Assume now $x \in \cap_{M \in M_1} \ker(M)$. By irreducibility of $M$ we have that the vector space $\text{span}\{M^kx \mid k \in \mathbb{N}\}$, which is invariant with respect to $M$, coincides with $\mathbb{R}^d$. This means that there exist $k \in \mathbb{N}$ and $\bar{\Pi} \in M^k$ such that $\bar{\Pi}x \notin \cap_{M \in M_1} \ker(M)$ (note that $\cap_{M \in M_1} \ker(M) \ne \mathbb{R}^d$ since $M_1$ contains a nonzero matrix). In particular $\bar{\Pi}x \notin \ker(\bar{M})$ for some $\bar{M} \in M_1$ and we can take $\Pi = \bar{M}\bar{\Pi}$. $\square$
For general weighted discrete-time switched systems, we need the following assumption.

**Assumption 12.** The set $\{A \mid (A, \tau) \in \mathcal{N}\}$ is irreducible.

**Proposition 13.** Assume that $\lambda(\Xi) \in \mathbb{R}$ (i.e., $\lambda(\Xi)$ is finite) and that Assumption 12 holds. Then there exists $C \ge 1$ such that $\|\Pi_{\omega_k}\| \le Ce^{\lambda(\Xi)|\omega_k|}$ for every $\omega \in \Omega$ and $k \in \mathbb{N}$.

*Proof.* Since $\lambda(\Xi)$ is finite, we can suppose without loss of generality that $\lambda(\Xi) = 0$ by Lemma 9. Consider the set

$$E := \{x \in \mathbb{R}^d \mid \sup_{\omega \in \Omega, k \in \mathbb{N}} |\Pi_{\omega_k} x| < \infty\}.$$

The set $E$ is a subspace of $\mathbb{R}^d$ which is invariant with respect to $\{A \in M_d(\mathbb{R}) \mid (A, \tau) \in \mathcal{N}\}$, which is irreducible by Assumption 12. Hence, $E$ is either $\{0\}$ or $\mathbb{R}^d$.

We will prove that $E = \mathbb{R}^d$. By contradiction, assume that $E = \{0\}$. Then we claim that there exists a finite subset $\mathcal{W}$ of $\Omega \times \mathbb{N}$ such that, for every nonzero $x \in \mathbb{R}^d$, there exists $(\omega, k) \in \mathcal{W}$ such that $|\omega_k| > 0$ and $|\Pi_{\omega_k} x| > 2|x|$. Note that we can choose the same pair $(\omega, k)$ for all vectors proportional to $x$. Then, we just need to prove the claim for all $x$ belonging to the unit sphere of $\mathbb{R}^d$. To prove the claim we first note that we can apply Lemma 11 with $\mathcal{M} = \{A \in M_d(\mathbb{R}) \mid (A, \tau) \in \mathcal{N}\}$ and $\mathcal{M}_1 = \{A \in M_d(\mathbb{R}) \mid (A, \tau) \in \mathcal{N}, \tau > 0\}$ (the fact that $\mathcal{M}_1$ contains a nonzero matrix follows from the fact that, by assumption, $\lambda(\Xi) > -\infty$). We then deduce from Lemma 11 that, for every $x \in \mathbb{R}^d$ with $|x| = 1$ there exists $\omega^1 \in \Omega$ and $k^1 \in \mathbb{N}$ such that $|\omega_{k^1}^1| > 0$ and $\Pi_{\omega_{k^1}^1} x \ne 0$. Then, since $E = \{0\}$, there exists $\omega^2 \in \Omega$ and $k^2 \in \mathbb{N}$ such that $|\Pi_{\omega_{k^2}^2} \Pi_{\omega_{k^1}^1} x| > 2|x|$. In other words we have constructed $\omega \in \Omega$ such that, setting $k = k^1 + k^2$ we have $|\Pi_{\omega_k} x| > 2|x|$ and $|\omega_k| = |\omega_{k^1}^1| + |\omega_{k^2}^2| > 0$. For $x \in \mathbb{R}^d$ with $|x| = 1$ we can find $(\omega^x, k^x) \in \Omega \times \mathbb{N}$ such that $|\omega_{k^x}^x| > 0$ and $|\Pi_{\omega_{k^x}^x} y| > 2|y|$ for every $y \in U_{\omega^x, k^x}$, $U_{\omega^x, k^x}$ being a open neighborhood of $x$. The sets $U_{\omega^x, k^x}$ form an open covering of the unit sphere and, by compactness of the latter, we can extract a finite covering associated with a finite set $\mathcal{W}$ of elements of $\Omega \times \mathbb{N}$ which satisfies the claim.

Let $\delta = \min_{(\omega, k) \in \mathcal{W}} |\omega_k|$ and $\Delta = \max_{(\omega, k) \in \mathcal{W}} |\omega_k|$. Fix $x_0 \in \mathbb{R}^d \setminus \{0\}$. Then we can construct recursively a sequence $\bar{\omega} \in \Omega$ by concatenating finite sequences $\omega_{k^i}$, with $(\omega^i, k^i) \in \mathcal{W}$ for $i \in \mathbb{N}$, in such a way that, setting $x_i = \Pi_{\omega_{k^{i-1}}^i} \cdots \Pi_{\omega_{k^0}^0} x_0$, one has $|\Pi_{\omega_{k^i}^i} x_i| \ge 2|x_i|$. Setting $\ell^n = \sum_{i=0}^{n-1} |\omega_{k^i}^i|$ we then have $\lim_{n \to +\infty} |\bar{\omega}_{\ell^n}| \ge \lim_{n \to +\infty} n\delta = +\infty$ so that

$$0 = \lambda(\Xi) \ge \limsup_{n \to +\infty} \frac{\log \|\Pi_{\bar{\omega}_{\ell^n}}\|}{|\bar{\omega}_{\ell^n}|} \ge \limsup_{n \to +\infty} \frac{\log |\Pi_{\bar{\omega}_{\ell^n}} x_0|}{|\bar{\omega}_{\ell^n}|} \\ \ge \limsup_{n \to +\infty} \frac{\log (2^n |x_0|)}{n\Delta} = \frac{\log 2}{\Delta},$$

which is a contradiction. We have therefore proved that $E$ is equal to $\mathbb{R}^d$.

Now, assume by contradiction that $\sup_{\omega \in \Omega, k \in \mathbb{N}} \|\Pi_{\omega_k}\|$ is not finite. Then, there exist a sequence $\{(\omega^n, k^n)\}_{n \in \mathbb{N}}$ of elements of $\Omega \times \mathbb{N}$ and a sequence of unit vectors $\{x^n\}_{n \in \mathbb{N}}$ in $\mathbb{R}^d$ so that $|\Pi_{\omega_{k^n}^n} x^n| = \|\Pi_{\omega_{k^n}^n}\|$ tends to infinity as $n$ goes to infinity. Up to a subsequence, one has that $\lim_{n \to +\infty} x^n = x^* \in \mathbb{R}^d$, and

$$|\Pi_{\omega_{k^n}^n} x^*| \ge |\Pi_{\omega_{k^n}^n} x^n| - |\Pi_{\omega_{k^n}^n} (x^n - x^*)| \\ \ge \|\Pi_{\omega_{k^n}^n}\| (1 - |x^n - x^*|) \ge \frac{\|\Pi_{\omega_{k^n}^n}\|}{2},$$
where the last inequality holds true for $n$ large enough. Hence, $x \notin E$, which is a contradiction with the fact that $E = \mathbb{R}^d$. Hence $\sup_{\omega \in \Omega, k \in \mathbb{N}} \|\Pi_{\omega_k}\|$ is finite, which concludes the proof of the proposition. $\square$

**Proposition 14.** Assume that $\hat{\lambda}(\Xi) < \lambda(\Xi) < +\infty$ and that Assumption 12 holds. Then for every $t > 0$ and every $x \in \mathbb{R}^n$, there exists $(\omega, k) \in \Omega \times \mathbb{N}$ satisfying $|\omega_k| \ge t$ and $|\Pi_{\omega_k} x| \ge c e^{\lambda(\Xi)|\omega_k|} |x|$ for some positive constant $c$ only depending on $\mathcal{N}$.

