# Stability criteria for hybrid linear systems with singular perturbations

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

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Ihab Haidar, Yacine Chitour, Jamal Daafouz, Paolo Mason, Mario Sigalotti. Stability criteria for hybrid linear systems with singular perturbations. 2025. hal-05137473

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# Stability criteria for hybrid linear systems with singular perturbations

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

June 30, 2025

## Abstract

We study a class of singularly perturbed impulsive linear switched systems exhibiting switching between slow and fast dynamics. To analyze their behavior, we construct auxiliary switched systems evolving in a single time scale. The stability or instability of these auxiliary systems directly determines that of the original system in the regime of small singular perturbation parameters.

## 1 Introduction

Consider the linear system evolving in $\mathbb{R}^d$

$$ \Sigma^{\epsilon} : \left\{ \begin{array}{ll} D_k^{\epsilon} \dot{X}(t) = \Lambda_k X(t), & t \in [t_k, t_{k+1}), \\ X(t_{k+1}) = R_k \lim_{t \searrow t_{k+1}} X(t), & k \ge 0, \end{array} \right. $$

where $\Lambda_k, R_k$ take values in a compact subset of $d \times d$ real matrices and $D_k^{\epsilon}$ is a diagonal matrix with diagonal entries in $\{1, \epsilon\}$, $\epsilon$ being a small positive

<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>†Yacine Chitour is with Laboratoire des Signaux et Systèmes (L2S), Université Paris-Saclay, CNRS, CentraleSupélec, Gif-sur-Yvette, France, yacine.chitour@l2s.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>§Paolo Mason is with Laboratoire des Signaux et Systèmes (L2S), Université Paris-Saclay, CNRS, CentraleSupélec, Gif-sur-Yvette, France, paolo.mason@l2s.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>
parameter. In this paper we deal with the problem of understanding the asymptotic behavior of this type of systems as $t$ goes to infinity in the regime where $\varepsilon$ is arbitrarily small.

System $\Sigma^\varepsilon$ represents a class of impulsive linear switched systems characterized by two time-scale dynamics: *fast variables*, whose velocities are modulated by $\frac{1}{\varepsilon}$, and the other ones called *slow variables*. The system is characterized by the dynamic interchange between slow and fast variables over time, governed by the switching signal $k \mapsto D_k^\varepsilon$. While singularly perturbed hybrid systems with fixed slow-fast variables have been extensively studied in the literature (see, e.g., [1, 13, 19, 18, 20, 15, 17, 16]), systems exhibiting switching slow/fast behaviors remain largely overlooked. Motivated by an industrial application in steel production (see [10]), stability properties of the system $\Sigma^\varepsilon$ were first investigated in [11] in terms of LMI characterizations. The approach that we adopt here has been first explored in [5], where some preliminary results to the present work were exhibited. Analyzing the stability of $\Sigma^\varepsilon$ is challenging as existing frameworks for singularly perturbed impulsive switched systems do not necessarily cover this class of systems. This work aims to address this gap by providing a comprehensive stability analysis for systems with switching slow/fast dynamics. It is important to emphasize that even in cases where the mappings $k \mapsto D_k^\varepsilon$ and $k \mapsto R_k$ are constant, with $R_k$ equal to the identity matrix, the classical singular perturbation theory [7] cannot be applied in its standard form. In particular, the stability of the full system cannot be deduced directly from the stability of its individual components. To address this challenge, various stability criteria have been proposed in the literature (see, e.g., [3, 4, 9, 12]). For example, in [3], upper and lower bounds were derived for the maximal Lyapunov exponent of singularly perturbed linear switched systems as $\varepsilon$ tends to zero. In [12], stability was established under a dwell-time condition, which, importantly, does not explicitly depend on the time-scale parameter. Additionally, a recent study in [16] explores the stabilization of switched affine singularly perturbed systems with state-dependent switching laws.

The purpose of this paper is twofold: first, to provide necessary or sufficient conditions ensuring a specific time-asymptotic behavior for $\Sigma^\varepsilon$ in the regime where $\varepsilon \sim 0$, and second, to establish upper and lower bounds for the limit of the maximal Lyapunov exponent of $\Sigma^\varepsilon$ as $\varepsilon$ tends to 0. Recall that the maximal Lyapunov exponent of a linear switched system represents the largest asymptotic exponential rate, as time tends to infinity, among all trajectories of the system. Stability conditions then emerge as special cases: specifically, a positive lower bound guarantees instability for all sufficiently small $\varepsilon$, while a negative upper bound ensures exponential stability for all
$\epsilon$ in a right-neighborhood of zero. This is provided after identifying some auxiliary discrete- and continuous-time single scale dynamics.

To carry out our analysis, we first rewrite system $\Sigma^\epsilon$ in a new coordinate system that preserves the slow and fast nature of the variables over time. This is achieved through a mode-dependent variable reordering transformation, leading to a time-varying dimension for the slow and fast variables. Starting from this new representation, we follow the classical Tikhonov approach to introduce auxiliary impulsive switched systems. In particular we introduce two continuous-time impulsive switched systems $\bar{\Sigma}$ and $\tilde{\Sigma}$ with reduced dimensions approximating the slow dynamics of $\Sigma^\epsilon$. System $\bar{\Sigma}$ is obtained by neglecting the transient behavior during mode transitions while system $\tilde{\Sigma}$ is obtained by including the transient dynamics into the jump part of $\bar{\Sigma}$. Based on these two auxiliary systems and under suitable assumptions, we give bounds on the limit as $\epsilon$ tends to 0 of the maximal Lyapunov exponent of $\Sigma^\epsilon$ as the following

$$ \lambda(\bar{\Sigma}) \le \liminf_{\epsilon \searrow 0} \lambda(\Sigma^\epsilon) \le \limsup_{\epsilon \searrow 0} \lambda(\Sigma^\epsilon) \le \lambda(\tilde{\Sigma}). \quad (1) $$

Observe that the left-hand side inequality in (1) yields a necessary condition for the stability of $\Sigma^\epsilon$, in the sense that if $\bar{\Sigma}$ is exponentially unstable then there exists $\epsilon_0 > 0$ such that for every $\epsilon \in (0, \epsilon_0)$ system $\Sigma^\epsilon$ is exponentially unstable as well. On the other hand, the right-hand side inequality in (1) yields a sufficient condition for the stability of $\Sigma^\epsilon$, in the sense that if $\tilde{\Sigma}$ is exponentially stable then there exists $\epsilon_0 > 0$ such that for every $\epsilon \in (0, \epsilon_0)$ system $\Sigma^\epsilon$ is exponentially stable as well. Under a dwell-time constraint, given that switching occurs slowly with respect to the time-scale $1/\epsilon$, the transient phase is too short to affect the dynamics of slow variables. Hence, in this case, systems $\bar{\Sigma}$ and $\tilde{\Sigma}$ have the same asymptotic behavior, leading to a complete characterisation of the limit as $\epsilon$ tends to 0 of the maximal Lyapunov exponent of $\Sigma^\epsilon$. Another auxiliary single-scale dynamics denoted by $\hat{\Sigma}$ representing the transient behavior of $\Sigma^\epsilon$ is also introduced. Based on $\hat{\Sigma}$, the limit as $\epsilon$ tends to 0 of the maximal Lyapunov exponent of $\Sigma^\epsilon$ satisfies the inequality

$$ \lambda(\hat{\Sigma}) \le \max\{0, \liminf_{\epsilon \searrow 0} \epsilon \lambda(\Sigma^\epsilon)\}, \quad (2) $$

giving a necessary condition for the stability of $\Sigma^\epsilon$ in terms of $\hat{\Sigma}$. In fact, from (2), it follows that the exponential instability of $\hat{\Sigma}$ implies the exponential instability of $\Sigma^\epsilon$ for every $\epsilon > 0$ sufficiently small. Moreover, $\lambda(\Sigma^\epsilon)$ is at least at order $1/\epsilon$ as $\epsilon$ tends to 0.
The paper is organised as follows. In Section 2, we reformulate system $\Sigma^\epsilon$ within a suitable mathematical class and introduce the notion of stability for impulsive linear switched systems. We also state a stability theorem from [2] concerning the stability of impulsive linear switched systems, which serves as a central tool for the subsequent analysis. Section 3 introduces the auxiliary switched systems $\bar{\Sigma}_\tau$, $\hat{\Sigma}$, and $\tilde{\Sigma}$, and presents the main contributions through two theorems. The proofs of these theorems are detailed in Sections 5 and 6. They rely on a series of auxiliary results, provided in Section 4, enabling the reformulation of systems $\bar{\Sigma}_\tau$, $\hat{\Sigma}$, and $\tilde{\Sigma}$ within the impulsive switched system framework. Additional technical details are provided in the Appendix. Section 7 addresses a particular class of $\Sigma^\epsilon$ called the *complementary case*, and presents an illustrative example.

## 1.1 Notation

By $\mathbb{R}$ we denote the set of real numbers and by $\mathbb{R}_{>\tau}$ 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 $I_n$. By $\text{GL}(n, \mathbb{R})$ we denote the set of $n \times n$ invertible real matrices. For $Q \in M_{n,m}(\mathbb{R})$ and $\ell \le n$, $c \le m$, we denote by $(Q)_{\ell,c}$ the $\ell \times c$ matrix obtained by truncating $Q$ and keeping only its first $\ell$ lines and first $c$ columns. 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 $M \in M_n(\mathbb{R})$.

Given $x : \mathbb{R}_{\ge 0} \to \mathbb{R}^n$ and $t > 0$, we set $x(t^-) := \lim_{s \uparrow t} x(s)$ if such limit exists.

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 $n$, $t_{n+1} = +\infty$. The value of $Z$ on $[t_k, t_{k+1})$ is denoted by $Z_k$. 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 $k \in \Theta(Z) := \{0\} \cup \Theta(Z)$).

Given a positive integer $d$, for every $\ell \in \{1, \dots, d\}$ and $\varepsilon > 0$ we use $E_{\ell}^{\varepsilon}$ to denote the $d \times d$ diagonal matrix with diagonal coefficients equal to 1 over the $\ell$ first lines and $\varepsilon$ elsewhere. We denote by $E_{\ell c}^{\varepsilon}$ the $d \times d$ diagonal matrix with diagonal coefficients equal to $\varepsilon$ over the $\ell$ first lines and 1 elsewhere. Note that $\varepsilon(E_{\ell}^{\varepsilon})^{-1} = E_{\ell c}^{\varepsilon}$.

# 2 Problem formulation and main assumption

## 2.1 Singularly perturbed switched system

We begin this section by establishing a detailed reformulation of system $\Sigma^{\varepsilon}$. Let us fix an integer $d \ge 2$ and a compact subset $K$ of $\{1, \dots, d-1\} \times \text{GL}(d, \mathbb{R}) \times M_d(\mathbb{R}) \times M_d(\mathbb{R})$. We will use $\sigma$ to denote either an element of $K$ or a signal in $S_K$, always specifying which case we are considering. The components of $\sigma$ will be denoted by $(\ell, P, \Lambda, R)$. In particular, if $\sigma$ is in $S_K$, then $\ell, P, \Lambda$, and $R$ are themselves signals.

For $\varepsilon > 0$, $\tau \ge 0$ and $\sigma = (\ell, P, \Lambda, R) \in S_{K,\tau}$, we introduce the system

$$ \Sigma_{K,\tau}^{\varepsilon} : \begin{cases} E_{\ell_k}^{\varepsilon} P_k \dot{X}(t) = \Lambda_k X(t), & t \in [t_k, t_{k+1}), k \in \Theta(\sigma), \\ X(t_k) = R_{k-1} X(t_k^{-}), & k \in \Theta^{*}(\sigma), \end{cases} $$

where $E_{\ell_k}^{\varepsilon}$ is a diagonal matrix with diagonal coefficients equal to 1 over the $\ell_k$ first lines and $\varepsilon$ elsewhere. The matrix $E_{\ell_k}^{\varepsilon} P_k$ identifies on each interval of time $[t_k, t_{k+1})$ the slow and fast variables of the system. The sets $\Theta(\sigma)$ and $\Theta^{*}(\sigma)$, introduced in Section 1.1, are used to parameterize the switching instants of the signal $\sigma \in S_{K,\tau}$.

We denote by $\Phi_{\sigma}^{\varepsilon}(t, 0)$ the flow at time $t$ of system $\Sigma_{K,\tau}^{\varepsilon}$ corresponding to the switching signal $\sigma \in S_{K,\tau}$, i.e., the matrix such that $X_0 \mapsto \Phi_{\sigma}^{\varepsilon}(t, 0)X_0$ maps the initial condition $X(0) = X_0$ to the evolution at time $t$ of the corresponding solution of $\Sigma_{K,\tau}^{\varepsilon}$.

In analogy with the equality $S_{K,0} = S_K$, System $\Sigma_{K,0}^{\varepsilon}$ will be denoted simply by $\Sigma_K^{\varepsilon}$.

**Remark 1.** The case $\ell = d$, i.e., when all variables are slow, can be addressed by adding an extra fast variable $X_{d+1}$ in $\Sigma_{K,\tau}^{\varepsilon}$, for example, defined by $\varepsilon \dot{X}_{d+1} = -X_{d+1}$. The stability analysis of this augmented system is equivalent to that of the original system, thus covering the case $\ell = d$.
For a fixed $\varepsilon > 0$, $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ is a special case of the class of impulsive linear switched systems studied in [2]. In next section we recall how such systems are defined and some crucial results about their exponential stability.

## 2.2 Impulsive linear switched systems

The definition of impulsive switched linear system and the main notions concerning its stability are recalled by the following definition.

