text:
\section{Introduction}

We consider in this paper a two-time-scales linear switching  system,
that is, a linear switching  system for which some variables evolve on a much faster rate than the others.
This class of systems appears in several industrial and engineering applications (see, e.g., \cite{4342105,Malloci2009,9199288}) where simplified models can be formulated by neglecting the effects of fast variables on the overall system. From control point of view, this allows to design a controller based on a reduced order model. However, the design based on a simplified model may not guarantee the stability of the overall system. To avoid this problem, a well-established framework developed in the mathematical and control community is that of singular perturbations \cite{KKO}. The singular perturbation theory allows the separation between slow and fast variables where different controllers for different time-scale variables can be designed in order to lead the overall system to its desired performance.

In mathematical terms, we study the behavior of
\begin{align*}
\dot x(t)&=A(t)x(t)+B(t)y(t),\\
\varepsilon\dot y(t)&=C(t)x(t)+D(t)y(t),
\end{align*}
where $\varepsilon$ denotes a small positive parameter and $A,B,C,D$ are matrix-valued
{signals}
undergoing arbitrary switching within a prescribed bounded range.
We deal, therefore, with a {1-parameter}
family
of linear switching systems $\Sigma:\varepsilon\mapsto \Sigma_\varepsilon$.
One of the main issues for such families of systems consists in understanding the time-asymptotic behavior of $\Sigma_\varepsilon$ as $t\to+\infty$ in the regime where $\varepsilon$ is small.
For instance, by saying that \emph{$\Sigma$ is exponentially stable} we refer to the fact that for every $\varepsilon>0$ sufficiently small, the corresponding linear switching system $\Sigma_\varepsilon$ is exponentially stable.
More refined notions, accounting for the uniform exponential behavior with respect to $\varepsilon$, are proposed in the paper (cf.~Definition~\ref{0-GES def}).
Few stability criteria  for singularly perturbed switching systems in the regime $\varepsilon\sim0$ have been obtained
in the literature: among them, let us mention \cite{Malloci2009CDC}, where conditions are obtained based on the existence of a common quadratic Lyapunov function, \cite{Hachemi2011} characterizing the stability in dimension two
based on the corresponding criteria in the non-singularly-perturbed case \cite{Balde2009,Boscain2005}, and \cite{Alwan2008}, where  stability
for time-delay singularly perturbed switching systems
is based on  dwell-time criteria.

A major mathematical difficulty
relies on the fact that we are interested in characterizing a doubly asymptotic regime,
where the order in which the limits are taken is crucial.
Indeed, a limit as $\varepsilon\to 0$ of the considered dynamics is well known to be given, through the Tikhonov decomposition, by
\begin{equation}\label{slow-intro}
\dot x(t)=\left(A(t)-B(t)D(t)^{-1}C(t)\right)x(t),
\end{equation}
with $y(t)=-D(t)^{-1}C(t)x(t)$.
 Heuristically, such a decomposition is
obtained by saying that $y$ tends instantaneously to $-D(t)^{-1}C(t)x(t)$ (the equilibrium of the equation for $y$ when $x\equiv x(t)$).
However, as it has already been observed in the literature,
 the switching system \eqref{slow-intro} may be exponentially stable even when, for every $\varepsilon>0$, $\Sigma_\varepsilon$ is unstable \cite{Malloci2009CDC}.
Notice that, in order to justify  the Tikhonov decomposition, we are assuming here that each $D(t)$ is a Hurwitz matrix. Actually, a standing assumption in this work is that
the fast dynamics are exponentially stable, i.e., the trajectories of
\begin{equation}\label{eq:fast-dym}
\dot y(t)=D(t)y(t)
\end{equation}
converge to the origin
with a uniform exponential
rate. It should be noticed that this condition is necessary for
the exponential stability of $\Sigma_\varepsilon$ in the regime $\varepsilon\sim 0$, since if
\eqref{eq:fast-dym} admits trajectories diverging exponentially, then there is no hope for $\Sigma_\varepsilon$ to be exponentially  stable for $\varepsilon$ small (a precise mathematical statement of this fact is proved in Proposition~\ref{prop:toto1}).
In the limit situation where \eqref{eq:fast-dym} is stable but not exponentially stable, we can still have exponential stability of $\Sigma_\varepsilon$ for $\varepsilon\sim 0$ \cite[Section V.A]{Hachemi2011}, but this is a rather degenerate
situation that we do not consider here.