*Proof.* As in the proof of Proposition 13 we assume, without loss of generality, that $\lambda(\Xi) = 0$. Consider the set

$$ \mathcal{R}_{\infty} := \cap_{t \ge 0} \mathcal{R}_t, $$

where $\mathcal{R}_t = \overline{\{\Pi_{\omega_k} | (\omega, k) \in \Omega \times \mathbb{N} \text{ s.t. } |\omega_k| \ge t\}}$. The sequence of sets $\mathcal{R}_t$ is decreasing, in the sense that $\mathcal{R}_t \supset \mathcal{R}_s$ whenever $s \ge t$, and each $\mathcal{R}_t$ is closed and, by Proposition 13, bounded. Hence $\mathcal{R}_{\infty}$ is closed and bounded, that is, it is a compact subset of $M_d(\mathbb{R})$. Moreover, $\mathcal{R}_{\infty}$ is nonempty by Cantor intersection theorem.

We claim that $\mathcal{R}_{\infty} \neq \{0\}$. Assume by contradiction that $\mathcal{R}_{\infty} = \{0\}$. Then

$$ \begin{aligned} \cap_{t \ge 0} (\mathcal{R}_t \cap \{A \in M_d(\mathbb{R}) \mid \|A\| \ge 1/2\}) &= \\ \mathcal{R}_{\infty} \cap \{A \in M_d(\mathbb{R}) \mid \|A\| \ge 1/2\} &= \emptyset. \end{aligned} $$

It follows from Cantor intersection theorem that $\mathcal{R}_T \cap \{A \in M_d(\mathbb{R}) \mid \|A\| \ge \frac{1}{2}\} = \emptyset$ for some $T$ large enough, that is,

$$ \|\Pi_{\omega_k}\| < \frac{1}{2}, \quad \forall (\omega, k) \in \Omega \times \mathbb{N} \text{ s.t. } |\omega_k| \ge T. $$

Next, we derive further estimates of $\|\Pi_{\omega_k}\|$ for finite sequences $\omega_k$ in two special cases.

(A) Consider a finite sequence $\omega_k = \{(A_n, \tau_n)\}_{n=1,...,k}$ with $\tau_n < T$ for every $n$. Then $\omega_k$ can be written as the concatenation of finite sequences $\omega_k^i$, $i = 1, ..., \ell$ (for some $\ell \ge 0$), of elements of $\mathcal{N}$ such that $|\omega_k^i| \in [T, 2T)$ together with a sequence $\bar{\omega}_k$ such that $|\bar{\omega}_k| \le T$. In particular $\ell$ satisfies $|\omega_k|/(2T) - 1/2 \le \ell \le |\omega_k|/T$. Then

$$ \|\Pi_{\omega_k}\| \le \|\Pi_{\bar{\omega}_k}\| \prod_{i=1}^{\ell} \|\Pi_{\omega_k^i}\| \le C 2^{-\ell} \le \sqrt{2C} 2^{-\frac{|\omega_k|}{2T}}, \quad (16) $$

where $C$ is as in Proposition 13.

(B) Consider a finite sequence $\omega_k = \{(A_n, \tau_n)\}_{n=1,...,k}$ with $\tau_n \ge T$ for every $n$. Pick $\gamma$ in the open interval $(\hat{\lambda}(\Xi), 0)$. By definition of $\hat{\lambda}(\Xi)$ and the inequality $\hat{\lambda}(\Xi) < \gamma$, it follows that, up to increasing $T$, $\|A\| \le \frac{1}{\sqrt{2C}} e^{\gamma\tau}$ for every $(A, \tau) \in \mathcal{N}$ with $\tau \ge T$. In particular,

$$ \|\Pi_{\omega_k}\| \le \frac{1}{\sqrt{2C}} e^{\gamma|\omega_k|}. \quad (17) $$

We observe now that for every $\omega$ in $\Omega$ with $\lim_{k \to +\infty} |\omega_k| = +\infty$ and every positive integer $\ell$ the finite sequence $\omega_\ell$ can be written as a concatenation of finite sequences which alternate between type (A) and type (B). By submultiplicativity of the matrix norm and the estimates (16) and (17) it follows that $\|\Pi_{\omega_\ell}\| \le \sqrt{2C} e^{-\alpha|\omega_\ell|}$, where $\alpha = \min\{-\gamma, \log 2/(2T)\} > 0$. Since $\omega$ and $\ell$ have been chosen arbitrarily, using the definition (12) of $\lambda(\Xi)$ we obtain $\lambda(\Xi) \le -\alpha < 0$, which is a contradiction. This concludes the proof that $\mathcal{R}_{\infty} \neq \{0\}$.
To conclude the proof it is enough to show that the map

$$x \mapsto v_t(x) := \max_{R \in \mathcal{R}_t} |Rx|$$

is a norm on $\mathbb{R}^d$ for every $t \ge 0$. Indeed, in this case, by the equivalence of norms on finite-dimensional spaces, there exists $\kappa > 0$ such that with every nonzero $x \in \mathbb{R}^d$ and $t \ge 0$ one can associate an element $R \in \mathcal{R}_t$ satisfying $|Rx| \ge \kappa|x|$. Then, from the definition of $\mathcal{R}_t$, it follows that there exists $(\omega, k) \in \Omega \times \mathbb{N}$ with $|\omega_k| \ge t$ such that $|\Pi_{\omega_k}x| \ge \frac{\kappa}{2}|x|$.

Let us prove that $v_t$ is a norm. By compactness of $\mathcal{R}_t$ one has that $v_t$ is well defined. Furthermore, by definition, it is clear that $v_t$ is absolutely homogeneous and satisfies the triangle inequality. It remains to show that $v_t$ is strictly positive outside the origin. Note that, by definition, each $\mathcal{R}_t$ is invariant with respect to right multiplication by elements of $\{A \in M_d(\mathbb{R}) | (A, \tau) \in \mathcal{N}\}$. It follows that the vector space

$$\{x \in \mathbb{R}^d | Rx = 0 \quad \forall R \in \mathcal{R}_t\},$$

which is a strict subspace of $\mathbb{R}^d$ since $\mathcal{R}_t \ne \{0\}$, is invariant with respect to the set $\{A \in M_d(\mathbb{R}) | (A, \tau) \in \mathcal{N}\}$ and therefore, by Assumption 12, it coincides with $\{0\}$. This means that $v_t$ is strictly positive outside the origin. We have therefore shown that $v_t$ is a norm.

This concludes the proof of the proposition. □

**Remark 15.** Proposition 14 cannot be extended, in general, to the case where $\hat{\lambda}(\Xi) = \lambda(\Xi)$. Consider, for example, the case where $\mathcal{N} = \{(\frac{1}{n}e^{-\frac{\tau}{n}}, \tau) | n \ge 2, \tau \ge 0\}$. In this case it is easy to verify that $\hat{\lambda}(\Xi) = \lambda(\Xi) = 0$.

On the other hand, consider a general $\omega \in \Omega$, that is a sequence $((\frac{1}{n_k}e^{-\frac{\tau_k}{n_k}}, \tau_k))_{k \in \mathbb{N}}$ in $\mathcal{N}$. Notice that

$$\|\Pi_{\omega_k}\| \le \frac{1}{\max\{n_1, \dots, n_k\}} e^{-\frac{|\omega_k|}{\max\{n_1, \dots, n_k\}}}, \quad \forall k \in \mathbb{N}.$$

Assume by contradiction that there exists $c > 0$ as in Proposition 14. In order to have $|\Pi_{\omega_k}x| \ge c|x|$ for a given $x \ne 0$, one must have $\frac{1}{\max\{n_1, \dots, n_k\}} > c$, that is, $\max\{n_1, \dots, n_k\} < \frac{1}{c}$. This implies that $e^{-\frac{|\omega_k|}{\max\{n_1, \dots, n_k\}}}$ is smaller than $e^{-c|\omega_k|}$, so that $e^{-c|\omega_k|}|x| \ge c|x|$. Since in Proposition 14 the sequence $\omega$ and the integer $k$ are taken in such a way that $|\omega_k| > t$ and $t$ can be arbitrarily large, a contradiction is reached.