**Definition 2.** Let $\tau \ge 0$, $d \in \mathbb{N}$, and $\mathcal{Z}$ be a bounded subset of $M_d(\mathbb{R}) \times M_d(\mathbb{R})$. An impulsive switched linear system is a switched system with state jumps of the form

$$ \Delta_{\mathcal{Z},\tau}: \begin{cases} \dot{x}(t) = Z_1(t_k)x(t), & t \in [t_k, t_{k+1}), k \in \Theta(\mathcal{Z}), \\ x(t_k) = Z_2(t_{k-1})x(t_k^-), & k \in \Theta^*(\mathcal{Z}), \end{cases} $$

where $Z \in S_{\mathcal{Z},\tau}$. Denote by $\Phi_Z(t, 0)$ the flow from time 0 to time $t$ of $\Delta_{\mathcal{Z},\tau}$ corresponding to the switching signal $Z$. System $\Delta_{\mathcal{Z},\tau}$ is said to be

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

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

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

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

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

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

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

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

Notice that for $\mu \in \mathbb{R}$, setting $\mathcal{Z}^{\mu} = \{(Z_1 + \mu I_d, Z_2) \mid (Z_1, Z_2) \in \mathcal{Z}\}$, we have $\lambda(\Delta_{\mathcal{Z}^{\mu},\tau}) = \lambda(\Delta_{\mathcal{Z},\tau}) + \mu$.

Let us introduce the notation $\Xi_y$ for a discrete-time switched system with set of modes $\mathcal{Y} \subset M_d(\mathbb{R})$, that is,

$$ \Xi_y: \quad x(k) = Y_k x(k-1), \quad k \in \mathbb{N}, Y \in \mathcal{Y}^{\mathbb{N}}. $$
Recall that $\Xi_y$ is said to be *bounded* if there exists a constant $C > 0$ such that for every $k \in \mathbb{N}$ and every $Y_1, \dots, Y_k \in \mathcal{Y}$, $\|Y_k \cdots Y_1\| \le C$. Otherwise, it is said to be *unbounded*. We will also say that $\Xi_y$ is *exponentially unstable* (EU) if there exist $c > 0, \delta > 0, x_0 \in \mathbb{R}^d \setminus \{0\}$, and a sequence of matrices $\{Y_k\}_{k \ge 0}$ in $\mathcal{Y}$ such that $\|Y_k \cdots Y_1 x_0\| \ge c e^{\delta k} \|x_0\|$ for every $k \ge 1$.

The next theorem provides an alternative characterization of the exponential stability of an impulsive linear switched system, formulated through its Lyapunov exponent.

**Theorem 3** ([2, Theorems 3 and 4, and Remark 20]). Let $\mathcal{Y} = \{Z_2 \mid (Z_1, Z_2) \in \mathcal{Z}\}$. Then $\lambda(\Delta_{\mathcal{Z},\tau}) = +\infty$ if and only if $\tau = 0$ and $\Xi_y$ is unbounded. Moreover, if $\tau > 0$ or system $\Xi_y$ is bounded, then the following properties hold:

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

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

3. $\Delta_{\mathcal{Z},\tau}$ is EU if and only if $\lambda(\Delta_{\mathcal{Z},\tau}) > 0$.

## 2.3 Problem statement and first stability result

When $\varepsilon > 0$ is fixed, $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ is clearly an impulsive linear switched system. Our goal is to characterize when $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ is ES or EU for all values of $\varepsilon > 0$ small enough, that is, according to Theorem 3, when $\lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon})$ is negative or positive for all values of $\varepsilon > 0$ small enough. A first trivial remark that can be done is that, since $S_{\mathcal{K},\tau_1} \subset S_{\mathcal{K},\tau_2}$ for $\tau_1 \ge \tau_2$, then if $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ is ES (respectively, EU) then $\Sigma_{\mathcal{K},\tilde{\tau}}^{\varepsilon}$ is ES for every $\tilde{\tau} \in [\tau, +\infty)$ (respectively, EU for every $\tilde{\tau} \in [0, \tau]$).

In order to present some further remark on the exponential stability of $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$, let us introduce the following notation: given $\sigma = (\ell, P, \Lambda, R) \in \mathcal{K}$, we set

$$ \begin{pmatrix} A(\sigma) & B(\sigma) \\ C(\sigma) & D(\sigma) \end{pmatrix} = \Lambda P^{-1}, \qquad (3) $$

where $A(\sigma) \in M_{\ell}(\mathbb{R})$ and $B(\sigma), C(\sigma), D(\sigma)$ have the corresponding dimensions. When it is clear from the context, we simply write $A, B, C, D$ instead of $A(\sigma), B(\sigma), C(\sigma), D(\sigma)$.

Let us introduce

$$ \mathcal{R} = \{R \mid (\ell, P, \Lambda, R) \in \mathcal{K}\} $$

and provide a first result.
**Proposition 4.** It holds that

$$ \liminf_{\varepsilon \searrow 0} \varepsilon \lambda(\Sigma_K^\varepsilon) \geq \sup_{(\ell, P, \Lambda, R) \in \mathcal{K}} \alpha(P^{-1} E_{\ell c}^0 \Lambda) \geq 0. \quad (4) $$

If, moreover, $\Xi_R$ is bounded then

$$ \lim_{\varepsilon \searrow 0} \varepsilon \lambda(\Sigma_K^\varepsilon) = \max(0, \lambda(\Delta_{\mathcal{Z},0})), $$

where $\mathcal{Z} = \{(P^{-1} E_{\ell c}^0 \Lambda, R) \mid (\ell, P, \Lambda, R) \in \mathcal{K}\}$.

*Proof.* First notice that $\sup_{(\ell, P, \Lambda, R) \in \mathcal{K}} \alpha(P^{-1} E_{\ell c}^0 \Lambda)$ is nonnegative because it is larger than or equal to the real part of each eigenvalue of each matrix $P^{-1} E_{\ell c}^0 \Lambda = P^{-1} \begin{pmatrix} 0 & 0 \\ C & D \end{pmatrix} P$, which is nonnegative because $\ell < d$.

Consider now, for a given $\varepsilon \geq 0$, the impulsive linear switched system $\Delta_{\mathcal{Z}^\varepsilon,0}$ with $\mathcal{Z}^\varepsilon = \{(P^{-1} E_{\ell c}^0 \Lambda, R) \mid (\ell, P, \Lambda, R) \in \mathcal{K}\}$. Then notice that, for every $\varepsilon > 0$, the time rescaling $t \mapsto t/\varepsilon$ yields

$$ \Phi_Z(t, 0) = \Phi_\sigma^\varepsilon(\varepsilon t, 0), $$

where $\sigma$ is an arbitrary signal in $S_K$ and $Z$ is the corresponding signal in $S_{\mathcal{Z}^\varepsilon}$. This implies at once that $\varepsilon\lambda(\Sigma_K^\varepsilon) = \lambda(\Delta_{\mathcal{Z}^\varepsilon,0})$. Next, by Theorem 3 we have that $\lambda(\Delta_{\mathcal{Z}^\varepsilon,0}) \geq \sup_{(\ell, P, \Lambda, R) \in \mathcal{K}} \alpha(P^{-1} E_{\ell c}^0 \Lambda)$. It follows that $\varepsilon\lambda(\Sigma_K^\varepsilon) \geq \sup_{(\ell, P, \Lambda, R) \in \mathcal{K}} \alpha(P^{-1} E_{\ell c}^0 \Lambda)$. The proof of (4) is completed by letting $\varepsilon$ go to zero on both sides of the last inequality.

The last part of the statement comes from the fact that, if $\Xi_R$ is bounded then $\lambda(\Delta_{\mathcal{Z}^\varepsilon,0}) < +\infty$ for every $\varepsilon \geq 0$ (Theorem 3). The convergence of $\lambda(\Delta_{\mathcal{Z}^\varepsilon,0})$ to $\lambda(\Delta_{\mathcal{Z}^0,0})$ as $\varepsilon$ tends to 0 is then a consequence of [2, Proposition 24 and Remark 28]. $\square$

**Proposition 4** immediately yields a sufficient condition for the exponential instability of $\Sigma_K^\varepsilon$, namely that $\alpha(P^{-1} E_{\ell c}^0 \Lambda) > 0$ for some $(\ell, P, \Lambda, R) \in \mathcal{K}$. Notice that for each $\sigma = (\ell, P, \Lambda, R) \in \mathcal{K}$ one has $P^{-1} E_{\ell c}^0 \Lambda = P^{-1} \begin{pmatrix} 0 & 0 \\ C(\sigma) & D(\sigma) \end{pmatrix} P$. This motivates the introduction of the following assumption.

**D-Hurwitz assumption.** For each $\sigma \in \mathcal{K}$, the matrix $D(\sigma)$ defined in (3) is Hurwitz.

# 3 Auxiliary switched systems and statement of the main results

The stability of $\Sigma_{K,\tau}^\varepsilon$ will be studied by comparing it with that of single-scale auxiliary systems, which are introduced in this section.
## 3.1 Block diagonalization

Following a classical approach (see e.g. [8]), for $\sigma = (\ell, P, \Lambda, R) \in K$ we introduce the transformation matrix $T^\varepsilon = T^\varepsilon(\sigma)$ given by

$$T^\varepsilon = \begin{pmatrix} I_\ell & 0 \\ D^{-1}C + \varepsilon Q^\varepsilon & I_{d-\ell} \end{pmatrix} P,$$

and the upper triangular matrix $\Gamma^\varepsilon = \Gamma^\varepsilon(\sigma)$ given by

$$\Gamma^\varepsilon = \begin{pmatrix} A - B D^{-1} C - \varepsilon B Q^\varepsilon & B \\ 0 & \frac{D}{\varepsilon} + (D^{-1} C + \varepsilon Q^\varepsilon) B \end{pmatrix},$$

where $Q^\varepsilon = Q^\varepsilon(\sigma)$ is chosen in such a way that

$$\frac{1}{\varepsilon} T^\varepsilon P^{-1} E_{\ell c}^\varepsilon \Lambda(T^\varepsilon)^{-1} = \Gamma^\varepsilon,$$

and $\|Q^\varepsilon(\sigma)\|$ is upper bounded uniformly with respect to $\sigma \in K$ and $\varepsilon$ small enough.

Notice that the coordinate transformation just introduced makes sense only if the matrix $D$ is invertible. The proof of the existence of $Q^\varepsilon$ can be found in [8].

Let us stress that the expression for $T^\varepsilon$ makes sense also for $\varepsilon = 0$, and we will write simply $T(\sigma)$ for $T^0(\sigma)$. Note that the matrices $T(\sigma)$ belong to a compact subset of invertible matrices.

The transformation above allows one to introduce the variables $x(t)$ and $z(t)$ of dimensions $\ell(t)$ and $d - \ell(t)$, respectively, such that

$$\begin{pmatrix} x(t) \\ z(t) \end{pmatrix} = T_k^\varepsilon X(t), \quad \forall t \in [t_k, t_{k+1}), \ k \in \Theta(\sigma), \qquad (5)$$

and system $\Sigma_{K,\tau}^\varepsilon$ can be equivalently represented in terms of the triangular matrices $\Gamma^\varepsilon = \Gamma^\varepsilon(\sigma)$, for $\sigma \in S_{K,\tau}$, as

$$\left\{ \begin{array}{l} \begin{pmatrix} \dot{x}(t) \\ \dot{z}(t) \end{pmatrix} = \Gamma_k^\varepsilon \begin{pmatrix} x(t) \\ z(t) \end{pmatrix}, \quad t \in [t_k, t_{k+1}), \ k \in \Theta(\sigma) \\ \\ \begin{pmatrix} x(t_k) \\ z(t_k) \end{pmatrix} = T_k^\varepsilon R_{k-1} (T_{k-1}^\varepsilon)^{-1} \begin{pmatrix} x(t_k^-) \\ z(t_k^-) \end{pmatrix}, \quad k \in \Theta^*(\sigma). \end{array} \right. \qquad (6)$$

Observe from (6) that even if the starting model $\Sigma_{K,\tau}^\varepsilon$ does not include jumps in its dynamics, i.e., if $R = I_d$ for every $(\ell, P, \Lambda, R) \in K$, the change of variables (5) leads anyway to a singularly perturbed switched system with jumps.
## 3.2 Slow dynamics by Tikhonov's approach

The Tikhonov decomposition of a singularly perturbed system consists in analyzing the limit behavior of the slow dynamics by setting $\varepsilon = 0$ and replacing in the equation of the slow dynamics the limit value of the fast variable. This can be done when the fast dynamics has a stable equilibrium (as long as the switching signal stays constant), that is, when $D(\sigma)$ is Hurwitz, for $\sigma \in \mathcal{K}$. Assuming that the D-Hurwitz assumption holds and applying this approach to (6) leads to the formulation of the system

$$ \bar{\Sigma}_{\tau} : \left\{ \begin{array}{ll} \dot{\bar{x}}(t) = M_k \bar{x}(t), & t \in [t_k, t_{k+1}), k \in \Theta(\sigma) \\ \bar{x}(t_k) = J(k) \bar{x}(t_k^-), & k \in \Theta^*(\sigma), \end{array} \right. $$

where $\sigma \in S_{\mathcal{K},\tau}$, $M_k = A_k - B_k D_k^{-1} C_k$ and $J(k) = (T_k R_{k-1} T_{k-1}^{-1})_{\ell_k, \ell_{k-1}}$.

In what follows, we write $\bar{\Sigma}$ for $\bar{\Sigma}_0$. We also introduce the subset of $M_d(\mathbb{R})$ given by

$$ \bar{\mathcal{R}} = \left\{ RT^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & 0 \end{pmatrix} T \mid (\ell, P, \Lambda, R) \in \mathcal{K} \right\}, $$

which is related to the jumps of system $\bar{\Sigma}_{\tau}$.

When dwell-time is active ($\tau > 0$) the reduced system $\bar{\Sigma}_{\tau}$ allows to establish a necessary and a sufficient condition for the stability of $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$. More precisely, we will prove that if system $\bar{\Sigma}_{\tau}$ is ES (respectively, EU) then $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ is ES (respectively, EU) for every $\varepsilon > 0$ small enough (cf. Theorems 5 and 6).