The goal of this paper is not only to give
necessary or sufficient conditions ensuring that $\Sigma$ has a certain time-asymptotic behavior in the regime $\varepsilon\sim 0$,
but also to prescribe upper and lower bounds
on the limit as $\varepsilon\to 0$
of the maximal Lyapunov exponent of $\Sigma_\varepsilon$. We recall that the maximal Lyapunov exponent of a linear switching system is the largest asymptotic exponential rate as the time goes to $+\infty$ among all trajectories of the system.
Necessary or sufficient conditions for stability then follow as particular cases: indeed, a positive lower bound ensures that, for every $\varepsilon$ small enough, system $\Sigma_\varepsilon$ is unstable, while a negative upper bound guarantees that $\Sigma_\varepsilon$ is exponentially stable for all $\varepsilon$ in a right-neighborhood of zero.

The bounds on the limit as $\varepsilon\to 0$
of the maximal Lyapunov exponent of $\Sigma_\varepsilon$ are obtained by identifying
suitable auxiliary switching systems for the variable $x$ (with a single time scale) that are either sub- or super-approximations of the asymptotic dynamics of $\Sigma_\varepsilon$ as $\varepsilon\to 0$, in the following sense. A sub-approximation $\bar \Sigma$ of $\Sigma$ is a  system such that each of its  trajectories  can be approximated arbitrarily well, as $\varepsilon\to 0$, by the $x$-component of a trajectory of $\Sigma_\varepsilon$. Conversely, a super-approximation $\hat \Sigma$ of $\Sigma$ is
a system such that the $x$-component of each trajectory of $\Sigma_\varepsilon$
can be approximated arbitrarily well, as $\varepsilon\to 0$, by a trajectory  of $\hat \Sigma$.

The first and rather natural choice for $\bar \Sigma$ is system \eqref{slow-intro} since it corresponds to the singular perturbation approach, see for instance \cite{kokotovic84}.
The heuristic explanation is that, if switching occurs ``slowly'' with respect to the time-scale $1/\varepsilon$, then the  variable $y(t)$  converges fast enough to $-D(t)^{-1}C(t)x(t)$, and the transient phase is too short to affect the dynamics of $x(t)$.
As for a choice of super-approximation $\hat \Sigma$, the idea is to consider the switching parameter as evolving on a time scale possibly much faster than $1/\varepsilon$. Hence the dynamics of $y$ can be seen, on a short time interval, as a switching system with affine vector fields $y\mapsto C(t)\bar x+D(t)y$, where $\bar x$ is fixed. This kind of switching systems have been studied, for instance, in \cite{DellaRossa2022,Nilsson2013}. The exponential convergence of the solutions of \eqref{eq:fast-dym}
implies that the trajectories of such an  affine switching system converge exponentially towards a compact set $K(\bar x)$ \cite{DellaRossa2022}. A natural choice for $\hat \Sigma$ is then the
{differential inclusion}
\begin{equation*
\dot x(t)\in A(t)x(t)+B(t)K(x(t)).
\end{equation*}
We show that, indeed, this system allows to identify an upper bound for the limit of the maximal Lyapunov exponent of $\Sigma_\varepsilon$ as $\varepsilon\to 0$.