**Proposition 16.** Let Assumption 12 hold. Assume, moreover, that $0 < \lambda(\Xi) < +\infty$. Then, either $\hat{\lambda}(\Xi) = \lambda(\Xi)$ or there exists $\omega \in \Omega$ and $k \in \mathbb{N}$ such that $\rho(\Pi_{\omega_k}) > 1$.

*Proof.* By definitions of $\hat{\lambda}(\Xi)$ and $\lambda(\Xi)$, we have either $\hat{\lambda}(\Xi) = \lambda(\Xi)$ or $\hat{\lambda}(\Xi) < \lambda(\Xi)$.

Assume $\hat{\lambda}(\Xi) < \lambda(\Xi)$. In this case, by Lemma 9, we can equivalently prove that $\sup_{(\omega,k) \in \Omega \times \mathbb{N}} \rho(\Pi_{\omega_k}) \ge 1$ whenever $\lambda(\Xi) = 0$. By Proposition 13, there exists $C > 0$ such that

$$\|\Pi_{\omega_k}\| \le C, \quad \text{for every } \omega \in \Omega \text{ and } k \in \mathbb{N}. \qquad (18)$$

Moreover, by Proposition 14, there exists a sequence $(\omega^n, k_n)_n$ in $\Omega \times \mathbb{N}$ with $|\omega_{k_n}^n| \to +\infty$ such that $\|\Pi_{\omega_{k_n}^n}\| \ge c$ for some $c > 0$ independent of $n$.

Denote by $\tau_n$ the maximal weight of an element of $\omega_{k_n}^n$ and by $(A_n, \tau_n)$ the corresponding element of $\mathcal{N}$. We claim that

$$\sup_n \tau_n < +\infty. \qquad (19)$$
Indeed, if this were not the case, write $\Pi_{\omega_{k_n}^n}$ as $\Pi_{\nu_n} A_n \Pi_{\mu_n}$, for some finite sequences $\nu_n$ and $\mu_n$ in $\mathcal{N}$. Applying (18) to $\mu_n$ and $\nu_n$ and using the relation $\tau_n = |\omega_{k_n}^n| - |\nu_n| - |\mu_n|$, we deduce that

$$ \|A_n\| \geq \frac{c}{C^2}. $$

Since $\tau_n \to +\infty$ by the contradiction assumption, we have, by definition of $\hat{\lambda}(\Xi)$, that $\hat{\lambda}(\Xi) \geq 0$, which is impossible given that $\hat{\lambda}(\Xi) < \lambda(\Xi) = 0$. This concludes the proof of (19).

Let us now define the vectors

$$ y_{k,n} = \Pi_{\omega_k^n} x_n, \quad k = 1, \dots, k_n, \qquad (20) $$

where $(x_n)_n$ is a sequence of unit vectors such that $\|\Pi_{\omega_{k_n}^n}\| = |\Pi_{\omega_{k_n}^n} x_n|$. Since

$$ |\Pi_{\omega_k^n \to \omega_{k_n}^n} y_{k,n}| = |\Pi_{\omega_{k_n}^n} x_n| \geq c, $$

we deduce from (18) that each $y_{k,n}$ belongs to $K$, where

$$ K := \{x \in \mathbb{R}^d \mid \frac{c}{C} \leq |x| \leq C\}. $$

We define the set

$$ I_n = \{(j_1, j_2) \mid 1 \leq j_1 < j_2 \leq k_n\}. \qquad (21) $$

By (19), we have $\#I_n \to +\infty$. For $n \geq 1$, let $j_1^n, j_2^n$ be such that

$$ (j_1^n, j_2^n) \in \arg \min_{(j_1, j_2) \in I_n} |y_{j_1, n} - y_{j_2, n}| \qquad (22) $$

and notice that

$$ \lim_{n \to +\infty} |y_{j_1^n, n} - y_{j_2^n, n}| \to 0 \qquad (23) $$

by the boundedness of $K$ and unboundedness of $\#I_n$. We have

$$ y_{j_2^n, n} = \Pi_{\omega_{j_1^n \to j_2^n}^n} y_{j_1^n, n}, \quad \forall n \geq 1. \qquad (24) $$

Up to extracting a subsequence, we can assume that $y_{j_1^n, n}$ converges to some $y^*$. Notice that $y^* \neq 0$ (by definition of $K$) and that $\lim_{n \to +\infty} y_{j_2^n, n} = y^*$ by (23).

By (18), we can extract a subsequence of $\Pi_{\omega_{j_1^n \to j_2^n}^n}$ converging to some $M \in M_d(\mathbb{R})$ as $n$ tends to infinity. By (24) we deduce that $My^* = y^*$, implying that $\rho(M) \geq 1$. Therefore,

$$ \lim_{n \to +\infty} \rho(\Pi_{\omega_{j_1^n \to j_2^n}^n}) = \rho(M) \geq 1, \qquad (25) $$

concluding the proof. $\square$

**Corollary 17.** Let Assumption 12 hold and suppose that $\lambda(\Xi) < +\infty$. Then

$$ \lambda(\Xi) = \max\{\hat{\lambda}(\Xi), \mu(\Xi)\}. \qquad (26) $$

*Proof.* Observe that $\rho(\Pi_{\omega_k}) \leq \|\Pi_{\omega_k}\|$ for every $(\omega, k) \in \Omega \times \mathbb{N}$. Hence, $\mu(\Xi) \leq \lambda(\Xi)$. Using the inequality $\hat{\lambda}(\Xi) \leq \lambda(\Xi)$, we get that $\max\{\hat{\lambda}(\Xi), \mu(\Xi)\} \leq \lambda(\Xi)$.

We are left to show that $\max\{\hat{\lambda}(\Xi), \mu(\Xi)\} \geq \lambda(\Xi)$. If $\hat{\lambda}(\Xi) = \lambda(\Xi)$ or $\lambda(\Xi) = -\infty$ the conclusion holds true.

Let us consider $\varepsilon > 0$ and apply Lemma 9 with $\xi = \varepsilon - \lambda(\Xi)$, noticing that $\lambda(\Xi)$ is finite by assumption. Since $\lambda(\Xi^\xi) = \varepsilon > 0$, we deduce from Proposition 16 that either $\hat{\lambda}(\Xi^\xi) = \lambda(\Xi^\xi)$ or $\mu(\Xi^\xi) > 0$, i.e., either $\hat{\lambda}(\Xi) = \lambda(\Xi)$ or $\mu(\Xi) + \varepsilon > \lambda(\Xi)$. Since this holds for any $\varepsilon > 0$ and we have reduced our analysis to the case $\hat{\lambda}(\Xi) < \lambda(\Xi)$, we conclude that $\mu(\Xi) \geq \lambda(\Xi)$. Therefore, $\max\{\hat{\lambda}(\Xi), \mu(\Xi)\} \geq \lambda(\Xi)$, which completes the proof. $\square$
## 4 System $\Sigma_{\mathcal{Z},\tau}$ seen as a weighted discrete-time switched system

In order to study the stability of System $\Sigma_{\mathcal{Z},\tau}$ introduced in Section 2, we will consider the weighted discrete-time switched system $\Xi_{\mathcal{Z},\tau} := \Xi_{\mathcal{N}_{\mathcal{Z},\tau}}$, where $\mathcal{N}_{\mathcal{Z},\tau} := \{(Z_2e^{tZ_1}, t) | t \ge \tau, (Z_1, Z_2) \in \mathcal{Z}\}$. Denote also $\Omega_{\mathcal{Z},\tau} := \Omega_{\mathcal{N}_{\mathcal{Z},\tau}}$, where we again use the notation introduced at the beginning of Section 3. It is important to note, before presenting the analysis here, that Assumption 12 is no longer required for this particular class of weighted discrete-time switched systems.