## 3.3 Transient dynamics

If there is no dwell-time constraint (i.e., if $\tau = 0$), the transient dynamics governed by the fast dynamics must be considered. To capture such transient dynamics, a rescaling of time is needed and new variables are introduced: $s = t/\varepsilon$, $\hat{x}(s) = x(\varepsilon s)$ and $\hat{z}(s) = z(\varepsilon s)$. After rewriting the dynamics (6) in terms of this new scale, the limit problem at $\varepsilon = 0$ is given, for $\sigma = (\ell, P, \Lambda, R) \in S_{\mathcal{K}}$, by

$$ \hat{\Sigma} : \left\{ \begin{array}{ll} \left( \begin{array}{c} \dot{\hat{x}}(s) \\ \dot{\hat{z}}(s) \end{array} \right) = \begin{pmatrix} 0 & 0 \\ 0 & D_k \end{pmatrix} \left( \begin{array}{c} \hat{x}(s) \\ \hat{z}(s) \end{array} \right), & s \in [s_k, s_{k+1}), k \in \Theta(\sigma) \\ \\ \left( \begin{array}{c} \hat{x}(s_k) \\ \hat{z}(s_k) \end{array} \right) = T_k R_{k-1} T_{k-1}^{-1} \left( \begin{array}{c} \hat{x}(s_k^-) \\ \hat{z}(s_k^-) \end{array} \right), & k \in \Theta^*(\sigma). \end{array} \right. $$
System $\hat{\Sigma}$ allows to establish a necessary condition for the stability of $\Sigma_{\mathcal{K}}^{\varepsilon}$, in the sense that its instability implies the instability of $\Sigma_{\mathcal{K}}^{\varepsilon}$ (Theorem 5).

Finally, we introduce an additional auxiliary system, obtained from $\bar{\Sigma}$ by allowing more complex jumps, which take into account the transient dynamics described by $\hat{\Sigma}$. Consider

$$ \tilde{\Sigma}: \begin{cases} \dot{\tilde{x}}(t) = M_k \tilde{x}(t), & t \in [t_k, t_{k+1}), k \in \Theta(\sigma) \\ \tilde{x}(t_k) = \tilde{J}(k) \tilde{x}(t_k^-) & k \in \Theta^*(\sigma), \end{cases} $$

for $\sigma \in S_{\mathcal{K}}$, where $\tilde{J}(k) = (T_k F_{k-1} R_{k-1} T_{k-1}^{-1})_{\ell_k, \ell_{k-1}}$ and $F_{k-1}$ is any element of $\hat{\mathcal{F}}$, where

$$ \hat{\mathcal{F}} = \{I_d\} \cup \left\{ \prod_{i=1}^{n} R_i T_i^{-1} \begin{pmatrix} I_{\ell_i} & 0 \\ 0 & e^{s_i D_i} \end{pmatrix} T_i \mid n \in \mathbb{N}, s_i > 0, \right. \\ \left. (\ell_i, P_i, \Lambda_i, R_i) \in \mathcal{K} \text{ for } i = 1, \dots, n \right\}. $$

Intuitively speaking, $\tilde{\Sigma}$ takes into account at once the two cases in which the difference between subsequent switching times is much larger or comparable to the parameter $\varepsilon$.

Although $\bar{\Sigma}_{\tau}$, $\hat{\Sigma}$ and $\tilde{\Sigma}$ are not formally impulsive linear switched system in the sense of Definition 2 (their jump dynamics depend at each time $t_k$ on the value of $\sigma$ both on $[t_k, t_{k+1})$ and $[t_{k-1}, t_k)$) and that their state dimensions may vary with time (as in the case of $\bar{\Sigma}_{\tau}$ and $\tilde{\Sigma}$), their stability properties can be defined in analogy with Definition 2. In particular, for $\Sigma = \bar{\Sigma}_{\tau}$ ($\hat{\Sigma}$, $\tilde{\Sigma}$, respectively), we denote by $\Phi_{\sigma}^{\Sigma}(t, 0)$ the flow from time 0 to time $t \ge 0$ of $\Sigma$ associated with a signal $\sigma \in S = S_{\mathcal{K},\tau}$ ($S_{\mathcal{K}}$, respectively) and introduce

$$ \lambda(\Sigma) = \limsup_{t \to +\infty} \sup_{\sigma \in S} \frac{\log(\|\Phi_{\sigma}^{\Sigma}(t, 0)\|)}{t}. $$

We also introduce for $\hat{\Sigma}$ the Lyapunov-like exponent

$$ \tilde{\lambda}(\hat{\Sigma}) = \limsup_{s \to +\infty} \sup_{\sigma \in S_{\mathcal{K}}, k \in \Theta^*(\sigma), s=s_k} \frac{\log(\|\Phi_{\sigma}^{\hat{\Sigma}}(s, 0)\|)}{s}, $$

obtained by considering the evolution only at switching times.

The relation between the exponents defined above and the corresponding stability properties will be discussed in detail in Section 4.
## 3.4 Main results

Our main results are summarized in the following two theorems. The first one contains, in particular, conditions under which $\Sigma_{K,\tau}^{\epsilon}$ is EU for every $\epsilon$ small enough.

**Theorem 5.** Assume that the D-Hurwitz assumption holds true. The following statements hold:

1. For every $\tau > 0$, we have

$$ \lambda(\bar{\Sigma}_{\tau}) \leq \liminf_{\epsilon \searrow 0} \lambda(\Sigma_{K,\tau}^{\epsilon}). \quad (7) $$

If, moreover, $\bar{\Sigma}_{\tau}$ is EU then $\Sigma_{K,\tau}^{\epsilon}$ is EU for every $\epsilon > 0$ small enough.

2. If $\tau = 0$ and both $\Xi_R$ and $\Xi_{\bar{R}}$ are bounded then inequality (7) holds true and if, moreover, $\bar{\Sigma}$ is EU then $\Sigma_K^{\epsilon}$ is EU for every $\epsilon > 0$ small enough.

3. If $\Xi_R$ is bounded, we have

$$ \lambda(\hat{\Sigma}) \leq \max\{0, \liminf_{\epsilon \searrow 0} \epsilon \lambda(\Sigma_K^{\epsilon})\}. \quad (8) $$

In particular, if $\hat{\Sigma}$ is EU then, for every $\epsilon > 0$ small enough, $\Sigma_K^{\epsilon}$ is EU and $\lambda(\Sigma_K^{\epsilon})$ is at least at order $1/\epsilon$ as $\epsilon$ tends to 0.

The second theorem collects results containing sufficient conditions for the ES of $\Sigma_{K,\tau}^{\epsilon}$ for $\epsilon$ small enough.

**Theorem 6.** Assume that the D-Hurwitz assumption holds true. The following statements hold:

1. For every $\tau > 0$, we have

$$ \lambda(\bar{\Sigma}_{\tau}) \geq \lim_{\epsilon \searrow 0} \lambda(\Sigma_{K,\tau}^{\epsilon}). \quad (9) $$

In particular, if $\bar{\Sigma}_{\tau}$ is ES then $\Sigma_{K,\tau}^{\epsilon}$ is ES for every $\epsilon > 0$ small enough.

2. Assume that $\tilde{\lambda}(\hat{\Sigma}) < 0$. Then,

$$ \limsup_{\epsilon \searrow 0} \lambda(\Sigma_K^{\epsilon}) \leq \lambda(\tilde{\Sigma}). \quad (10) $$

In particular, if $\tilde{\Sigma}$ is ES then $\Sigma_K^{\epsilon}$ is ES for every $\epsilon > 0$ small enough.
As a direct consequence of Theorems 5 and 6, we obtain the following corollary.

**Corollary 7.** Assume that the D-Hurwitz assumption holds true and that $\tau > 0$. Then

$$ \lambda(\bar{\Sigma}_{\tau}) = \lim_{\varepsilon \searrow 0} \lambda(\Sigma_{K,\tau}^{\varepsilon}). $$

In the simplified case of switched singular perturbations with constant $\ell$ and $P, R \equiv I_d$, the corollary takes the following form, which completes the results obtained in [3].

**Corollary 8.** Let $\tau > 0$ and $\mathcal{M}$ be a compact subset of $M_d(\mathbb{R})$. Consider the singularly perturbed linear switched system

$$ \Upsilon_{\tau}^{\varepsilon}: \begin{cases} \dot{x}(t) = A_k x(t) + B_k y(t), \\ \varepsilon \dot{y}(t) = C_k x(t) + D_k y(t), \end{cases} \quad t \in [t_k, t_{k+1}), k \in \Theta(M), $$

for $M = (\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}) \in \mathcal{S}_{M,\tau}$. Suppose that $D$ is Hurwitz for every $(\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}) \in \mathcal{M}$. Consider the reduced order system

$$ \tilde{\Upsilon}_{\tau}: \dot{\bar{x}}(t) = M_k \bar{x}(t), \quad t \in [t_k, t_{k+1}), \ k \in \Theta(M), $$

where $M_k = A_k - B_k D_k^{-1} C_k$, for $M = (\begin{smallmatrix} A & B \\ C & D \end{smallmatrix}) \in \mathcal{S}_{M,\tau}$. Then

$$ \lambda(\tilde{\Upsilon}_{\tau}) = \lim_{\varepsilon \searrow 0} \lambda(\Upsilon_{\tau}^{\varepsilon}). $$

In particular, if $\tilde{\Upsilon}_{\tau}$ is ES (respectively, EU) then $\Upsilon_{\tau}^{\varepsilon}$ is ES (respectively, EU) for every $\varepsilon > 0$ small enough.

The proofs of Theorems 5 and 6 are provided in Sections 5.1, 5.2, 5.3, 6.1, and 6.2. In order to obtain these proofs, we introduce a series of results that allow us to reformulate the auxiliary switched systems $\bar{\Sigma}_{\tau}$, $\hat{\Sigma}$, and $\tilde{\Sigma}$ in the framework of impulsive switched systems, as defined in Definition 2. This re-formulation is essential for properly characterizing the Lyapunov exponents of the auxiliary switched systems, which exhibit time-varying dimensions and “multi-mode-dependent” jumping parts. These preliminary results are developed in Section 4. Moreover, additional technical preliminaries are required. In particular, Section 5 presents an approximation result that links the flow of the singularly perturbed system to the auxiliary systems introduced. Additionally, Section 6 includes converse-type theorems for impulsive switched systems, originally established in [2] and adapted to our context, which play a key role in the proofs of Theorems 5 and 6.
## 4 Lyapunov exponents of singularly perturbed and auxiliary systems

Given $\mu \in \mathbb{R}$, $\sigma = (\ell, P, \Lambda, R) \in \mathcal{K}$, and $\varepsilon > 0$, we introduce

$$M^{\mu}(\sigma) = M(\sigma) + \mu I_{\ell} \quad \text{and} \quad \Gamma^{\varepsilon, \mu}(\sigma) = \Gamma^{\varepsilon}(\sigma) + \mu I_{d}. \qquad (11)$$

By $\Sigma_{\mathcal{K},\tau}^{\varepsilon,\mu}$ we denote the $\mu$-shifted system associated with (6) and corresponding to $\Gamma^{\varepsilon,\mu}$. Notice that $\Gamma^{\varepsilon,0}$ coincides with $\Gamma^{\varepsilon}$ and $\Sigma_{\mathcal{K},\tau}^{\varepsilon,0}$, up to the choice of coordinates, with $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$. Notice also that $\lim_{\varepsilon \searrow 0} \alpha(\Gamma^{\varepsilon,\mu}) = \alpha(M^{\mu})$ for every $\sigma \in \mathcal{K}$.

Let us recall some notation from [2]. Given $n \in \mathbb{N}$ and a subset $\mathcal{N}$ of $M_n(\mathbb{R}) \times \mathbb{R}_{\ge 0}$, we denote by $\Omega_{\mathcal{N}}$ the set of all sequences $\omega = ((N_j, \tau_j))_{j \in \mathbb{N}}$ in $\mathcal{N}$ such that $\sum_{j \in \mathbb{N}} \tau_j = +\infty$. For every $\omega = ((N_j, \tau_j))_{j \in \mathbb{N}} \in \Omega_{\mathcal{N}}$ and $k \in \mathbb{N}$, we set

$$\omega_k = ((N_j, \tau_j))_{j=1}^k, \quad |\omega_k| = \tau_1 + \dots + \tau_k, \quad \Pi_{\omega_k} = N_k \dots N_1.$$

Given $\mu \in \mathbb{R}$, $\tau \ge 0$, and $\varepsilon > 0$, we define

$$\mathcal{N}_{\tau}^{\varepsilon, \mu} = \left\{ \left( R(T^{\varepsilon})^{-1} e^{t \Gamma^{\varepsilon, \mu}} T^{\varepsilon}, t \right) \mid \sigma \in \mathcal{K}, t \ge \tau \right\},$$

which is a subset of $M_d(\mathbb{R}) \times \mathbb{R}_{\ge 0}$. We denote $\mathcal{N}_{\tau}^{\varepsilon,0}$ simply by $\mathcal{N}_{\tau}^{\varepsilon}$, $\mathcal{N}_{0}^{\varepsilon,\mu}$ simply by $\mathcal{N}^{\varepsilon,\mu}$, and $\mathcal{N}_{0}^{\varepsilon,0}$ simply by $\mathcal{N}^{\varepsilon}$.

**Lemma 9.** If $\Xi_R$ is unbounded then $\lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon}) = +\infty$ for every $\varepsilon > 0$. If $\tau > 0$ or system $\Xi_R$ is bounded, then, for every $\mu \in \mathbb{R}$,

$$\lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon,\mu}) = \max \left( \sup_{\sigma \in \mathcal{K}} \alpha(\Gamma^{\varepsilon,\mu}), \sup_{\substack{\omega \in \Omega_{\mathcal{N}_{\tau}^{\varepsilon,\mu}} \\ k \in \mathbb{N}}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right) < +\infty.$$

*Proof.* First notice that $\mu$ can be taken equal to zero, since $\lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon,\mu}) = \lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon}) + \mu$ and the terms in the right-hand side of (9) scale analogously. Observe that $\Sigma_{\mathcal{K},\tau}^{\varepsilon}$ can be equivalently written as

$$\left\{ \begin{array}{ll} \dot{X}(t) = (T_k^{\varepsilon})^{-1} \Gamma_k^{\varepsilon} T_k^{\varepsilon} X(t), & t \in [t_k, t_{k+1}), k \in \Theta(\sigma), \\ X(t_k) = R_{k-1} X(t_k^{-}), & k \in \Theta^*(\sigma), \end{array} \right.$$

for $\sigma = (\ell, P, \Lambda, R) \in S_{\mathcal{K},\tau}$, that is, as the impulsive linear switched system $\Delta_{\mathcal{Z},\tau}$, with

$$\mathcal{Z} = \{((T^{\varepsilon}(\sigma))^{-1} \Gamma^{\varepsilon}(\sigma) T^{\varepsilon}(\sigma), R(\sigma)) \mid \sigma \in \mathcal{K}\}.$$
Applying Theorem 3 we have that if $\Xi_R$ is unbounded then $\lambda(\Sigma_{\mathcal{K}}^{\varepsilon}) = +\infty$ for every $\varepsilon > 0$, while if $\Xi_R$ is bounded or $\tau > 0$ then

$$ 
\begin{aligned}
+ \infty > \lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon}) &= \\
& \max \left( \sup_{\sigma \in \mathcal{K}} \alpha((T^{\varepsilon})^{-1} \Gamma^{\varepsilon} T^{\varepsilon}), \sup_{\sigma \in \mathcal{S}_{\mathcal{K},\tau}, k \in \mathbb{N}} \frac{\log(\rho(\Phi_{\sigma}^{\varepsilon}(t_k, 0)))}{t_k} \right) \\
&= \max \left( \sup_{\sigma \in \mathcal{K}} \alpha(\Gamma^{\varepsilon}), \sup_{\omega \in \Omega_{\mathcal{N}_{\mathcal{F}},k}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k})))}{|\omega_k|} \right),
\end{aligned}
$$

concluding the proof.