As mentioned in the previous paragraph, the two choices of $\bar \Sigma$ and $\hat \Sigma$ just described correspond to  signals switching at a rate much slower or much faster than $1/\varepsilon$.
We also consider a third regime, namely, the situation in which the switching occurs at rate exactly $1/\varepsilon$. This corresponds to considering a  signal $t\mapsto (A(t),B(t),C(t),D(t))=:\sigma(t)$ defined on an interval $[0,T]$ and all its reparameterizations $\sigma_\varepsilon:t\mapsto \sigma(t/\varepsilon)$ defined on $[0,\varepsilon T]$.
The flow of $\Sigma_\varepsilon$ corresponding to $\sigma_\varepsilon$, evaluated at time $\varepsilon T$,
has an effect of order $O(\varepsilon)$ on the coordinate $x$ (since the velocities are of order 1 and the length of the time interval is of order $\varepsilon$) and of order $O(1)$ on the coordinate $y$.
Suitable computations show that the evolution of $x$ can actually be described as
\[x\mapsto x+\varepsilon T\Lambda(T,\sigma)x+O(\varepsilon^2),\]
where the term $\Lambda(T,\sigma)$ does not depend on the initial condition of the variable $y$.
We then propose another choice of sub-approximation of $\Sigma$, denoted $\check \Sigma$, which coincides with the switching system having as possible modes all the matrices of the type $\Lambda(T,\sigma)$.
We provide an explicit expression for such matrices and we prove that, indeed, each
trajectory of $\check \Sigma$ can be approximated arbitrarily well, as $\varepsilon\to 0$, by the $x$-component of a trajectory of $\Sigma_\varepsilon$.
Moreover, we show that the modes of
$\bar \Sigma$ are also modes of $\check\Sigma$, implying that the
maximal Lyapunov exponents of $\bar \Sigma$ is upper bounded by that of $\check\Sigma$.
The main result of our paper, therefore,
consists
in
providing an interval containing all limits points of the maximal Lyapunov exponents of $\Sigma_\varepsilon$ as $\varepsilon$ goes to zero: the upper and lower bounds are given by the maximal Lyapunov exponents of $\hat \Sigma$ and $\check\Sigma$, respectively (see Theorem~\ref{thm:main}).

The paper is organized as follows:
In Section~\ref{sec:statement} we present precise definitions of the systems $\bar \Sigma$, $\check \Sigma$, $\hat \Sigma$, and of their maximal Lyapunov exponents. We also provide the  statement of the main result, Theorem~\ref{thm:main}, whose proof is given in the reminder of the paper. In particular, in Section~\ref{sec:bar} we compare the
maximal Lyapunov exponents of $\bar \Sigma$ and the
asymptotic {values}
as $\varepsilon$ goes to zero of the maximal Lyapunov exponent of $\Sigma_\varepsilon$ (Proposition~\ref{epsbehavior}). The comparison with the maximal Lyapunov exponent of $\check\Sigma$ is the subject of
Section~\ref{czech} (Proposition~\ref{check}), while Section~\ref{sec:hat} discusses the comparison with the maximal Lyapunov exponent of $\hat\Sigma$.

\subsection{Notations}
By $\mathbb{R}$ we denote the set of real numbers, by $|\cdot|$ the Euclidean norm of a real vector, and by $\|\cdot\|$ the induced matrix norm.
We use $\mathbb{R}_+$ to denote the set of non-negative real numbers and we write $\floor{x}$ to denote the
evaluation of the
floor function at a real number $x$.
By $M_n(\mathbb{R})$ we denote the set of $n\times n$ real matrices. The $n\times n$ identity matrix is denoted by $I_{n}$. We use $\rho(M)$ to denote the spectral radius of a matrix $M\in M_n(\mathbb{R})$, defined as the largest modulus among the eigenvalues of $M$.
By $B_r(x)$
  we denote the closed ball of radius $r>0$ and center $x\in\mathbb{R}^n$. 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 $d(x,Y)=\displaystyle\inf_{y\in Y}|x-y|$ and $d(y,X)=\displaystyle\inf_{x\in X}|x-y|$. Given a subset $\mathcal{N}$ of $M_n(\mathbb{R})$,
we denote
by $\mathcal{S}_{\mathcal{N}}$  the set of all measurable functions from $\mathbb{R}_+$ to
 $\mathcal{N}$.

 \section{Problem statement and main results}\label{sec:statement}

\subsection{Stability notions for singularly perturbed  linear switching  systems}

Fix $n,m\in \mathbb{N}$ and a compact set of matrices $\mathcal{M}\subset M_{n+m}(\mathbb{R})$.
Let
$\Sigma=(\Sigma_\varepsilon)_{\varepsilon>0}$ be the family of  linear switching systems
\begin{equation*
\Sigma_\varepsilon: \quad  \begin{array}{rll}
\quad \dot x(t)&=&A(t)x(t)+B(t)y(t),\\
\varepsilon \dot y(t)&=&C(t)x(t)+D(t)y(t),
\end{array}
\end{equation*}
where $\varepsilon$ denotes a small positive parameter and
\begin{equation*
t\mapsto \begin{pmatrix}
A(t)&B(t)\\
C(t) &  D(t)
\end{pmatrix}
\end{equation*}
is an arbitrary element of
the set
$\mathcal{S}_{\mathcal{M}}$ of measurable functions from $\mathbb{R}_+$ to $\mathcal{M}$.