The two systems $\Sigma_{\mathcal{Z},\tau}$ and $\Xi_{\mathcal{Z},\tau}$ are strongly related but not completely equivalent. With every $Z \in S_{\mathcal{Z},\tau}$ with infinitely many switchings and every initial condition $x_0$, we can associate a trajectory of $\Xi_{\mathcal{Z},\tau}$ given by the evaluation at the switching times of $Z$ of the trajectory of $\Sigma_{\mathcal{Z},\tau}$ starting from $x_0$ and corresponding to $Z$. However, one cannot always associate with a trajectory of $\Sigma_{\mathcal{Z},\tau}$ corresponding to a signal $Z \in S_{\mathcal{Z},\tau}$ having finitely many switchings a trajectory of $\Xi_{\mathcal{Z},\tau}$ having the same asymptotic behavior. Moreover, in the case where $\tau = 0$, $\Xi_{\mathcal{Z},\tau}$ may contain more trajectories than those corresponding to trajectories of $\Sigma_{\mathcal{Z},\tau}$, since $\mathcal{N}_{\mathcal{Z},0}$ contains also elements of the type $(Z_2, 0)$ for $(Z_1, Z_2) \in \mathcal{Z}$, while the distance between two switching times is always positive. (In this sense, $\mathcal{N}_{\mathcal{Z},0}$ can be used to study also switched systems that can jump several times at the same time instant, provided that there are finitely many jumps on any positive time-interval).

Notice that

$$ \lambda(\Xi_{\mathcal{Z},\tau}) \le \lambda(\Sigma_{\mathcal{Z},\tau}), \qquad (27) $$

as it can be deduced from the definition of the two maximal Lyapunov exponents, by noticing that for every $\omega \in \Omega_{\mathcal{Z},\tau}$ and every $k \in \mathbb{N}$, there exist a sequence $(Z^n)_{n \in \mathbb{N}}$ in $S_{\mathcal{Z},\tau}$ and a sequence $(t_n)_{n \in \mathbb{N}}$ in $[0, +\infty)$ such that $\lim_{n \to \infty} t_n = |\omega_k|$ and $\lim_{n \to \infty} \Phi_{Z^n}(t_n, 0) = \Pi_{\omega_k}$. Each $Z^n$ can be constructed by associating with $\omega_k = ((Z_{j,2}e^{t_j Z_{j,1}}, t_j))_{j=1}^k$ the piecewise constant signal whose jth piece is equal to $(Z_{j,1}, Z_{j,2})$ on an interval of length $t_j + \frac{1}{n}$.

**Remark 18.** Notice that both $\lambda(\Xi_{\mathcal{Z},\tau}) = -\infty$ and $\lambda(\Xi_{\mathcal{Z},\tau}) = +\infty$ may occur. For instance, $\lambda(\Xi_{\mathcal{Z},\tau}) = -\infty$ when $Z_2 = 0$ for every $(Z_1, Z_2) \in \mathcal{Z}$. As for $\lambda(\Xi_{\mathcal{Z},\tau}) = +\infty$, a necessary condition for it to happen is that $\tau = 0$. Indeed, if $\tau$ is positive and $\gamma_1, \gamma_2 \in (0, +\infty)$ are taken so that $\|Z_1\| \le \gamma_1$ and $\|Z_2\| \le e^{\gamma_2\tau}$ for every $(Z_1, Z_2) \in \mathcal{Z}$ (which is possible because $\mathcal{Z}$ is bounded), then $\|\Pi_{\omega_k}\| \le e^{(\gamma_1+\gamma_2)|\omega_k|}$ for every $k \in \mathbb{N}$ and $\omega \in \Omega_{\mathcal{Z},\tau}$, yielding that $\lambda(\Xi_{\mathcal{Z},\tau}) \le \gamma_1 + \gamma_2$. Lemma 19 below characterizes the case where $\lambda(\Xi_{\mathcal{Z},0}) = +\infty$.

**Lemma 19.** The following three properties are equivalent:

1. $\lambda(\Xi_{\mathcal{Z},0}) < +\infty$;

2. The set $\{Z_2^1 \cdots Z_2^k | k \ge 1, (Z_1^1, Z_2^1), \dots, (Z_1^k, Z_2^k) \in \mathcal{Z}\}$ is bounded;

3. There exist $C > 0$ and $\gamma \in \mathbb{R}$ such that $\|\Phi_Z(t, 0)\| \le Ce^{\gamma t}$ for every $t \ge 0$ and every $Z \in S_{\mathcal{Z},0}$.

*Proof.* Let us first prove that Property 1 implies Property 2. For that, we assume that $\{Z_2^1 \cdots Z_2^k | k \ge 1, (Z_1^1, Z_2^1), \dots, (Z_1^k, Z_2^k) \in \mathcal{Z}\}$ is unbounded and we are going to prove that $\lambda(\Xi_{\mathcal{Z},0}) = +\infty$. Let $C > 0$ be such that $\|Z_1\| \le C$ for every $(Z_1, Z_2) \in \mathcal{Z}$. Then $\|e^{tZ_1}\| \le e^{C|t|}$ for every $(Z_1, Z_2) \in \mathcal{Z}$ and $t \in \mathbb{R}$. For every $n \in \mathbb{N}$, let $(Z_1^{1,n}, Z_2^{1,n}), \dots, (Z_1^{k,n}, Z_2^{k,n}) \in \mathcal{Z}$ be such that

$$ \|Z_2^{k,n} \cdots Z_2^{1,n}\| \ge e^{n^2C}. $$
Then $Z_2^{k,n} \cdots Z_2^{1,n} e^{nZ_1^{1,n}} = \Pi_{\omega_{k_n}^n}$ for some $\omega^n \in \Omega_{\mathbb{Z},0}$ with $|\omega_{k_n}^n| = n$. Notice that

$$ 
\begin{aligned}
e^{n^2 C} &\le \|Z_2^{k,n} \cdots Z_2^{1,n}\| \\
&\le \|\Pi_{\omega_{k_n}^n}\| \|e^{-nZ_1^{1,n}}\| \\
&\le \|\Pi_{\omega_{k_n}^n}\| e^{nC}.
\end{aligned}
$$

Hence

$$ \lambda(\Xi_{\mathbb{Z},0}) \ge \limsup_{n \to +\infty} \frac{\log(\|\Pi_{\omega_{k_n}^n}\|)}{n} = +\infty. $$

Assume now that Property 2 holds true and let us prove Property 3. Define

$$ v(x) = \sup_{k \ge 0, (Z_1^n, Z_2^n)_{n \in \mathbb{N}} \in \mathcal{Z}^N} \|Z_2^k \cdots Z_2^1 x\|, $$

with the convention that $Z_2^k \cdots Z_2^1 = I_d$ if $k = 0$. Notice that $v$ is finite by Property 2 and that $v(x) \ge \|x\| > 0$ for $x \ne 0$. Moreover, $v$ is homogeneous and satisfies the triangle inequality, hence it is a norm. Denote by $\|\cdot\|_v$ the matrix norm induced by $v$. Then $\|Z_2\|_v \le 1$ for every $(Z_1, Z_2) \in \mathcal{Z}$. This implies that there exists $\gamma > 0$ such that $\|\Phi_Z(t, 0)\|_v \le e^{\gamma t}$ for every $t \ge 0$ and $Z \in S_{\mathbb{Z},0}$. Property 3 follows.

The fact that Property 3 implies Property 1 follows from inequality (27) with $\tau = 0$. $\square$

**Remark 20.** By Remark 18 and Lemma 19, $\lambda(\Sigma_{\mathbb{Z},\tau}) = +\infty$ if and only if $\tau = 0$ and $\{Z_2^1 \cdots Z_2^k | k \ge 1, (Z_1^1, Z_2^1), \dots, (Z_1^k, Z_2^k) \in \mathcal{Z}\}$ is unbounded.

We are ready to prove Theorem 3, which we restate as follows (notice that the equality between the first and third term in the statement below corresponds to the statement of Theorem 3 thanks of Remark 8).