□

We introduce the subset of $M_d(\mathbb{R}) \times \mathbb{R}_{\ge 0}$ given by

$$ \hat{\mathcal{N}} = \left\{ \left( RT^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & e^{sD(\sigma)} \end{pmatrix} T, s \right) \mid \sigma \in \mathcal{K}, s > 0 \right\}. \quad (12) $$

**Lemma 10.** If $\Xi_R$ is unbounded then $\lambda(\hat{\Sigma}) = +\infty$. On the other hand, if $\Xi_R$ is bounded, then

$$ \lambda(\hat{\Sigma}) = \max \left( 0, \sup_{\omega \in \Omega_{\hat{\mathcal{N}}}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k})))}{|\omega_k|} \right) < +\infty, \quad (13) $$

and $\hat{\Sigma}$ is ES (respectively, EU) if and only if $\lambda(\hat{\Sigma}) < 0$ (respectively, $\lambda(\hat{\Sigma}) > 0$).

*Proof.* Consider

$$ \hat{\Delta} : \begin{cases} \dot{X}(s) = T_k^{-1} \begin{pmatrix} 0 & 0 \\ 0 & D_k \end{pmatrix} T_k X(s), s \in [s_k, s_{k+1}), k \in \Theta(\sigma), \\ X(s_k) = R_{k-1} X(s_k^{-}), \quad k \in \Theta^*(\sigma), \end{cases} $$

for $\sigma = (\ell, P, \Lambda, R) \in \mathcal{S}_K$. First notice that $\hat{\Delta}$ is an impulsive linear switched system in the sense of Definition 2. Moreover we have that the trajectories of $\hat{\Sigma}$ and $\hat{\Delta}$ only differ by a mode-dependent change of coordinates belonging to a compact set of invertible matrices, given as follows: for $t \ge 0$ there exists $\sigma \in \mathcal{K}$ so that

$$ X(t) = T(\sigma)^{-1} \begin{pmatrix} \hat{x}(t) \\ \hat{z}(t) \end{pmatrix}. $$

As a consequence, we have that $\lambda(\hat{\Sigma}) = \lambda(\hat{\Delta})$ and $\hat{\Sigma}$ is ES (respectively, EU) if and only if the same is true for $\hat{\Delta}$.
Applying Theorem 3, we have that if $\Xi_R$ is unbounded then $\lambda(\hat{\Delta}) = +\infty$, while if $\Xi_R$ is bounded then

$$ 
\begin{aligned}
+ \infty > \lambda(\hat{\Delta}) &= \\
&\max \left( \sup_{\sigma \in \mathcal{K}} \alpha \left( T^{-1} \begin{pmatrix} 0 & 0 \\ 0 & D(\sigma) \end{pmatrix} T \right), \sup_{\omega \in \Omega_{\mathcal{N}'}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right) \\
&= \max \left( 0, \sup_{\omega \in \Omega_{\mathcal{N}'}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right)
\end{aligned}
$$

and $\hat{\Delta}$ is ES (respectively, EU) if and only if $\lambda(\hat{\Delta}) < 0$ (respectively, $\lambda(\hat{\Delta}) > 0$), concluding the proof. □

For $\mu \in \mathbb{R}$ and $\tau \ge 0$, we introduce the subset of $M_d(\mathbb{R}) \times \mathbb{R}_{\ge 0}$ given by

$$ \bar{\mathcal{N}}_{\tau}^{\mu} = \left\{ \left( RT^{-1} \begin{pmatrix} e^{tM^{\mu}} & 0 \\ 0 & 0 \end{pmatrix} T, t \right) \mid \sigma \in \mathcal{K}, t \ge \tau \right\}. $$

We denote $\bar{\mathcal{N}}_{\tau}^0$ simply by $\bar{\mathcal{N}}_{\tau}$. Moreover, consider the system $\bar{\mathcal{N}}_{\tau}^{\mu}$ built as $\bar{\mathcal{N}}_{\tau}$ where we replace the matrix $M_k$ by the matrix $M_k^{\mu} = M_k + \mu I_{\ell_k}$.

**Lemma 11.** If $\Xi_{\bar{\mathcal{R}}}$ is unbounded then $\lambda(\bar{\mathcal{S}}) = +\infty$. On the other hand, if $\tau > 0$ or system $\Xi_{\bar{\mathcal{R}}}$ is bounded, then

$$ \lambda(\bar{\mathcal{S}}_{\tau}^{\mu}) = \max \left( \sup_{\sigma \in \mathcal{K}} \alpha(M^{\mu}), \sup_{\substack{\omega \in \Omega_{\bar{\mathcal{N}}_{\tau}^{\mu}} \\ k \in \mathbb{N}}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right) < +\infty, \quad (14) $$

and $\bar{\mathcal{S}}_{\tau}^{\mu}$ is ES (respectively, EU) if and only if $\lambda(\bar{\mathcal{S}}_{\tau}^{\mu}) < 0$ (respectively, $\lambda(\bar{\mathcal{S}}_{\tau}^{\mu}) > 0$).

*Proof.* As in the proof of Lemma 9, we assume without loss of generality that $\mu = 0$.

For each $\sigma \in \mathcal{K}$, let us define $\bar{R} = \bar{R}(\sigma)$ as $\bar{R} = RT^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & 0 \end{pmatrix} T$.

Observe that, for every $\delta \in \mathbb{R}$, $\bar{\mathcal{N}}_{\tau}$ can be equivalently written as

$$ \bar{\mathcal{N}}_{\tau} = \left\{ \left( \left( \begin{array}{cc} tT^{-1} & M \\ \bar{R}e & 0 \end{array} \right)^T \begin{array}{c} \delta I_{d-\ell} \\ 0 \end{array} \right), t \mid \sigma \in \mathcal{K}, t \ge \tau \right\}. $$
Consider the associated linear impulsive system

$$ \bar{\Delta}_{\tau}^{\delta}: \left\{ \begin{array}{l} \dot{X}(t) = T_k^{-1} \begin{pmatrix} M_k & 0 \\ 0 & \delta I_{d-\ell_k} \end{pmatrix} T_k X(t), \\ t \in [t_k, t_{k+1}), k \in \Theta(\sigma), \\ X(t_k) = \bar{R}_{k-1} X(t_k^{-}), \\ k \in \Theta^{*}(\sigma), \end{array} \right. \quad (15) $$

for $\sigma \in S_{\mathcal{K},\tau}$, and let us denote by $\Phi_{\sigma}^{\bar{\Delta}}(t, 0)$ the corresponding flow at time $t$ associated with a signal $\sigma \in S_{\mathcal{K}}$.

If $\Xi_{\bar{R}}$ is unbounded then, by Theorem 3, $\lambda(\bar{\Delta}_{0}^{\delta}) = +\infty$. Actually, applying [2, Lemma 19] one can find sequences $(\omega^n)_{n \in \mathbb{N}}$ in $\Omega_{\bar{N}_0}$ and $(k_n)_{n \in \mathbb{N}}$ in $\mathbb{N}$ such that $\lim_{n \to \infty} |\omega_{k_n}^n| = +\infty$ and

$$ \limsup_{n \to +\infty} \frac{\log(\|\Pi_{\omega_{k_n}}^n\|)}{|\omega_{k_n}^n|} = +\infty. \quad (16) $$

Now, observe that for every $\tau \ge 0$, $\omega \in \Omega_{\bar{N}_{\tau}}$, and $k \in \mathbb{N}$ we have

$$ \Pi_{\omega_k} = R_k T_k^{-1} \begin{pmatrix} \Phi_{\sigma_{\omega}}^{\bar{\Sigma}} (|\omega_k|^{-}, 0) & 0 \\ 0 & 0 \end{pmatrix} T_0 \quad (17) $$

where we recall that $\Phi_{\sigma_{\omega}}^{\bar{\Sigma}}(|\omega_k|^{-}, 0)$ denotes the limit as $s \nearrow |\omega_k|$ of the flow of system $\bar{\Sigma}_{\tau}$ from time 0 to time $s$, associated with the signal $\sigma_{\omega}$ that corresponds to $\omega$. We can then deduce from (16) and (17) that

$$ \limsup_{n \to +\infty} \frac{\log(\|\Phi_{\sigma_{\omega}}^{\bar{\Sigma}}(|\omega_{k_n}^n|^{-}, 0)\|)}{|\omega_{k_n}^n|} = +\infty, $$

yielding $\lambda(\bar{\Sigma}) = +\infty$.

If either $\Xi_{\bar{R}}$ is bounded or $\tau > 0$, we deduce from Theorem 3 that

$$ \lambda(\bar{\Delta}_{\tau}^{\delta}) = \max \left( \delta, \sup_{\sigma \in \mathcal{K}} \alpha(M), \sup_{\omega \in \Omega_{\bar{N}_{\tau}}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k})))}{|\omega_k|} \right). $$

Now, fix $\delta < \sup_{\sigma \in \mathcal{K}} \alpha(M)$ and let us prove that $\lambda(\bar{\Sigma}_{\tau}) = \lambda(\bar{\Delta}_{\tau}^{\delta})$. According to [2, Theorem 21] we can characterize $\lambda(\bar{\Delta}_{\tau}^{\delta})$ also as

$$ \lambda(\bar{\Delta}_{\tau}^{\delta}) = \max \left( \sup_{\sigma \in \mathcal{K}} \alpha(M), \sup_{\omega \in \Omega_{\bar{N}_{\tau}}} \limsup_{k \to \infty} \frac{\log(\|\Pi_{\omega_k}\|)}{|\omega_k|} \right). $$
By consequence, using (17), we have

$$
\begin{align*}
\lambda(\bar{\Delta}_{\tau}^{\delta}) &\le \\
&\max \left( \sup_{\sigma \in \mathcal{K}} \alpha(M), \sup_{\omega \in \Omega_{\bar{\mathcal{N}}_{\tau}}} \limsup_{k \to \infty} \frac{\log(\|\Phi_{\sigma_{\omega}}^{\bar{\Sigma}}(|\omega_k|^{-}, 0)\|)}{|\omega_k|} \right) \\
&\le \lambda(\bar{\Sigma}_{\tau}),
\end{align*}
$$

where the last inequality follows from the definition of $\lambda(\bar{\Sigma}_{\tau})$ (considering a constant signal to deduce that $\alpha(M) \le \lambda(\bar{\Sigma}_{\tau})$ for every $\sigma \in \mathcal{K}$).

On the other hand, for $\sigma \in S_{\mathcal{K},\tau}$, $k \in \mathbb{N}$, and $t \in [t_k, t_{k+1})$ we have

$$
\Phi_{\sigma}^{\bar{\Delta}}(t, 0) = T_k^{-1} \begin{pmatrix} \Phi_{\sigma}^{\bar{\Sigma}}(t, 0) & 0 \\ * & 0 \end{pmatrix} T_0
$$

from which we get the inequality

$$
\| \Phi_{\sigma}^{\bar{\Sigma}}(t, 0) \| \le \left\| \begin{pmatrix} \Phi_{\sigma}^{\bar{\Sigma}}(t, 0) & 0 \\ * & 0 \end{pmatrix} \right\| \le C \| \Phi_{\sigma}^{\bar{\Delta}}(t, 0) \|, \quad \forall t \ge t_1, \quad (18)
$$

for some $C > 0$ depending only on $\mathcal{K}$. From the definition of Lyapunov exponent of $\bar{\Sigma}_{\tau}$ and $\bar{\Delta}_{\tau}^{\delta}$ it follows that

$$
\begin{align*}
\lambda(\bar{\Sigma}_{\tau}) &= \limsup_{t \to +\infty} \sup_{\sigma \in S_{\mathcal{K},\tau}} \frac{\log(\|\Phi_{\sigma}^{\bar{\Sigma}}(t, 0)\|)}{t} \\
&\le \limsup_{t \to +\infty} \sup_{\sigma \in S_{\mathcal{K},\tau}} \frac{\log(\|\Phi_{\sigma}^{\bar{\Delta}}(t, 0)\|)}{t} = \lambda(\bar{\Delta}_{\tau}^{\delta}),
\end{align*}
$$

concluding the proof that $\lambda(\bar{\Sigma}_{\tau}) = \lambda(\bar{\Delta}_{\tau}^{\delta})$.

Notice that $\lambda(\bar{\Sigma}_{\tau}^{\mu}) = \lambda(\bar{\Delta}_{\tau}^{\delta,\mu})$, where $\bar{\Delta}_{\tau}^{\delta,\mu}$ is the natural shifted version of $\bar{\Delta}_{\tau}^{\delta}$.

Let us conclude the proof by showing that, under the assumption that $\Xi_{\bar{\mathcal{R}}}$ is bounded, $\bar{\Sigma}_{\tau}^{\mu}$ is ES (respectively, EU) if and only if $\lambda(\bar{\Sigma}_{\tau}^{\mu}) < 0$ (respectively, $\lambda(\bar{\Sigma}_{\tau}^{\mu}) > 0$).

One implication being trivial by definition of $\lambda(\bar{\Sigma}_{\tau}^{\mu})$, let us assume that
$\lambda(\bar{\Sigma}_{\tau}^{\mu}) < 0$ (respectively, $\lambda(\bar{\Sigma}_{\tau}^{\mu}) > 0$) and prove that $\bar{\Sigma}_{\tau}^{\mu}$ is ES (respectively, EU). On the one hand, if $\lambda(\bar{\Sigma}_{\tau}^{\mu}) < 0$ then, since $\lambda(\bar{\Sigma}_{\tau}^{\mu}) = \lambda(\bar{\Delta}_{\tau}^{\delta,\mu})$ and thanks to Theorem 3, $\bar{\Delta}_{\tau}^{\delta,\mu}$ is ES. The exponential stability of $\bar{\Sigma}_{\tau}^{\mu}$ follows then from (18) and the fact that, according to (14), $\alpha(M^{\mu}(\sigma)) < 0$ for every $\sigma \in \mathcal{K}$. On the other hand, if $\lambda(\bar{\Sigma}_{\tau}^{\mu}) > 0$ then (14) immediately identifies a constant or periodic signal yielding the exponential instability of $\bar{\Sigma}_{\tau}^{\mu}$. $\square$
**Remark 12.** As a consequence of Lemma 11, if $\Xi_{\bar{R}}$ is bounded, then $\lim_{\tau \searrow 0} \lambda(\bar{\Sigma}_{\tau}) = \lambda(\bar{\Sigma})$. Indeed, first notice that if $\Xi_{\bar{R}}$ is bounded then $\lambda(\bar{\Sigma}_{\tau})$ is characterized by (14) for every $\tau \ge 0$. Notice also that $\lambda(\bar{\Sigma}) \ge \lambda(\bar{\Sigma}_{\tau})$ for every $\tau > 0$. On the other hand, if $\lambda(\bar{\Sigma}) = \sup_{\sigma \in \mathcal{K}} \alpha(M)$ then $\lambda(\bar{\Sigma}) \le \lambda(\bar{\Sigma}_{\tau})$ for every $\tau > 0$. We are left to prove that $\liminf_{\tau \searrow 0} \lambda(\bar{\Sigma}_{\tau}) \ge \lambda(\bar{\Sigma})$ when for every $\delta > 0$ there exist $\omega \in \Omega_{\bar{N}}$ and $k \in \mathbb{N}$ such that $\lambda(\bar{\Sigma}) \le \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} + \delta$. In that case, for every $\tau > 0$ small enough $\omega_k$ can be completed to a k-periodic sequence in $\Omega_{\bar{N}_{\tau}}$, so that $\lambda(\bar{\Sigma}_{\tau}) \ge \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \ge \lambda(\bar{\Sigma}) - \delta$, completing the proof of the claim.

**Remark 13.** Another consequence of Lemma 11 is that, if $\Xi_{\bar{R}}$ is EU, then there exists $\tau > 0$ such that $\bar{\Sigma}_{\tau}$ is EU. Indeed, if $\Xi_{\bar{R}}$ is EU then there exist $\bar{R}(0), \dots, \bar{R}(L-1) \in \bar{R}$ such that $\rho(\bar{R}(L-1) \cdots \bar{R}(0)) > 1$ (see, e.g., [6]). Let $\tau > 0$ and $\omega \in \Omega_{\bar{N}_{\tau}}$ be the L-periodic sequence given by $\Pi_{\omega_L} = \bar{R}(L-1)e^{\tau \bar{M}_{L-1}} \cdots \bar{R}(0)e^{\tau \bar{M}_0}$, where $\bar{M}_k = e^{T_k \begin{pmatrix} M_k & 0 \\ 0 & 0 \end{pmatrix} T_k}$ and $\bar{R}(k) = R_k T_k^{-1} \begin{pmatrix} I_{\ell_k} & 0 \\ 0 & 0 \end{pmatrix} T_k$, for $k = 0, \dots, L-1$. By the continuity of the spectral radius, it follows that $\rho(\Pi_{\omega_L}) > 1$ for $\tau$ small enough. Observe that $\sup_{\omega \in \Omega_{\bar{N}_{\tau}}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \ge \frac{\log(\rho(\Pi_{\omega_L}))}{L\tau} > 0$, the conclusion follows from Lemma 11.

We introduce the subset of $M_d(\mathbb{R}) \times \mathbb{R}_{\ge 0}$ given by

$$ \tilde{\mathcal{N}}^{\mu} = \left\{ \left( FRT^{-1} \begin{pmatrix} e^{tM^{\mu}} & 0 \\ 0 & 0 \end{pmatrix} T, t \right) \mid \sigma \in \mathcal{K}, F \in \hat{\mathcal{F}}, t \ge 0 \right\} $$

and the subset of $M_d(\mathbb{R})$ given by

$$ \tilde{\mathcal{R}} = \left\{ FRT^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & 0 \end{pmatrix} T \mid \sigma \in \mathcal{K}, F \in \hat{\mathcal{F}} \right\}. $$

We denote $\tilde{\mathcal{N}}^0$ simply by $\tilde{\mathcal{N}}$.

**Lemma 14.** If $\Xi_{\bar{R}}$ is unbounded then $\lambda(\bar{\Sigma}) = +\infty$. On the other hand, if $\Xi_{\bar{R}}$ is bounded, then

$$ \lambda(\tilde{\Sigma}^{\mu}) = \max \left( \sup_{\sigma \in \mathcal{K}} \alpha(M^{\mu}), \sup_{\substack{\omega \in \Omega_{\tilde{\mathcal{N}}^{\mu}} \\ k \in \mathbb{N}}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right) < +\infty, $$

and $\tilde{\Sigma}_{\tau}^{\mu}$ is ES (respectively, EU) if and only if $\lambda(\tilde{\Sigma}^{\mu}) < 0$ (respectively, $\lambda(\tilde{\Sigma}^{\mu}) > 0$).

*Proof.* The proof follows the same lines of that of Lemma 11. □
# 5 Proof of Theorem 5

Let us start this section by stating an approximation result of the exponential of $\Gamma^{\epsilon, \mu}$, whose proof is given in the appendix.

**Lemma 15.** Let the D-Hurwitz assumption holds. Let $\mu \in \mathbb{R}$ and $T \subset \mathbb{R}_{\ge 0}$. Assume that either $T$ is bounded or $\alpha(M^\mu) < 0$ for every $\sigma \in \mathcal{K}$. Then there exist $C > 1$ (independent of $\mu, T$) and $K > 0$ such that for every $t \in T$, $\sigma = (\ell, P, \Lambda, R) \in \mathcal{K}$, and every $\epsilon > 0$ small enough

* if $t \ge C\epsilon|\log(\epsilon)|$ then

$$ \left\| (T^{\epsilon})^{-1} e^{t\Gamma^{\epsilon, \mu}} T^{\epsilon} - T^{-1} \begin{pmatrix} e^{tM^{\mu}} & 0 \\ 0 & 0 \end{pmatrix} T \right\| \le K\epsilon; \quad (19) $$

* if $t < C\epsilon|\log(\epsilon)|$ then

$$ \left\| (T^{\epsilon})^{-1} e^{t\Gamma^{\epsilon, \mu}} T^{\epsilon} - T^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & e^{\frac{t}{\epsilon}D} \end{pmatrix} T \right\| \le K\epsilon|\log(\epsilon)|, \quad (20) $$

where $M^\mu = M^\mu(\sigma)$ and $\Gamma^{\epsilon, \mu} = \Gamma^{\epsilon, \mu}(\sigma)$ are defined in (11), and $D = D(\sigma)$ is given in (3).

## 5.1 Proof of item 1 of Theorem 5

Let $\mu \in \mathbb{R}$ be such that $\mu > -\lambda(\bar{\Sigma}_\tau)$, so that $\lambda(\bar{\Sigma}_\tau^\mu) > 0$. From Lemma 11, there exist either $\sigma \in \mathcal{K}$ such that $\alpha(M^\mu(\sigma)) > 0$ or $\omega \in \Omega_{\bar{\Sigma}_\tau^\mu}$ and $k \in \mathbb{N}$ such that the spectral radius of $\Pi_{\omega_k}$ is greater than one. In the first case, by continuity of the spectral abscissa, we have $\alpha(\Gamma^{\epsilon, \mu}) > 0$ for sufficiently small $\epsilon > 0$. In the second case, thanks to (19) and the fact that $\bar{\Sigma}_\tau^\mu$ is bounded, there exists $K_\omega > 0$ such that for $\epsilon > 0$ sufficiently small we have

$$ \|\Pi_{\omega_k^\epsilon} - \Pi_{\omega_k}\| \le K_\omega \epsilon |\log(\epsilon)|, $$

where $\omega^\epsilon \in \Omega_{\bar{\Sigma}_\tau^\epsilon, \mu}$ is the sequence corresponding to $\omega$, in the sense that if the $j$th element of $\omega$ is the pair $(R(\sigma)T(\sigma)^{-1}(e^{tM^\mu(\sigma)} \begin{pmatrix} 0 \\ 0 \end{pmatrix})T(\sigma), t)$ then the $j$th element of $\omega^\epsilon$ is $(R(\sigma)T^\epsilon(\sigma)^{-1}e^{t\Gamma^{\epsilon, \mu}}T^\epsilon(\sigma), t)$. By consequence, from the continuity of the spectral radius, we have $\rho(\Pi_{\omega_k^\epsilon}) > 1$ for $\epsilon$ small enough. Let $\tilde{\omega}^\epsilon \in \Omega_{\bar{\Sigma}_\tau^\epsilon, \mu}$ be $k$-periodic such that $\tilde{\omega}_k^\epsilon$ is given by $\omega_k^\epsilon$. Using Lemma 9, it follows that $\lambda(\bar{\Sigma}_\tau^\epsilon, \mu) > 0$ for every $\epsilon > 0$ small enough. This proves, in particular, the last part of the statement.
Since $\lambda(\bar{\Sigma}_{\tau}^{\mu}) = \mu + \lambda(\bar{\Sigma}_{\tau})$ and $\lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon,\mu}) = \mu + \lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon})$, inequality (7) is obtained by letting $\mu \searrow -\lambda(\bar{\Sigma}_{\tau})$.

The rest of the proof follows immediately from Lemma 11 together with item 3 of Theorem 3.

## 5.2 Proof of item 2 of Theorem 5

If $\tau = 0$ and both $\Xi_R$ and $\Xi_R$ are bounded then $\max\{\lambda(\bar{\Sigma}), \lambda(\Sigma_{\mathcal{K},\tau}^{\varepsilon})\} < +\infty$ for every $\varepsilon > 0$, as it follows from Theorem 3 and Lemmas 9 and 11. The remainder of the proof proceeds analogously to that of item 1.

## 5.3 Proof of item 3 of Theorem 5

Since $\Xi_R$ is bounded, $\lambda(\hat{\Sigma})$ is characterized by (13) in Lemma 10. Let $\delta > 0$. There exist $\bar{\omega} \in \Omega_{\mathcal{N}}$ and $j \in \mathbb{N}$ such that

$$ \frac{\log(\rho(\Pi_{\bar{\omega}_j}))}{|\bar{\omega}_j|} > \sup_{\omega \in \Omega_{\mathcal{N}}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} - \delta. \quad (21) $$

Let $\bar{\omega}^{\varepsilon} \in \Omega_{\mathcal{N}}^{\varepsilon}$ be the sequence corresponding to $\bar{\omega}$, in the sense that if the $j$th element of $\bar{\omega}$ is the pair $(R(\sigma)T(\sigma)^{-1}(e^{\int_{0}^{\ell(\sigma)} \frac{1}{s^{D(\sigma)}}})T(\sigma), s)$ then the $j$th element of $\bar{\omega}^{\varepsilon}$ is $(R(\sigma)T^{\varepsilon}(\sigma)^{-1}e^{s\varepsilon\Gamma^{\varepsilon,\mu}}T^{\varepsilon}(\sigma), s\varepsilon)$. Notice that $|\bar{\omega}_k^{\varepsilon}| = \varepsilon|\bar{\omega}_k|$ for every $k \in \mathbb{N}$. Thanks to Lemma 15 and the continuity of the spectral radius, for $\varepsilon$ small enough it holds that

$$ \frac{\log(\rho(\Pi_{\bar{\omega}_j^{\varepsilon}}))}{|\bar{\omega}_j|} > \frac{\log(\rho(\Pi_{\bar{\omega}_j}))}{|\bar{\omega}_j|} - \delta. \quad (22) $$

Using the fact that $\sup_{\omega \in \Omega_{\mathcal{N}}^{\varepsilon}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \ge \frac{\log(\rho(\Pi_{\bar{\omega}_j^{\varepsilon}}))}{\varepsilon|\bar{\omega}_j|}$, we deduce from (21) together with (22) that

$$ \varepsilon \sup_{\omega \in \Omega_{\mathcal{N}}^{\varepsilon}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} > \sup_{\omega \in \Omega_{\mathcal{N}}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} - 2\delta. $$

By arbitrariness of $\delta$, it follows that

$$ \liminf_{\varepsilon \searrow 0} \left\{ \varepsilon \sup_{\omega \in \Omega_{\mathcal{N}}^{\varepsilon}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right\} \ge \sup_{k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|}. $$
By consequence, from Lemmas 9 and 10 we get

$$ \lambda(\hat{\Sigma}) \le \max \left( 0, \liminf_{\varepsilon \searrow 0} \left\{ \varepsilon \sup_{\omega \in \Omega_{\mathcal{N}}^{\varepsilon}, k \in \mathbb{N}} \frac{\log(\rho(\Pi_{\omega_k}))}{|\omega_k|} \right\} \right) \\ \le \max \left( 0, \liminf_{\varepsilon \searrow 0} \varepsilon \lambda(\Sigma_{\mathcal{K}}^{\varepsilon}) \right), $$

concluding the proof of inequality (8).

The rest of the proof follows immediately from Lemma 10 together with item 3 of Theorem 3.

# 6 Proof of Theorem 6

We begin this section by recalling some converse Lyapunov results established in [2] for impulsive linear switched systems and establishing some consequences.