**Theorem 21.** Assume that $\lambda(\Sigma_{\mathbb{Z},\tau}) < +\infty$. Then

$$ 
\begin{aligned}
\lambda(\Sigma_{\mathbb{Z},\tau}) &= \max \left\{ \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \lambda(\Xi_{\mathbb{Z},\tau}) \right\} \\
&= \max \left\{ \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \mu(\Xi_{\mathbb{Z},\tau}) \right\}.
\end{aligned}
$$

In the proof of the theorem, we will make use of the following technical result, providing a useful bound on the norm of an exponential matrix, which is a variation of [26, Equation (2.11)].

**Lemma 22.** If $M \in M_d(\mathbb{R})$ and $t \ge 0$ then

$$ \|e^{tM}\| \le e^{t\alpha(M)} \sum_{k=0}^{d-1} \frac{t^k d^k \|M\|^k}{k!}. $$

*Proof.* By Schur triangularization theorem [16, Theorem 2.3.1], there exists a unitary matrix $U$ such that we can write

$$ M = U^*TU $$
for some upper triangular matrix $T \in M_d(\mathbb{C})$. We can write $T = D + N$ where $D$ is the diagonal part of $T$ and $N$ is strictly upper triangular. As in [26, Equation (2.11)] we have

$$ \|e^{tM}\| \le e^{t\alpha(M)} \sum_{k=0}^{d-1} \frac{t^k \|N\|^k}{k!}. \qquad (28) $$

Moreover, considering the matrix norm $\|A\|_{\infty} = \max_{i,j} |A_{ij}|$, we have

$$ \|N\| \le d\|N\|_{\infty} \le d\|D + N\|_{\infty} \le d\|D + N\| = d\|M\|, \qquad (29) $$

where the first inequality is obtained as a simple application of Cauchy-Schwarz inequality, and the last equality follows from the fact that the transformation $U$ is unitary. The lemma follows by combining (28) with (29). $\square$

*Proof of Theorem 21.* We first notice that, for every $(Z_1, Z_2) \in \mathcal{Z}$, the flow corresponding to the constant signal $Z(\cdot) \equiv (Z_1, Z_2)$, without switchings, satisfies $\Phi_Z(t, 0) = e^{tZ_1}$ for every $t \ge 0$. Hence $\alpha(Z_1) \le \lambda(\Sigma_{\mathcal{Z},\tau})$. Thus

$$ \sup_{(Z_1, Z_2) \in \mathcal{Z}} \alpha(Z_1) \le \lambda(\Sigma_{\mathcal{Z},\tau}). \qquad (30) $$

Moreover, since $\rho(M) \le \|M\|$ for every $M \in M_d(\mathbb{R})$, and by (27), it follows that

$$ \mu(\Xi_{\mathcal{Z},\tau}) \le \lambda(\Xi_{\mathcal{Z},\tau}) \le \lambda(\Sigma_{\mathcal{Z},\tau}). \qquad (31) $$

Furthermore, by definition, there exist sequences of elements $Z_1^n \in M_d(\mathbb{R})$, $\tau_n > 0$, $\omega^n \in \Omega$, and $k_n \in \mathbb{N}$ with $\lim_{n \to \infty} (\tau_n + |\omega_{k_n}^n|) = +\infty$ such that

$$ \lambda(\Sigma_{\mathcal{Z},\tau}) = \lim_{n \to \infty} \frac{\log(\|e^{\tau_n Z_1^n} \Pi_{\omega_{k_n}^n}\|)}{\tau_n + |\omega_{k_n}^n|} \\ = \lim_{n \to \infty} \frac{\log(\|e^{\tau_n Z_1^n}\|) + \log(\|\Pi_{\omega_{k_n}^n}\|)}{\tau_n + |\omega_{k_n}^n|}. $$

If the sequence $\tau_n$ is bounded then the previous limit is equal to $\limsup_{n \to \infty} \frac{\log(\|\Pi_{\omega_{k_n}^n}\|)}{|\omega_{k_n}^n|}$, hence it is bounded by $\lambda(\Xi_{\mathcal{Z},\tau})$, while if the sequence $|\omega_{k_n}^n|$ is bounded then the limit is equal to $\limsup_{n \to \infty} \alpha(Z_1^n)$ by Lemma 22, and it is bounded by $\sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1)$. If both $\tau_n$ and $|\omega_{k_n}^n|$ tend to infinity (up to a subsequence) then, using the fact that any ratio $\frac{a_1+a_2}{b_1+b_2}$ with $b_1, b_2$ positive is smaller than $\max\{\frac{a_1}{b_1}, \frac{a_2}{b_2}\}$, we still get

$$ \lambda(\Sigma_{\mathcal{Z},\tau}) \le \limsup_{n \to \infty} \max \left\{ \frac{\log(\|e^{\tau_n Z_1^n}\|)}{\tau_n}, \frac{\log(\|\Pi_{\omega_{k_n}^n}\|)}{|\omega_{k_n}^n|} \right\} \\ = \max \left\{ \limsup_{n \to \infty} \frac{\log(\|e^{\tau_n Z_1^n}\|)}{\tau_n}, \limsup_{n \to \infty} \frac{\log(\|\Pi_{\omega_{k_n}^n}\|)}{|\omega_{k_n}^n|} \right\} \\ \le \max \left\{ \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \lambda(\Xi_{\mathcal{Z},\tau}) \right\}. $$
Combining with (30) and (31) we get

$$ \lambda(\Sigma_{\mathcal{Z},\tau}) = \max \left\{ \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \lambda(\Xi_{\mathcal{Z},\tau}) \right\}, $$

and we are left to prove that $\mu(\Xi_{\mathcal{Z},\tau})$ is equal to $\lambda(\Xi_{\mathcal{Z},\tau}) = \lambda(\Sigma_{\mathcal{Z},\tau})$ whenever

$$ \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1) < \lambda(\Xi_{\mathcal{Z},\tau}). \qquad (32) $$

For this purpose, consider a flag of subspaces $\{0\} \subsetneq E_1 \subsetneq \dots \subsetneq E_r = \mathbb{R}^d$ such that each $E_j$ is invariant for $\{Z_2e^{tZ_1} | t \ge \tau, (Z_1, Z_2) \in \mathcal{Z}\}$ and $r$ is maximal among all flags with the same property. The flag induces a block triangularization

$$ PAP^{-1} = \begin{pmatrix} A_{11} & A_{12} & \dots \\ 0 & A_{22} & A_{23} & \dots \\ 0 & 0 & A_{33} & A_{34} & \dots \\ \vdots & \ddots & \ddots & \ddots & \\ 0 & \dots & \dots & 0 & A_{rr} \end{pmatrix}, $$

with $P$ invertible and independent of $A \in \{Z_2e^{tZ_1} | t \ge \tau, (Z_1, Z_2) \in \mathcal{Z}\}$. Up to a linear change of coordinates, we can assume that $P = I_d$.

Let $\mathcal{N}_{i,\tau} := \{(Z_2e^{tZ_1})_{ii} | t \ge \tau, (Z_1, Z_2) \in \mathcal{Z}\}$ and consider the corresponding weighted discrete-time switched system $\Xi_i := \Xi_{\mathcal{N}_{i,\tau}}$. Notice that, by maximality of $r$, $\mathcal{N}_{i,\tau}$ is irreducible.

For $\omega \in \Omega_{\mathcal{Z},\tau}$ and $k \in \mathbb{N}$, the spectrum of $\Pi_{\omega_k}$ is given by the union of the spectra of $(\Pi_{\omega_k})_{11}, \dots, (\Pi_{\omega_k})_{rr}$. Hence, $\mu(\Xi_{\mathcal{Z},\tau}) = \max_{i=1,\dots,r} \mu(\Xi_i)$.