**Theorem 16** ([2, Theorem 2]). An impulsive linear system $\Delta_{\mathcal{Z},\tau}$ is ES if and only if $\sup_{(Z_1,Z_2) \in \mathcal{Z}} \alpha(Z_1) < 0$ and there exist $c > 1$, $\gamma > 0$, and $V: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ 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

$$ |x| \le V(x) \le c|x|, \qquad (23) $$

$$ V(Z_2 e^{tZ_1} x) \le e^{-\gamma t} V(x). \qquad (24) $$

The second result concerns the quantity $\tilde{\lambda}(\Delta_{\mathcal{Z},\tau})$ associated with an impulsive linear switched system $\Delta_{\mathcal{Z},\tau}$, defined as

$$ \tilde{\lambda}(\Delta_{\mathcal{Z},\tau}) = \limsup_{t \to +\infty} \frac{1}{t} \sup_{Z \in S_{\mathcal{Z},\tau}, k \in \Theta^*(Z), t=t_k} \log(\|\Phi_Z(t,0)\|). $$

**Proposition 17** ([2, Remark 25]). Let $\tilde{\lambda}(\Delta_{\mathcal{Z},\tau}) < 0$. Then there exist $c > 1$, $\gamma > 0$, and $V: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ 1-homogeneous and Lipschitz continuous such that (23) and (24) hold true.

As a corollary of Theorem 16 and Proposition 17, we have the following result.

**Corollary 18.** Let $\tau \ge 0$ and $\mu \in \mathbb{R}$. Consider one of the following three cases:
$$ (C1) \quad \bar{\Sigma}_{\tau}^{\mu} \text{ is ES and } \mathcal{N} = \bar{\mathcal{N}}_{\tau}^{\mu}, $$

$$ (C2) \quad \tilde{\Sigma}^{\mu} \text{ is ES and } \mathcal{N} = \tilde{\mathcal{N}}^{\mu}, $$

$$ (C3) \quad \tilde{\lambda}(\hat{\Sigma}) < 0 \text{ and } \mathcal{N} = \tilde{\mathcal{N}}. $$

Then there exist $c > 1$, $\gamma > 0$, and $V : \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ 1-homogeneous and Lipschitz continuous such that (23) holds true and

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

for every $(N, t) \in \mathcal{N}$.

*Proof.* In case (C1) observe that, as noticed in Lemma 11, the exponential stability of $\bar{\Sigma}_{\tau}^{\mu}$ is equivalent to that of $\bar{\Delta}_{\tau}^{\delta}$ introduced in (15), up to replacing $M$ by $M^{\mu}$ and choosing $\delta < \sup_{\sigma \in \mathcal{K}} \alpha(M^{\mu})$. The conclusion then follows from Theorem 16.

The argument for Case (C2) is similar, with the role of Lemma 11 played by Lemma 14.

As for Case (C3), it is enough to observe that $\tilde{\lambda}(\hat{\Sigma}) = \tilde{\lambda}(\hat{\Delta})$, where $\hat{\Delta}$ is the system introduced in the proof of Lemma 10, and to apply Proposition 17. $\square$

## 6.1 Proof of item 1 of Theorem 6

In order to prove (9), we can assume without loss of generality that $\lambda(\bar{\Sigma}_{\tau}) < +\infty$. Now, consider $\mu \in \mathbb{R}$ such that $\lambda(\bar{\Sigma}_{\tau}^{\mu}) < 0$ (or, equivalently, that $\bar{\Sigma}_{\tau}^{\mu}$ is ES by Lemma 11) and let $V$ be the Lyapunov function associated with system $\bar{\Sigma}_{\tau}^{\mu}$ by Corollary 18 in Case (C1), with corresponding constants $c > 1$, $\gamma > 0$. For $\sigma \in \mathcal{K}$ and $t \ge \tau$, let $(N^{\varepsilon}, t) \in \mathcal{N}_{\tau}^{\varepsilon, \mu}$ where $N^{\varepsilon} = R(T^{\varepsilon})^{-1}e^{t\Gamma^{\varepsilon, \mu}}T^{\varepsilon} \in \mathcal{N}_{\tau}^{\varepsilon, \mu}$ is the associated evolution. Since $\sup_{\sigma \in \mathcal{K}} \alpha(M^{\mu}) \le \lambda(\bar{\Sigma}_{\tau}^{\mu})$, by continuity of the spectral abscissa and compactness of $\mathcal{K}$, it follows that $\sup_{\sigma \in \mathcal{K}} \alpha(\Gamma^{\varepsilon, \mu}) < 0$ for every $\varepsilon > 0$ small enough, which implies

$$ \|N^{\varepsilon}\| \le C_0 e^{-\eta t}, \quad (25) $$

for some $C_0 > 1$ and $\eta > 0$ (independent of $\varepsilon, \sigma$). Hence, from (23),

$$ V(N^{\varepsilon}x) \le cC_0e^{-\eta t}V(x). $$

If we fix $t^* > \frac{\log(cC_0)}{\eta}$, we deduce from the inequality above that there exists $\beta > 0$ sufficiently small such that, if $t \ge t^*$ then

$$ V(N^{\varepsilon}x) \le e^{-\beta t}V(x). \quad (26) $$
Furthermore, thanks to Lemma 15, there exist $K > 0$ (independent of $\sigma, \varepsilon$) such that for $\varepsilon > 0$ small enough we have $\|N^\varepsilon - N\| \le K\varepsilon$, where $(N, t)$ is the corresponding element in $\mathcal{N}_\tau^\mu$. For $x \in \mathbb{R}^d$, we have

$$V(N^\varepsilon x) \le L \|N^\varepsilon - N\| |x| + V(Nx) \le (\varepsilon KL + e^{-\gamma t}) V(x),$$

where $L > 0$ is such that $V$ is $L$-Lipschitz continuous. It follows from the equation above that, for $\varepsilon$ sufficiently small and up to reducing $\beta > 0$, the inequality (26) holds true even if $t \in [\tau, t^*]$, hence for every $t \in \mathbb{R}_{\ge \tau}$. By Theorem 16, we deduce then that $\Sigma_\tau^{\varepsilon, \mu}$ is ES (and hence $\lambda(\Sigma_\tau^{\varepsilon, \mu}) < 0$) for $\varepsilon$ small enough. Since $\lambda(\Sigma_{K,\tau}^{\varepsilon, \mu}) = \mu + \lambda(\Sigma_{K,\tau}^{\varepsilon})$, and considering the limit as $\mu \nearrow -\lambda(\bar{\Sigma}_\tau)$, we deduce that $\limsup_{\varepsilon \searrow 0} \lambda(\Sigma_{K,\tau}^{\varepsilon}) \le \lambda(\bar{\Sigma}_\tau)$.

## 6.2 Proof of item 2 of Theorem 6

**Lemma 19.** Let the D-Hurwitz assumption hold and the set $\hat{\mathcal{F}}$ be bounded. Let $C > 1$ and $K$ be as in Lemma 15 for a fixed $\mu \in \mathbb{R}$ and $\mathcal{T} = [0, 1]$. Then, there exists $\kappa > 0$ such that for $\varepsilon$ small enough, for every $n \in \mathbb{N}$ and $(N_1^\varepsilon, t_1), \dots, (N_n^\varepsilon, t_n) \in \mathcal{N}^{\varepsilon, \mu}$ such that $t_1, \dots, t_n \le C\varepsilon|\log(\varepsilon)|$, denoting by $(N_1, s_1), \dots, (N_n, s_n)$ the corresponding elements in $\mathcal{N}$ with $s_1 = t_1/\varepsilon, \dots, s_n = t_n/\varepsilon$, we have

$$\|N_n^\varepsilon \cdots N_1^\varepsilon - N_n \cdots N_1\| \le \kappa\varepsilon|\log(\varepsilon)|.$$

*Proof.* Observe that

$$\begin{aligned} N_n^\varepsilon \cdots N_1^\varepsilon &= (N_n^\varepsilon - N_n + N_n) \cdots (N_1^\varepsilon - N_1 + N_1) \\ &= \sum_{i \in \{0,1\}^n} (N_1^\varepsilon - N_1)^{i_1} (N_1)^{1-i_1} \cdots (N_n^\varepsilon - N_n)^{i_n} (N_n)^{1-i_n}. \end{aligned}$$

By consequence, we have

$$N_n^\varepsilon \cdots N_1^\varepsilon - N_n \cdots N_1 = \sum_{i \in \{0,1\}^n, i \neq 0} (N_1^\varepsilon - N_1)^{i_1} (N_1)^{1-i_1} \cdots (N_n^\varepsilon - N_n)^{i_n} (N_n)^{1-i_n}.$$

Observe that each term in the sum is the product of elements of type $(N_j^\varepsilon - N_j) \cdots (N_k^\varepsilon - N_k)$, for some $1 \le j \le k \le n$, and elements of type $N_i \cdots N_\ell$, for some $1 \le i \le \ell \le n$. Thanks to Lemma 15, we have

$$\| (N_j^\varepsilon - N_j) \cdots (N_k^\varepsilon - N_k) \| \le (KC\varepsilon|\log(\varepsilon)|)^{k-j+1}$$
for $\varepsilon$ small enough. In addition, from the fact that $\hat{\mathcal{F}}$ is bounded, there exists $c > 0$ such that, for every $1 \le i \le \ell \le n$,

$$ \|N_i \cdots N_\ell\| \le c. $$

For $m \in \{1, \dots, n\}$, denote by $S_m$ the subset of $\{0, 1\}^n$ composed of all elements with $m$ components equal to 1 and $n-m$ equal to 0. We have

$$ \left\| \sum_{i \in S_m} (N_1^\varepsilon - N_1)^{i_1} (N_1)^{1-i_1} \cdots (N_n^\varepsilon - N_n)^{i_n} (N_n)^{1-i_n} \right\| \le c(cKC\varepsilon|\log(\varepsilon)|)^m $$

for $\varepsilon$ small enough. By consequence, we have

$$ \| N_n^\varepsilon \cdots N_1^\varepsilon - N_n \cdots N_1 \| \le c \sum_{m=1}^{n} (cKC\varepsilon|\log(\varepsilon)|)^m $$

for $\varepsilon > 0$ small enough. The conclusion follows with $\kappa = 2c^2KC$. $\square$

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

**Remark 20.** A sufficient condition for $\hat{\mathcal{F}}$ to be bounded is that $\tilde{\lambda}(\hat{\Sigma}) < 0$. This can be deduced, for instance, from [2, Lemma 19] applied to the system $\hat{\Delta}$ introduced in the proof of Lemma 10.

*Proof of item 2 of Theorem 6.* First notice that if $\Xi_{\hat{\mathcal{R}}}$ is unbounded, then, by Lemma 14, $\lambda(\hat{\Sigma}) = +\infty$ and there is nothing to prove.

Assume then that $\Xi_{\hat{\mathcal{R}}}$ is bounded. Let $\mu \in \mathbb{R}$ be such that $\lambda(\hat{\Sigma}^\mu) < 0$. Let $V$ be the Lyapunov function associated with system $\hat{\Sigma}^\mu$ by Corollary 18 in Case (C2), with corresponding constants $c > 1$ and $\gamma > 0$. Let, moreover, $W$ be the Lyapunov function associated with system $\hat{\Sigma}$ by Corollary 18 in Case (C3), with corresponding constants $c_W > 1$ and $\gamma_W > 0$.

Fix $\varepsilon > 0$ and $\sigma \in S_K$ and associate with them the corresponding sequence $\omega^\varepsilon \in \Omega_{N^\varepsilon, \mu}$. Let $(t_i)_{i \in \mathbb{N}}$ be the switching times of $\sigma$, $(\sigma_i)_{i \in \mathbb{N}}$ its switching values, and $(N_i^\varepsilon, t_{i+1} - t_i)_{i \in \mathbb{N}}$ the corresponding values of $\omega^\varepsilon$. Let $C > 1$ and $K > 0$ be as in Lemma 15 with $T = \mathbb{R}_{\ge 0}$ (which is possible because $\lambda(\hat{\Sigma}^\mu) < 0$ implies that $\alpha(M^\mu) < 0$ for every $\sigma \in K$, according to Lemma 14). Fix $\bar{s} = 2\log(cc_W)/\gamma_W > 0$, so that, for every $s \ge \bar{s}$,

$$ \begin{aligned} cc_W e^{-\gamma_W s} &= cc_W e^{-\gamma_W \bar{s}} e^{-\gamma_W (s-\bar{s})} = e^{-\frac{\gamma_W}{2} \bar{s}} e^{-\gamma_W (s-\bar{s})} \\ &\le e^{-\frac{\gamma_W}{2} s}. \end{aligned} \qquad (27) $$

We say that an interval $[t_i, t_{i+1})$ is of
type (a) if $t_{i+1} - t_i \ge C|\epsilon \log(\epsilon)|$,

type (b) if $\epsilon \bar{s} \le t_{i+1} - t_i < C|\epsilon \log(\epsilon)|$,

type (c) if $0 < t_{i+1} - t_i < \epsilon \bar{s}$.

For every $i \in \mathbb{N}$, define $N_i$ as follows: if $[t_i, t_{i+1})$ is of type (a) (respectively, (b) or (c)), let $(N_i, t_{i+1} - t_i) \in \bar{\mathcal{N}}^\mu$ (respectively, $(N_i, \frac{t_{i+1}-t_i}{\epsilon}) \in \hat{\mathcal{N}}$) be the pair corresponding to the same mode $\sigma_i$ as $N_i^\epsilon$.

By Lemma 15 and Lemma 19 (which can be applied thanks to Remark 20), there exist $K, \kappa > 0$ such that, for $\epsilon$ small enough, $\|N_i^\epsilon - N_i\| \le K\epsilon$ when $[t_i, t_{i+1})$ is of the type (a) and $\|N_i^\epsilon \cdots N_j^\epsilon - N_i \cdots N_j\| \le \kappa\epsilon|\log(\epsilon)|$ when each of the intervals $[t_j, t_{j+1})$, ..., $[t_i, t_{i+1})$ is of the type (b) or (c).

We now associate with every interval $[t_i, t_{i+1})$ of type (a), (b), or (c) the minimal index $k(i) \in \{0, \dots, i\}$ such that each interval $[t_j, t_{j+1})$ with $k(i) \le j < i$ is of type (c). We then regroup the intervals $[t_i, t_{i+1})$ as follows.

We say that $[t_j, t_{i+1})$ is of type (I) if $[t_i, t_{i+1})$ is of type (a) and one of the two following properties holds: either $j = i$ and $t_i - t_{k(i)} \ge \epsilon \bar{s}$, or $j = k(i)$ and $t_i - t_{k(i)} < \epsilon \bar{s}$.

We can then split the complement in $\mathbb{R}_{\ge 0}$ of the union of all the intervals of type (I) in intervals $[t_j, t_k)$ that we call of type (II), which are such that $t_k - t_j \ge \epsilon \bar{s}$ and each $[t_l, t_{l+1})$ with $j \le l < k$ is of type (b) or (c).

Fix an interval $[t_j, t_{i+1})$ of type (I). We distinguish two cases, depending on whether $t_{i+1} - t_i$ is larger than a constant $T > 0$ to be fixed later. Consider first the case where $t_{i+1} - t_i \le T$. Notice that, for every $x \in \mathbb{R}^d$,

$$V(N_i x) \le e^{-\gamma(t_{i+1}-t_i)} V(x)$$

and, in the case $j = k(i)$,

$$V(N_{i-1} \cdots N_j x) \le V(x).$$

Let $L_V > 0$ be such that $V$ is $L$-Lipschitz continuous. Consider also a constant $C_0 > 0$ such that $N_i^\epsilon$ and every matrix in $\hat{\mathcal{F}}$ has norm smaller than $\tilde{\kappa}$ (cf. (25)). Hence,

$$
\begin{align*}
V(N_i^\epsilon \cdots N_j^\epsilon x) & \\
&\le L_V \|N_i^\epsilon \cdots N_j^\epsilon - N_i \cdots N_j\| \|x\| + V(N_i \cdots N_j x) \\
&\le L_V (\|N_i^\epsilon\| \|N_{i-1}^\epsilon \cdots N_j^\epsilon - N_{i-1} \cdots N_j\| \\
&\quad + \|N_i^\epsilon - N_i\| \|N_{i-1} \cdots N_j\|) \|x\| + V(N_i \cdots N_j x) \\
&\le L_V C_0 (\kappa + K) \epsilon |\log(\epsilon)| \|x\| + e^{-\gamma(t_{i+1}-t_i)} V(x).
\end{align*}
$$
Let $\tilde{K} = cL_V C_0 (\kappa + K)$. Using inequality (23), it follows that

$$V(N_i^{\epsilon} \cdots N_j^{\epsilon} x) \leq (\tilde{K}\epsilon |\log(\epsilon)| + e^{-\gamma(t_{i+1}-t_i)}) V(x).$$

Let $\epsilon$ be sufficiently small such that $\tilde{K}\epsilon|\log(\epsilon)|+e^{-\gamma(t_{i+1}-t_i)} < e^{-\frac{\gamma}{2}(t_{i+1}-t_i-\bar{s}\epsilon)}$. This choice is possible because for sufficiently small $\epsilon$ we have $s \mapsto f(s) = \tilde{K}\epsilon|\log(\epsilon)|+e^{-\gamma s}-e^{-\frac{\gamma}{2}(s-\bar{s}\epsilon)} < 0$ over the interval $[C|\epsilon \log(\epsilon)|, T]$. By consequence, for $\epsilon$ sufficiently small we have

$$V(N_i^{\epsilon} \cdots N_j^{\epsilon} x) \leq e^{-\frac{\gamma}{2}(t_{i+1}-t_i-\bar{s}\epsilon)} V(x) \leq e^{-\frac{\gamma}{2}(t_{i+1}-t_j)} V(x). \quad (28)$$

Consider now the case where $[t_j, t_{i+1})$ is of type (I) and $t_{i+1}-t_i > T$. As we proved in Section 6.1, there exists $\nu > 0$ such that $V(N^{\epsilon}x) \leq e^{-\nu\gamma t}V(x)$ for every $x \in \mathbb{R}^d$ and $t \geq 1$, where $(N^{\epsilon}, t) \in \mathcal{N}^{\epsilon, \mu}$. In particular, assuming that $T \geq 1$,

$$ \begin{aligned} V(N_i^{\epsilon} \cdots N_j^{\epsilon} x) &\leq e^{-\nu\gamma(t_{i+1}-t_i)} V(N_{i-1}^{\epsilon} \cdots N_j^{\epsilon} x) \\ &\leq e^{-\nu\gamma(t_{i+1}-t_i)} c(\|N_{i-1}^{\epsilon} \cdots N_j^{\epsilon} - N_{i-1} \cdots N_j\| \\ &\quad + \|N_{i-1} \cdots N_j\|) |x| \\ &\leq e^{-\nu\gamma(t_{i+1}-t_i)} c(\kappa\epsilon |\log(\epsilon)| + C_0) |x|. \end{aligned} $$

Up to choosing $T$ large enough and $\epsilon$ small enough, $e^{-\nu\gamma(t_{i+1}-t_i)}c(\kappa\epsilon|\log(\epsilon)|+C_0) \leq e^{-\frac{\nu\gamma}{2}(t_{i+1}-t_i-\bar{s}\epsilon)}$ for every $t_{i+1}-t_i > T$, so that

$$V(N_i^{\epsilon} \cdots N_j^{\epsilon} x) \leq e^{-\frac{\nu\gamma}{2}(t_{i+1}-t_j)} V(x). \quad (29)$$

Now, fix an interval $[t_j, t_k)$ of type (II). Let $L_W > 0$ be such that $W$ is $L_W$-Lipschitz continuous. Notice that

$$ \begin{aligned} W(N_{k-1}^{\epsilon} \cdots N_j^{\epsilon} x) &\leq L_W \|N_{k-1}^{\epsilon} \cdots N_j^{\epsilon} - N_{k-1} \cdots N_j\| \|x\| \\ &\quad + W(N_{k-1} \cdots N_j x) \\ &\leq L_W \kappa\epsilon |\log(\epsilon)| \|x\| + e^{-\gamma W \frac{t_k-t_j}{\epsilon}} W(x). \end{aligned} $$

Letting $\hat{K} = c_W L_W \kappa$, we have

$$W(N_{k-1}^{\epsilon} \cdots N_j^{\epsilon} x) \leq \left( \hat{K}\epsilon |\log(\epsilon)| + e^{-\gamma W \frac{t_k-t_j}{\epsilon}} \right) W(x).$$

Let $\epsilon$ be sufficiently small such that $\hat{K}\epsilon|\log(\epsilon)|+e^{-\gamma W \frac{t_k-t_j}{\epsilon}} < e^{-\frac{\gamma W}{2} \frac{t_k-t_j}{\epsilon}}$. Thus, we have

$$W(N_{k-1}^{\epsilon} \cdots N_j^{\epsilon} x) \leq e^{-\frac{\gamma W}{2} \frac{t_k-t_j}{\epsilon}} W(x).$$
By consequence, using (23) both for V and W and (27), it follows that

$$
\begin{align}
V(N_{k-1}^{\epsilon} \cdots N_{j}^{\epsilon} x) &\le cW(N_{k-1}^{\epsilon} \cdots N_{j}^{\epsilon} x) \nonumber \\
&\le c e^{-\frac{\gamma W}{2} \frac{t_k - t_j}{\epsilon}} W(x) \le e^{-\frac{\gamma W}{4} \frac{t_k - t_j}{\epsilon}} |x| \nonumber \\
&\le e^{-\frac{\gamma W}{4} \frac{t_k - t_j}{\epsilon}} V(x). \tag{30}
\end{align}
$$

By combining inequalities (28), (29), and (30) and applying Theorem 16, we deduce that system Σ$^{ε,µ}$ is ES (the inequality α(Γ$^{ε,µ}$(σ)) < 0 for σ ∈ K and ε > 0 small is guaranteed since α(M$^{µ}$(σ)) < 0 by Lemma 14 and lim$_{ε}$↓0 α(Γ$^{ε,µ}$(σ)) = α(M$^{µ}$(σ))).

Inequality (10) is obtained by taking the limit as μ ↛ − λ( ̂ Σ) . □

**Remark 21.** Recall that λ ( ̂ Σ ) ≥ 0 by Lemma 10. The condition ̂ λ ( ̂ Σ ) < 0 hence implies that R ( σ ) x = 0 for every x ∈ R d such that ( 0 C ( σ ) 0 D ( σ ) ) P ( σ ) x = 0 , for every σ ∈ K (see [2, Proposition 26]).

7 Applications

7.1 The complementary case

In this section, we consider the complementary case, in which system Σ ε τ results from switching between two linear d -dimensional systems, the second one obtained by exchanging the slow and fast dynamics of the first system. We derive a simple necessary condition for stability when d ≥ 2, which is also sufficient in the particular case d = 2. This is formalized in the following proposition.

**Proposition 22.** Consider the switched system Σ ε τ defined by the switching under a dwell-time constraint τ > 0 between

$$
\begin{cases}
\dot{x} = M_{11}(t)x + M_{12}(t)y, \\
\varepsilon \dot{y} = M_{21}(t)x + M_{22}(t)y,
\end{cases}
$$

and

$$
\begin{cases}
\varepsilon \dot{x} = M_{11}(t)x + M_{12}(t)y, \\
\dot{y} = M_{21}(t)x + M_{22}(t)y,
\end{cases}
$$

where x ∈ R $^{l}$, y ∈ R d − ℓ with ℓ ∈ { 1 , . . . , d − 1 } , and M = ( M$_{11}$ M$_{12}$ M$_{21}$ M$_{22}$ ) ∈ S M ,τ , M being a bounded subset of R $^{d}$. One has the following:
(i) System $\Sigma_{\tau}^{\epsilon}$ is EU for every $\tau > 0$ and every $\epsilon > 0$ small enough if either their exists $M \in \mathcal{M}$ such that $\max\{\alpha(M_{11}), \alpha(M_{22})\} > 0$ or, in the case where the D-Hurwitz assumption is satisfied, there exist $M, N \in \mathcal{M}$ such that $\rho(M_{11}^{-1}M_{12}N_{22}^{-1}N_{21}) > 1$.

(ii) Conversely, in the case when $d = 2$ and $\ell = 1$, if the D-Hurwitz assumption is satisfied (i.e., $M_{11}, M_{22} < 0$ for every $M \in \mathcal{M}$) and $\rho(M_{11}^{-1}M_{12}N_{22}^{-1}N_{21}) < 1$ (i.e., $|M_{12}N_{21}| < |M_{11}N_{22}|$) for every $M, N \in \mathcal{M}$, then $\Sigma_{\tau}^{\epsilon}$ is ES for every $\tau > 0$ and every $\epsilon > 0$ small enough.

*Proof.* System $\Sigma_{\tau}^{\epsilon}$ can be equivalently written as system $\Sigma_{\mathcal{K},\tau}^{\epsilon}$ where in this case the compact set $\mathcal{K}$ is given by

$$ \mathcal{K} = \{\ell\} \times \{I_d\} \times \mathcal{M} \times \{I_d\} \\ \cup \{d-\ell\} \times \{J_d\} \times \{J_d\mathcal{M}\} \times \{I_d\}, $$

where $J_d = \begin{pmatrix} 0 & I_{d-\ell} \\ I_{\ell} & 0 \end{pmatrix}$ and $J_d\mathcal{M} = \{J_dM | M \in \mathcal{M}\}$. The first part of item i) is a direct consequence of Proposition 4. Concerning the second part of point i), in this case one can easily verify that $\Xi_{\bar{\mathcal{R}}}$, where $\bar{\mathcal{R}}$ is given by

$$ \bar{\mathcal{R}} = \left\{ \begin{pmatrix} I_{\ell} & 0 \\ -M_{22}^{-1}M_{21} & 0 \end{pmatrix}, \begin{pmatrix} 0 & -M_{11}^{-1}M_{12} \\ 0 & I_{d-\ell} \end{pmatrix} \mid M \in \mathcal{M} \right\}, $$

is unbounded, and then thanks to Remark 13, $\bar{\Sigma}_{\tau}$ is EU. By Theorem 5 it follows that $\Sigma_{\tau}^{\epsilon}$ is EU for every $\tau > 0$ and for every $\epsilon > 0$ small enough.

Concerning the point (ii), one can easily verify in this case that each matrix $M_k$ is Hurwitz. Given that $\bar{\Sigma}_{\tau}$ is one-dimensional, it is necessary ES, and the conclusion follows from Theorem 6. $\square$

**Remark 23.** Note that in general the condition that the spectral radius of $M_{11}^{-1}M_{12}N_{22}^{-1}N_{21}$ is smaller than one is not a sufficient condition for $\Sigma_{\tau}^{\epsilon}$ to be exponentially stable (see [5, Example 19]).

## 7.2 Numerical example

Here, we illustrate through a numerical example the use of the auxiliary system $\tilde{\Sigma}$ to give a stability criterion for the system $\Sigma_{\mathcal{K}}^{\epsilon}$.

For $r \in [0, 1]$, consider the switched system $\Sigma_{\mathcal{K}}^{\epsilon}$ in the case where $d = 2$ and

$$ \mathcal{K} = \left\{ \begin{pmatrix} 1, & \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, & \begin{pmatrix} -1 & 1 \\ -1 & -1 \end{pmatrix}, & \begin{pmatrix} 2r & 2r \\ r & r \end{pmatrix}, \\ & \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, & \begin{pmatrix} -1 & -1 \\ 1 & -1 \end{pmatrix}, & \begin{pmatrix} -2r & -2r \\ r & r \end{pmatrix} \end{pmatrix} \right\}. $$
In this case, one can easily verify that $\hat{\Sigma}$ is ES. In addition, when $r < 1/\sqrt{3}$ the system $\tilde{\Sigma}$ is also ES. By consequence, thanks to Theorem 6, we have that $\Sigma_K^\epsilon$ is ES for every $\epsilon > 0$ small enough (see Fig. 1).