Let us conclude the argument by assuming, for now, that

$$ \lambda(\Xi_{\mathcal{Z},\tau}) = \max_{i=1,\dots,r} \lambda(\Xi_i). \qquad (33) $$

By Corollary 17, for $i = 1, \dots, r$ one has $\lambda(\Xi_i) = \max\{\hat{\lambda}(\Xi_i), \mu(\Xi_i)\}$. Notice that, for every $(Z_1, Z_2) \in \mathcal{Z}$ and for $i = 1, \dots, r$,

$$ \|(Z_2e^{tZ_1})_{ii}\| \le \|Z_2e^{tZ_1}\| \le \|Z_2\| \|e^{tZ_1}\|. $$

Hence, $\hat{\lambda}(\Xi_i) \le \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1)$ for every $i \in \{1, \dots, r\}$.

Since $\lambda(\Sigma_{\mathcal{Z},\tau}) = \lambda(\Xi_{\mathcal{Z},\tau})$, picking $i$ such that $\lambda(\Xi_{\mathcal{Z},\tau}) = \lambda(\Xi_i)$, we have that either $\lambda(\Sigma_{\mathcal{Z},\tau}) = \mu(\Xi_i) \le \mu(\Xi_{\mathcal{Z},\tau})$ or $\lambda(\Sigma_{\mathcal{Z},\tau}) = \hat{\lambda}(\Xi_i) \le \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1)$, proving the desired inequality.

We are left to prove that, under assumption (32), equality (33) holds true. Notice that $\lambda(\Xi_i) \le \lambda(\Xi_{\mathcal{Z},\tau})$ for $i = 1, \dots, r$. The equality is proved by induction on $r$. The case $r = 1$ is trivial.

Assume that the equality holds true for some positive integer $r$ and consider $\Xi_{\mathcal{Z},\tau}$ with maximal flag $(E_1, \dots, E_{r+1})$ of length $r+1$. For $\omega \in \Omega_{\mathcal{Z},\tau}$ and $k \in \mathbb{N}$, write

$$ \Pi_{\omega_k} = \begin{pmatrix} (\Pi_{\omega_k})_{11} & (\Pi_{\omega_k})_{1R} \\ 0 & (\Pi_{\omega_k})_{RR} \end{pmatrix}, $$

where $R$ stands for the $r$uple of indices $(2, \dots, r+1)$. Applying the induction hypothesis, one deduces that for every $\varepsilon > 0$ there exists $C_1(\varepsilon) > 0$ independent of $\omega$ and $k$ such that $\|(\Pi_{\omega_k})_{RR}\| \le$
$C_1(\varepsilon)e^{(\nu+\varepsilon)|\omega_k|}$, where $\nu = \max_{i=2,...,r+1} \lambda(\Xi_i)$. On the other hand, for every $\varepsilon > 0$ there exists $C_2(\varepsilon)$ such that

$$ \|(\Pi_{\omega_{j \to k}})_{11}\| \le C_2(\varepsilon)e^{(\lambda(\Xi_1)+\varepsilon)(|\omega_k|-|\omega_j|)}. $$

Notice, moreover, that

$$ (\Pi_{\omega_k})_{1R} = \sum_{j=1}^{k} (\Pi_{\omega_{j \to k}})_{11} (Z_2^j e^{\tau_j Z_1^j})_{1R} (\Pi_{\omega_{j-1}})_{RR}, $$

and that there exists $C_3(\varepsilon) > 0$ such that $\|(Z_2^j e^{\tau_j Z_1^j})_{1R}\| \le C_3 e^{\tau_j (\varepsilon+\sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1))}$. One deduces that $\lambda(\Xi_{\mathcal{Z},\tau}) \le \max\{\lambda(\Xi_1), \nu, \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1)\} + \varepsilon$. Since $\varepsilon$ is arbitrary and we are assuming (32), this concludes the inductive step. $\square$

**Remark 23.** Theorem 21 may fail to hold when $\lambda(\Sigma_{\mathcal{Z},\tau}) = +\infty$. Consider for example $d = 2$, $\tau = 0$ and $\mathcal{Z}$ made by the single element $(Z_1, Z_2)$ with $Z_1 = (\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix})$ and $Z_2 = (\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix})$. Then it is easy to check that $\lambda(\Sigma_{\mathcal{Z},\tau}) = +\infty$, while $\alpha(Z_1) = 0$ and $\mu(\Xi_{\mathcal{Z},0}) = 0$.

As a corollary, we obtain Theorem 4, which we restate for convenience.

**Corollary 24.** Let $\lambda(\Sigma_{\mathcal{Z},\tau}) < +\infty$. We have the following properties.

1. System $\Sigma_{\mathcal{Z},\tau}$ is ES if and only if $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$;

2. System $\Sigma_{\mathcal{Z},\tau}$ is EU if and only if $\lambda(\Sigma_{\mathcal{Z},\tau}) > 0$.

*Proof.* We start by proving Item 1. On the one hand, if $\Sigma_{\mathcal{Z},\tau}$ is ES then clearly $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$. On the other hand, $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$ implies that, for every $\gamma \in (\lambda(\Sigma_{\mathcal{Z},\tau}), 0)$, there exists $T > 0$ such that, for every $t > T$ and every $Z \in S_{\mathcal{Z},\tau}$, $\|\Phi_Z(t, 0)\| \le e^{\gamma t}$. We are left to show that $\{\Phi_Z(t, 0) | Z \in S_{\mathcal{Z},\tau}, t \in [0, T]\}$ is bounded. In the case $\tau > 0$ this is straightforward (see Remark 18). If $\tau = 0$, since $\lambda(\Xi_{\mathcal{Z},0}) \le \lambda(\Sigma_{\mathcal{Z},0}) < +\infty$, we deduce the boundedness from Property 3 of Lemma 19.

Concerning Item 2, if $\Sigma_{\mathcal{Z},\tau}$ is EU then clearly $\lambda(\Sigma_{\mathcal{Z},\tau}) > 0$. On the other hand, if $\lambda(\Sigma_{\mathcal{Z},\tau}) > 0$ then, from Theorem 21, there exist either $(Z_1, Z_2) \in \mathcal{Z}$ such that $\alpha(Z_1) > 0$ or $\omega \in \Omega_{\mathcal{Z},\tau}$ and $k \in \mathbb{N}$ such that $\rho(\Pi_{\omega_k}) > 1$. In the first case, $\Sigma_{\mathcal{Z},\tau}$ is obviously EU by taking the signal constantly equal to $(Z_1, Z_2)$. In the second case, we consider the sequence $\omega^* \in \Omega_{\mathcal{Z},\tau}$ obtained by repeating $\omega_k$ infinitely many times. Let $Z^* \in S_{\mathcal{Z},\tau}$ be the signal associated with $\omega^*$ and $T = |\omega_k|$. There exist $c > 1$ and $x_0 \in \mathbb{R}^d \setminus \{0\}$ such that $|\Phi_{Z^*}(nT, 0)x_0| \ge c^n|x_0|$. Let $C$ and $\gamma$ be as in Property 3 of Lemma 19. Then for $t \in [nT, (n+1)T)$ one has

$$ \begin{aligned} & |\Phi_{Z^*}(t, 0)x_0| \ge \\ & |\Phi_{Z^*}((n+1)T, 0)x_0| \|\Phi_{Z^*}((n+1)T, s)^{-1}\| \ge \\ & c^{n+1}|x_0| \frac{e^{-|\gamma|T}}{C}. \end{aligned} $$

This concludes the proof that $\Sigma_{\mathcal{Z},\tau}$ is EU. $\square$
Proof of Theorem 2. Let us first assume that $\Sigma_{\mathcal{Z},\tau}$ is ES. Then $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$ and $\Xi_{\mathcal{Z},\tau}$ is ES a well. These two properties imply, respectively, that $\sup_{(Z_1,Z_2)\in\mathcal{Z}} \alpha(Z_1) < 0$ by Theorem 21 and that there exist $V$ as in the statement by Theorem 7.