<figure><img src="image_3.png" /><figcaption>Figure 1: Time evolution (left) and phase plane plot (right) of system $\Sigma_K^\epsilon$ of Section 7.2 with $r = 0.45$, $\epsilon = 0.1$ and periodic piecewise-constant switching signal of period 0.4.</figcaption></figure>

# 8 Conclusion

This paper develops a comprehensive stability analysis for a class of singularly perturbed impulsive linear switched systems characterized by mode-
dependent switching between slow and fast dynamics. Reduced-order single-scale systems are introduced in order to capture the properties resulting from the interaction of the slow and fast dynamics, as the singular perturbation parameter $\epsilon$ approaches zero. More precisely, the paper establishes upper and lower bounds on the maximal Lyapunov exponent of the original system, expressed in terms of the maximal Lyapunov exponents of these auxiliary systems, as $\epsilon$ tends to zero. As a consequence, necessary and sufficient conditions are derived for the exponential stability of the singularly perturbed system for $\epsilon$ small enough. Furthermore, a complete characterization of the exponential stability is obtained under a dwell-time constraint on the switching laws.

## 9 Appendix: approximations of the flow of singularly perturbed systems

We are interested in this section in approximating the flow of $\Sigma_{\mathcal{K}}^{\epsilon, \mu}$ on an interval where the signal $\sigma$ is constant, that is, in approximating $e^{t\Gamma_{\epsilon, \mu}(\sigma)}$ for some $\sigma \in \mathcal{K}$. We start our analysis by a useful Grönwall's type result.

**Lemma 24.** Let $A \in M_n(\mathbb{R})$ be Hurwitz and $B \in L^\infty([t_0, \infty), M_n(\mathbb{R}))$. Then there exist $\alpha, \delta, K, \epsilon_0 > 0$ depending continuously on A and $\|B\|_\infty$ such that, for every $\tilde{A} \in M_n(\mathbb{R})$ with $\|A - \tilde{A}\| < \delta$ and every $\epsilon \in (0, \epsilon_0)$ the solution of

$$ \dot{z}(t) = \left( \frac{\tilde{A}}{\epsilon} + B(t) \right) z(t), \quad \forall t \ge t_0, \qquad (31) $$

satisfies the inequalities $|z(t)| \le Ke^{-\frac{\alpha}{\epsilon}(t-t_0)}|z(t_0)|$ and $|z(t)| - e^{-\frac{\tilde{A}}{\epsilon}(t-t_0)}z(t_0)| \le K \min(\epsilon, t-t_0)|z(t_0)|$ for every $t \ge t_0$.

*Proof.* By applying the variation of constants formula to (31), we obtain

$$ z(t) = e^{\frac{\tilde{A}}{\epsilon}(t-t_0)}z(t_0) + \int_{t_0}^{t} e^{\frac{\tilde{A}}{\epsilon}(t-s)}B(s)z(s)ds. \qquad (32) $$

Thanks to the fact that A is Hurwitz, for $\delta$ small enough, there exist $c, \alpha > 0$ such that

$$ \|e^{\frac{\tilde{A}}{\epsilon}(t-s)}\| \le c e^{-\frac{2\alpha}{\epsilon}(t-s)}, \quad \forall t \ge s. \qquad (33) $$

From (33) together with (32) we get that

$$ |z(t)| \le c e^{-\frac{2\alpha}{\epsilon}(t-t_0)}|z(t_0)| + c \|B\|_{\infty} \int_{t_0}^{t} e^{-\frac{2\alpha}{\epsilon}(t-s)}|z(s)|ds, $$
for every $t \ge t_0$, that is,

$$ \zeta(t) \le c|z(t_0)| + c\|B\|_{\infty} \int_{t_0}^{t} \zeta(s)ds, \quad \forall t \ge t_0, $$

with $\zeta(t) = e^{\frac{2\alpha}{\epsilon}(t-t_0)}|z(t)|$. By using Grönwall's inequality, there exists $\epsilon_0$ so that, for every $\epsilon \in (0, \epsilon_0)$, we get that

$$ |z(t)| \le c e^{-\frac{\alpha}{\epsilon}(t-t_0)} |z(t_0)|, \quad \forall t \ge t_0. \quad (34) $$

The second inequality in the statement hence follows by bounding the integral term in (32) using (33) and (34). We first get

$$ \begin{aligned} |z(t) - e^{\frac{\bar{\lambda}}{\epsilon}(t-t_0)}z(t_0)| &\le c^2 \|B\|_{\infty} (t-t_0) e^{-\frac{\alpha}{\epsilon}(t-t_0)} |z(t_0)| \\ &\le c^2 \|B\|_{\infty} \min\{t-t_0, \epsilon \sup_{s \in \mathbb{R}_{\ge 0}} se^{-\alpha s}\} |z(t_0)|, \quad \forall t \ge t_0, \end{aligned} $$

and, since $s \mapsto se^{-\alpha s}$ is uniformly bounded on $\mathbb{R}_{\ge 0}$, this yields the conclusion. $\square$

We can now state the following.

**Lemma 25.** Let the D-Hurwitz assumption hold. Let $\mu \in \mathbb{R}$ and $T \subset \mathbb{R}_{\ge 0}$. Assume that either $T$ is bounded or $\alpha(M^\mu) < 0$ for every $\sigma \in \mathcal{K}$. Then there exists $K > 0$ such that for $(\ell, P, \Lambda, R) \in \mathcal{K}$, $t \in T$, and $\epsilon > 0$ small enough,

$$ \left\| e^{t\Gamma^{\epsilon, \mu}} - \begin{pmatrix} e^{tM^\mu} & 0 \\ 0 & e^{t\frac{D}{\epsilon}} \end{pmatrix} \right\| \le K \min(\epsilon, t), \quad (35) $$

where $M^\mu$ and $\Gamma^{\epsilon, \mu}$ are defined in (11), and $D = D(\sigma)$ is given in (3).

*Proof.* Let $\mu \in \mathbb{R}$ and $(x_0, z_0) \in \mathbb{R}^\ell \times \mathbb{R}^{d-\ell}$ be fixed. Consider the trajectory $t \mapsto (x(t), z(t))^T = e^{t\Gamma^{\epsilon, \mu}}(x_0, z_0)^T$. As proved in Lemma 24 in the Appendix, there exist $K, \alpha > 0$ independent of $(\ell, P, \Lambda, R) \in \mathcal{K}$ such that

$$ |z(t)| \le K e^{-\frac{\alpha}{\epsilon}t} |z_0|, \quad \text{for } t \ge 0, \quad (36) $$

and

$$ |z(t) - e^{\frac{t}{\epsilon}D} z_0| \le K \min(\epsilon, t) |z_0|, \quad \text{for } t \ge 0. $$

By a slight abuse of notation, in what follows we still use $K$ to denote possibly larger constants independent of $(\ell, P, \Lambda, R)$ and $\epsilon$.
Using that estimate in the dynamics of $x$, we deduce by a simple application of Grönwall's lemma that

$$|x(s)| \le K|(x_0, z_0)|, \quad \forall s \in [0, \sup T). \qquad (37)$$

By applying the variation of constant formula, we have

$$x(t) = e^{tM^{\mu}}x_0 - \varepsilon \int_0^t e^{(t-s)M^{\mu}} BQ^{\varepsilon}x(s)ds + \int_0^t e^{(t-s)M^{\mu}} Bz(s)ds.$$

Notice that, by (36) and (37), we have that

$$\left| \int_0^t e^{(t-s)M^\mu} Bz(s)ds \right| \le K \min(\varepsilon, t) |z_0|,$$

and

$$\left| \varepsilon \int_0^t e^{(t-s)M^\mu} BQ^\varepsilon x(s)ds \right| \le K\varepsilon t |(x_0, z_0)|.$$

Hence, inequality (35) holds.

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

*Proof of Lemma 15.* First observe that

$$T^{\varepsilon} = \begin{pmatrix} I_{\ell} & 0 \\ D^{-1}C + \varepsilon Q^{\varepsilon} & I_{d-\ell} \end{pmatrix} P = T + O(\varepsilon), \\ (T^{\varepsilon})^{-1} = P^{-1} \begin{pmatrix} I_{\ell} & 0 \\ -D^{-1}C - \varepsilon Q^{\varepsilon} & I_{d-\ell} \end{pmatrix} = T^{-1} + O(\varepsilon).$$

Hence, by the uniform boundedness of $e^{t\Gamma^{\varepsilon,\mu}}$ (see equations (36)-(37)),

$$(T^{\varepsilon})^{-1} e^{t\Gamma^{\varepsilon,\mu}} T^{\varepsilon} = T^{-1} e^{t\Gamma^{\varepsilon,\mu}} T + O(\varepsilon). \qquad (38)$$

By the D-Hurwitz assumption, there exist $c \ge 1$ and $\gamma > 0$ depending only on $\mathcal{K}$ such that $\|e^{sD}\| \le ce^{-\gamma s}$ for all $s \ge 0$. Set $C = \max\{1, 1/\gamma\}$ and let us consider the two cases $t \ge C\varepsilon|\log(\varepsilon)|$ and $t < C\varepsilon|\log(\varepsilon)|$.

If $t \ge C\varepsilon|\log(\varepsilon)|$ then $\|e^{t/D}\| \le ce^{-\gamma t} \le c\varepsilon^{\gamma C}$, and by consequence, thanks to (35), one has that

$$\left\| e^{t\Gamma^{\varepsilon,\mu}} - \begin{pmatrix} e^{tM^{\mu}} & 0 \\ 0 & 0 \end{pmatrix} \right\| = O(\varepsilon).$$

Inequality (19) follows from (38).
If $t < C\epsilon|\log(\epsilon)|$, then, by (38),

$$ (T^{\epsilon})^{-1} e^{t\Gamma^{\epsilon,\mu}} T^{\epsilon} - T^{-1} \begin{pmatrix} I_{\ell} & 0 \\ 0 & e^{\frac{t}{\epsilon}D} \end{pmatrix} T \quad (39) $$

$$ = T^{-1} \left( e^{t\Gamma^{\epsilon,\mu}} - \begin{pmatrix} I_{\ell} & 0 \\ 0 & e^{\frac{t}{\epsilon}D} \end{pmatrix} \right) T + O(\epsilon). \quad (40) $$

Thanks to (35), one has that

$$ \left\| e^{t\Gamma^{\epsilon,\mu}} - \begin{pmatrix} I_{\ell} & 0 \\ 0 & e^{\frac{t}{\epsilon}D} \end{pmatrix} \right\| = O(\epsilon|\log(\epsilon)|), $$

and the conclusion follows. □

## References

[1] Mahmoud Abdelgalil, Daniel E. Ochoa, and Jorge I. Poveda. Multi-time scale control and optimization via averaging and singular perturbation theory: From ODEs to hybrid dynamical systems. *Annual Reviews in Control*, 56:100926, 2023.

[2] Yacine Chitour, Jamal Daafouz, Ihab Haidar, Paolo Mason, and Mario Sigalotti. Stability characterization for impulsive linear switched systems. Preprint, 2025.

[3] Yacine Chitour, Ihab Haidar, Paolo Mason, and Mario Sigalotti. Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems. *Automatica*, 155:111151, 2023.

[4] Fouad El Hachemi, Mario Sigalotti, and Jamal Daafouz. Stability analysis of singularly perturbed switched linear systems. *IEEE Trans. Automatica. Control*, 57(8):2116-2121, 2012.

[5] Ihab Haidar, Yacine Chitour, Jamal Daafouz, Paolo Mason, and Mario Sigalotti. Necessary conditions for the stability of singularly perturbed linear systems with switching slow-fast behaviors. In *63rd IEEE Conference on Decision and Control, CDC 2024*, 2024.

[6] Raphaël Jungers. *The joint spectral radius: theory and applications*, volume 385. Springer Science & Business Media, 2009.
[7] Petar Kokotović, Hassan K. Khalil, and John O'Reilly. *Singular perturbation methods in control*, volume 25 of *Classics in Applied Mathematics*. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. Analysis and design, Corrected reprint of the 1986 original.

[8] Petar V. Kokotović. A Riccati equation for block-diagonalization of ill-conditioned systems. *IEEE Trans. Automatic Control*, AC-20(6):812–814, 1975.

[9] Ivan Malloci, Jamal Daafouz, and Claude Iung. Stabilization of continuous-time singularly perturbed switched systems. In *Proceedings of the 48th IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference*, pages 6371–6376, 2009.

[10] Ivan Malloci, Jamal Daafouz, Claude Iung, Rémi Bonidal, and Patrick Szczepanski. Switched system modeling and robust steering control of the tail end phase in a hot strip mill. *Nonlinear Analysis: Hybrid Systems*, 3(3):239–250, 2009.

[11] Jihene Ben Rejeb, Irinel-Constantin Morărescu, Antoine Girard, and Jamal Daafouz. Stability analysis of a general class of singularly perturbed linear hybrid systems. *Automatica*, 90:98–108, 2018.

[12] Ricardo G Sanfelice and Andrew R Teel. On singular perturbations due to fast actuators in hybrid control systems. *Automatica*, 47(4):692–701, 2011.

[13] H. Shim and D. Liberzon. Further results on stability of linear systems with slow and fast time variation and switching. *Proc. 27th ACM Int. Conf. Hybrid Systems: Computation and Control (HSCC 2024)*, 96:1–7, 2024.

[14] PS Simeonov and DD Bainov. Stability of the solutions of singularly perturbed systems with impulse effect. *Journal of mathematical analysis and applications*, 136(2):575–588, 1988.

[15] Irina Subotic, Dominic Grob, Marcello Colombino, and Florian Dorfler. A Lyapunov framework for nested dynamical systems on multiple time scales with application to converter-based power systems. *IEEE Transactions on Automatic Control*, 66(12):5909–5924, 2021.
[16] Ying Tang, Christophe Fiter, and Laurentiu Hetel. A study on switched affine system interconnected with fast LTI dynamics. Automatica, 160:111462, 2024.

[17] Aneel Tanwani, Hyungbo Shim, and Andrew R. Teel. Singularly perturbed hybrid systems for analysis of networks with frequently switching graphs. IEEE Transactions on Automatic Control, pages 1-16, 2024.

[18] Wei Wang, Andrew R. Teel, and Dragan Nešić. Analysis for a class of singularly perturbed hybrid systems via averaging. Automatica, 48(6):1057-1068, 2012.

[19] Weixuan Wang, Alejandro I Maass, Dragan Nešić, Ying Tan, Ro main Postoyan, and W.P.M.H. Heemels. Stabilization of singularly perturbed networked control systems over a single channel. Preprint arXiv:2502.18768, 2025.