Assume now that $\sup_{(Z_1,Z_2)\in\mathcal{Z}} \alpha(Z_1) < 0$ and that $V$ as in the statement of the theorem exists. Then $\lambda(\Xi_{\mathcal{Z},\tau}) < 0$ by Theorem 7. The conclusion follows using Theorem 21. $\square$

**Remark 25.** Notice that a Lyapunov function as in Theorem 2 exists also under the sole assumption that $\lambda(\Xi_{\mathcal{Z},\tau}) < 0$. Indeed, if the latter inequality holds, by definition of $\lambda(\Xi_{\mathcal{Z},\tau})$ and using Lemma 19, we deduce that $\Xi_{\mathcal{Z},\tau}$ is ES. The conclusion then follows by Theorem 7.

It follows from Theorem 21 that when $\lambda(\Xi_{\mathcal{Z},\tau}) < \lambda(\Sigma_{\mathcal{Z},\tau}) < +\infty$ there must exist $(Z_1, Z_2) \in \mathcal{Z}$ such that $\lambda(\Xi_{\mathcal{Z},\tau}) < \alpha(Z_1)$. The following proposition investigates such a situation.

**Proposition 26.** Assume that $\lambda(\Xi_{\mathcal{Z},\tau}) < \alpha(Z_1)$ for some $(Z_1, Z_2) \in \mathcal{Z}$. Then $Z_2x = 0$ for every generalized eigenvector $x$ of $Z_1$ associated with an eigenvalue of real part $\alpha(Z_1)$.

*Proof.* For simplicity of notation, we write the proof when $x$ is a generalized eigenvector of $Z_1$ associated with a real eigenvalue, the general case being similar. Recall that $x$ is a generalized eigenvector of $Z_1$ associated with the eigenvalue $\alpha(Z_1)$ if there exist $k \ge 1$ linearly independent vectors $x_1, \dots, x_k$ so that $x = x_k$ and $Z_1x_j = \alpha(Z_1)x_j + \sum_{i=1}^{j-1} x_i$ for $1 \le j \le k$. One says that $x_1, \dots, x_k$ is a Jordan chain of length $k$ associated with $x$.

We will prove the conclusion by induction on $k$. For $k = 1$, $x$ is simply an eigenvector of $Z_1$ associated with the eigenvalue $\alpha(Z_1)$ and we can assume without loss of generality that $|x| = 1$. Suppose by contradiction that $Z_2x \ne 0$. Using the fact that $e^{tZ_1}x = e^{\alpha(Z_1)t}x$ and based on the definition of $\lambda(\Xi_{\mathcal{Z},\tau})$, one has that

$$ \alpha(Z_1) > \lambda(\Xi_{\mathcal{Z},\tau}) \ge \limsup_{t \to +\infty} \frac{\log |Z_2 e^{tZ_1}x|}{t} \\ = \alpha(Z_1) + \limsup_{t \to +\infty} \frac{\log |Z_2x|}{t} = \alpha(Z_1), $$

yielding a contradiction. Therefore, we must have $Z_2x = 0$.

Assume now that the conclusion holds true for every $j$ with $1 \le j \le k-1$. Consider a generalized eigenvector $x$ with Jordan chain $x_1, \dots, x_k$ of length $k$. Notice that for every $j$ with $1 \le j \le k-1$, $x_j$ is a generalized eigenvector with Jordan chain $x_1, \dots, x_j$ of length $j \le k-1$. Applying the induction hypothesis one gets that $Z_2x_j = 0$. Moreover, for $t \ge 0$,

$$ e^{tZ_1}x = e^{\alpha(Z_1)t}\left(x + \sum_{j=1}^{k-1} \frac{t^{j-1}}{(j-1)!}x_j\right). $$

Then $Z_2e^{tZ_1}x = e^{\alpha(Z_1)t}Z_2x$. If $Z_2x \ne 0$, we argue as before and reach a contradiction. This ends the induction argument and concludes the proof of the proposition. $\square$

A consequence of Theorem 21 is the continuity of the maximal Lyapunov exponent, as detailed in the following proposition.

**Proposition 27.** Let $\tau > 0$ and $U$ be the set of bounded subsets $\mathcal{Z}$ of $M_d(\mathbb{R}) \times M_d(\mathbb{R})$. Endow $U$ with the topology induced by the Hausdorff distance. Then the function $\mathcal{Z} \mapsto \lambda(\Sigma_{\mathcal{Z},\tau})$ is continuous on $U$.
Proof. We begin by noting that the map $\mathcal{Z} \mapsto \sup_{Z \in \mathcal{Z}} \alpha(Z_1)$ is continuous on $\mathcal{U}$. This follows directly from the uniform continuity of $\alpha(\cdot)$ on bounded subsets of $M_d(\mathbb{R})$.

We now claim that, if $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$ for some $\mathcal{Z} \in \mathcal{U}$, then for $\varepsilon > 0$ small enough and every $\mathcal{Z}' \in \mathcal{U}$ with $d_H(\mathcal{Z}, \mathcal{Z}') < \varepsilon$ we have $\lambda(\Sigma_{\mathcal{Z}',\tau}) < 0$. Indeed, if $\lambda(\Sigma_{\mathcal{Z},\tau}) < 0$ then, from Theorem 3 together with Theorem 4, we have $\sup_{Z \in \mathcal{Z}} \alpha(Z_1) < 0$ and there exist $\gamma > 0$ and a Lyapunov function $V: \mathbb{R}^d \to \mathbb{R}_+$ satisfying

$$V(Z_2 e^{tZ_1} x) \le e^{-\gamma t} V(x), \quad \forall x \in \mathbb{R}^d, \qquad (34)$$

for every $(Z_1, Z_2) \in \mathcal{Z}$ and $t \in \mathbb{R}_{\ge \tau}$. Let $\mathcal{Z}'$ be sufficiently close to $\mathcal{Z}$ such that $\sup_{Z' \in \mathcal{Z}'} \alpha(Z'_1) < \frac{1}{2} \sup_{Z \in \mathcal{Z}} \alpha(Z_1) < 0$. Then, by Lemma 22, there exist $C > 0$ and $\tilde{\gamma} \in (0, \gamma)$ such that for every $t \ge 0$, $Z \in \mathcal{Z}$, and $Z' \in \mathcal{Z}'$, one has

$$\|e^{tZ_1}\| \le C e^{-\tilde{\gamma}t} \quad \text{and} \quad \|e^{tZ'_1}\| \le C e^{-\tilde{\gamma}t}. \qquad (35)$$

Let us now show that, for every $Z' \in \mathcal{Z}'$ and every $t \ge \tau$, there exists $Z \in \mathcal{Z}$ such that

$$e^{-\gamma t} + L \|Z'_2 e^{tZ'_1} - Z_2 e^{tZ_1}\| \le e^{-\frac{\tilde{\gamma}}{2}t}, \qquad (36)$$

where $L > 0$ is such that $V$ is $L$-Lipschitz continuous. To see that, notice that for every $Z \in \mathcal{Z}$, $Z' \in \mathcal{Z}'$, and $t \ge \tau$, we have

$$\begin{aligned} & e^{-(\gamma - \frac{\tilde{\gamma}}{2})t} + e^{\frac{\tilde{\gamma}}{2}t} L \|Z'_2 e^{tZ'_1} - Z_2 e^{tZ_1}\| \le \\ & e^{-\frac{\tilde{\gamma}}{2}\tau} + e^{\frac{\tilde{\gamma}}{2}t} \|Z'_2(e^{tZ'_1} - e^{tZ_1})\| + \|Z'_2 - Z_2\| \|e^{\frac{\tilde{\gamma}}{2}t} e^{tZ_1}\|. \end{aligned} \qquad (37)$$

Using (35), note that, for every $t \ge 0$, one has that

$$e^{\frac{\tilde{\gamma}}{2}t} \|Z'_2(e^{tZ'_1} - e^{tZ_1})\| \le C_1 e^{-\frac{\tilde{\gamma}}{2}t} \quad \text{and} \quad \|e^{\frac{\tilde{\gamma}}{2}t} e^{tZ_1}\| \le C.$$

Pick $\kappa \in (0, \frac{1 - e^{-\frac{\tilde{\gamma}}{2}\tau}}{2})$. Fix $T > 0$ so that $C_1 e^{-\frac{\tilde{\gamma}}{2}t} \le \kappa$ for all $t \ge T$ and choose $Z$ close enough to $Z'$ so that

$$\sup_{t \in [0,T]} \|Z'_2(e^{tZ'_1} - e^{tZ_1})\| \le \kappa.$$

Hence we deduce that for every $t \ge \tau$, $e^{\frac{\tilde{\gamma}}{2}t} \|Z'_2(e^{tZ'_1} - e^{tZ_1})\| \le \kappa$. Similarly, choose again $Z$ close enough to $Z'$ so that $C \|Z'_2 - Z_2\| \le \kappa$. Collecting all the above estimates, one gets that the left-hand side of (37) is upper bounded by $e^{-\frac{\tilde{\gamma}}{2}\tau} + 2\kappa$ for $t \ge \tau$, which implies (36). Using the $L$-Lipschitz continuity of $V$, it follows from (34) that, for every $x \in \mathbb{R}^d$, $(Z'_1, Z'_2) \in \mathcal{Z}'$, and $t \ge \tau$, one has

$$\begin{aligned} V(Z'_2 e^{tZ'_1} x) &\le V(Z_2 e^{tZ_1} x) + L \|Z'_2 e^{tZ'_1} x - Z_2 e^{tZ_1} x\| \\ &\le e^{-\gamma t} V(x) + L \|Z'_2 e^{tZ'_1} x - Z_2 e^{tZ_1} x\| V(x) \\ &\le e^{-\frac{\tilde{\gamma}}{2}t} V(x), \end{aligned}$$

where we have used the fact that $\|x\| \le V(x)$. The direct version of Theorem 3 implies that $\Xi_{\mathcal{Z}',\tau}$ is ES and by consequence Theorem 4 implies that $\lambda(\Sigma_{\mathcal{Z}',\tau}) < 0$. This concludes the proof of the claim.
The above claim actually proves the lower semi-continuity of $\mathcal{Z} \mapsto \lambda(\Sigma_{\mathcal{Z},\tau})$. Indeed, let $\delta > 0$ and fix $\mathcal{Z} \in \mathcal{U}$. Define $\xi = -\lambda(\Sigma_{\mathcal{Z},\tau}) - \delta$. Then, by Lemma 9, we have $\lambda(\Sigma_{\mathcal{Z}^\xi,\tau}) = -\delta < 0$. The claim guarantees that there exists a neighbourhood $\mathcal{W}$ of $\mathcal{Z}$ in $\mathcal{U}$ such that for all $\mathcal{Z}' \in \mathcal{W}$ we have $\lambda(\Sigma_{\mathcal{Z}'^\xi,\tau}) < 0$, which means that $\lambda(\Sigma_{\mathcal{Z}',\tau}) < \lambda(\Sigma_{\mathcal{Z},\tau}) + \delta$.

For the upper semi-continuity of $\mathcal{Z} \mapsto \lambda(\Sigma_{\mathcal{Z},\tau})$, consider $\delta > 0$ and $\mathcal{Z} \in \mathcal{U}$. According to Theorem 3, we have

$$ \lambda(\Sigma_{\mathcal{Z},\tau}) = \max \left( \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1), \mu(\Sigma_{\mathcal{Z},\tau}) \right). $$

If $\lambda(\Sigma_{\mathcal{Z},\tau}) = \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1)$, then, for every $\mathcal{Z}'$ close enough to $\mathcal{Z}$ in $\mathcal{U}$ we have

$$ \begin{aligned} \lambda(\Sigma_{\mathcal{Z}',\tau}) &\ge \sup_{(Z'_1,Z'_2) \in \mathcal{Z}'} \alpha(Z'_1) \ge \sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1) - \delta \\ &= \lambda(\Sigma_{\mathcal{Z},\tau}) - \delta. \end{aligned} $$

Otherwise, there exists $Z \in \mathcal{S}_{\mathcal{Z},\tau}$ and $k \in \Theta^*(Z)$ such that

$$ \frac{\log(\rho(\Phi_Z(t_k, 0)))}{t_k} > \lambda(\Sigma_{\mathcal{Z},\tau}) - \frac{\delta}{2}. $$

Let $\mathcal{W}$ be a neighbourhood of $\mathcal{Z}$ in $\mathcal{U}$ such that for every $\mathcal{Z}' \in \mathcal{W}$ there exists $Z' \in \mathcal{S}_{\mathcal{Z}',\tau}$ such that $\Theta^*(Z') = \Theta^*(Z)$ and

$$ \frac{\log(\rho(\Phi_{Z'}(t_k, 0)))}{t_k} \ge \frac{\log(\rho(\Phi_Z(t_k, 0)))}{t_k} - \frac{\delta}{2}. $$

It follows that

$$ \begin{aligned} \lambda(\Sigma_{\mathcal{Z}',\tau}) &\ge \sup_{\tilde{Z} \in \mathcal{S}_{\mathcal{Z}',\tau}, \tilde{k} \in \Theta^*(\tilde{Z})} \frac{\log(\rho(\Phi_{\tilde{Z}}(t_{\tilde{k}}, 0)))}{t_{\tilde{k}}} \\ &\ge \frac{\log(\rho(\Phi_{Z'}(t_k, 0)))}{t_k} \ge \frac{\log(\rho(\Phi_Z(t_k, 0)))}{t_k} - \frac{\delta}{2}. \end{aligned} $$

It follows that $\lambda(\Sigma_{\mathcal{Z}',\tau}) \ge \lambda(\Sigma_{\mathcal{Z},\tau}) - \delta$. In both cases, we conclude that there exists a neighbourhood $\mathcal{W}$ of $\mathcal{Z}$ in $\mathcal{U}$ such that $\lambda(\Sigma_{\mathcal{Z}',\tau}) > \lambda(\Sigma_{\mathcal{Z},\tau}) - \delta$ for every $\mathcal{Z}' \in \mathcal{W}$. $\square$

In the case where $\tau = 0$ we can extend Proposition 27 as follows.

**Remark 28.** Let $\tau = 0$ and $\lambda(\Sigma_{\mathcal{Z},0}) < +\infty$. Set $\mathcal{Z}_2 = \{Z_2 | (Z_1, Z_2) \in \mathcal{Z}\}$ and let $\mathcal{U}$ be the set of bounded subsets $\mathcal{Z}'$ of $M_d(\mathbb{R}) \times M_d(\mathbb{R})$ such that

$$ \{Z_2' | (Z_1', Z_2') \in \mathcal{Z}'\} = \mathcal{Z}_2. $$

Notice that, by Lemma 19, $\lambda(\Sigma_{\mathcal{Z}',0}) < +\infty$ for every $\mathcal{Z}' \in \mathcal{U}$. Then showing the continuity of $\mathcal{U} \ni \mathcal{Z}' \mapsto \lambda(\mathcal{Z}')$ follows the same lines as that of Proposition 27, where the argument to get (36) in the case $\tau = 0$ now requires the extra fact that for every $T > 0$ and every bounded subset $\mathcal{B}$ of $M_d(\mathbb{R})$ there exists $c > 0$ such that $\|e^{tZ_1} - e^{tZ_1'}\| \le ct\|Z_1 - Z_1'\|$ for every $Z_1, Z_1' \in \mathcal{B}$ and $t \in [0, T]$.
# 5 Conclusion

This paper addresses the stability analysis of impulsive linear switched systems by examining their equivalent representation as weighted discrete-time switched systems. We provide two contributions: a converse Lyapunov theorem that characterizes exponential stability via the existence of a Lyapunov function, and a Berger-Wang-type result that establishes the equality between two measures of asymptotic growth, one based on the operator norm and the other on the spectral radius. These results are first developed in the general context of weighted discrete-time switched systems and subsequently applied to impulsive linear switched systems. In particular, using the Berger-Wang formula, we establish a characterization of exponential stability for impulsive linear switched systems in terms of the sign of their maximal Lyapunov exponent.

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