For a given $\varepsilon>0$,
the usual stability notions, recalled in the following definition, apply to the
linear switching system
$\Sigma_\varepsilon$. \\

\begin{definition} \label{def:LSS-lyapunov}
Let $d\in \mathbb{N}$ and $\mathcal{N}$ be a bounded subset of $M_d(\mathbb{R})$. Consider the linear switching system
\begin{equation}\label{general}
    \Sigma_{\mathcal{N}}:\quad \dot x(t)=N(t)x(t),\qquad N\in \mathcal{S}_{\mathcal{N}},
\end{equation}
and denote by $\Phi_{N}(t,0)$ the flow from time $0$ to time $t$ of $\Sigma_{\mathcal{N}}$ associated with the switching signal $N$.
Then $\Sigma_{\mathcal{N}}$
is said to be
\begin{enumerate}
\item
\emph{exponentially  stable} (ES, for short) if there exist $C>0$ and $\delta>0$ such that
\[\|\Phi_{N}(t,0)\|\le C e^{-\delta t},\qquad \forall\,t\ge 0, \forall\, N\in \mathcal{S}_{\mathcal{N}};\]
\item
\emph{exponentially  unstable} (EU, for short) if there exist $C>0$, $\delta>0$, and a nonzero trajectory $t\mapsto x(t)$ of $\Sigma_{\mathcal{N}}$ such that
\[|x(t)|\ge C e^{\delta t}|x(0)|,\qquad \forall t\ge 0.\]
\end{enumerate}
The \emph{maximal Lyapunov exponent
of $\Sigma_{\mathcal{N}}$} is defined as
\begin{equation*
    \lambda(\Sigma_{\mathcal{N}})=\limsup_{t\to+\infty}\frac{1}{t}\sup_{N\in\mathcal{S}_{\mathcal{N}}}\log\|\Phi_{N}(t,0)\|.
\end{equation*}
\end{definition}

\begin{remark}
Notice that, for every $t>0$ and every $N\in \mathcal{S}_{\mathcal{N}}$, by Gelfand's formula,
\[
\rho(\Phi_{N}(t,0))=\lim_{k\to+\infty}\|\Phi_{N}(t,0)^k\|^{\frac1k}=\lim_{k\to+\infty}\|\Phi_{\tilde N}(kt,0)\|^{\frac1k}
\]
where $\tilde N$ denotes the $t$-periodic signal obtained by periodization of $N|_{[0,t]}$.
As a consequence,
\begin{equation}\label{eq:gelfand}
\rho(\Phi_{N}(t,0))\le e^{t \lambda(\Sigma_\mathcal{N})}.
\end{equation}
Actually, it  is well known (see, e.g., \cite{WIRTH200217}) that an equivalent definition of the maximal Lyapunov exponent is
\begin{equation}\label{eq:JSR}
\lambda(\Sigma_{\mathcal{N}})=\limsup_{t\to+\infty}\frac{1}{t}\sup_{N\in\mathcal{S}_{\mathcal{N}}}\log\rho(\Phi_{N}(t,0)).
\end{equation}

\end{remark}

\begin{remark} \label{rem:pseudo-fenichel}
It follows from the definitions that $\Sigma_{\mathcal{N}}$  is ES if and only if $\lambda(\Sigma_{\mathcal{N}})<0$. Also, by~\eqref{eq:JSR}, we have that $\Sigma_{\mathcal{N}}$ is EU  if and only if $\lambda(\Sigma_{\mathcal{N}})>0$.
Moreover, the properties of exponential stability/instability and the value of $\lambda(\Sigma_{\mathcal{N}})$ keep unchanged if we replace $\mathcal{S}_{\mathcal{N}}$ by the class of piecewise-constant functions from $\mathbb{R}_+$ to $\mathcal{N}$.
\end{remark}

In the following we
write
$\Phi^\varepsilon_{M}(t,0)$ to denote the flow from time $0$ to time $t$ of $\Sigma_\varepsilon$ associated with a signal $M\in \mathcal{S}_{\mathcal{M}}$ and
we
introduce the following stability notions for the 1-parameter family $\Sigma$ of switching systems. \\

\begin{definition}\label{0-GES def}
We say that the
1-parameter
family
of linear switching systems $\Sigma:\varepsilon\mapsto \Sigma_\varepsilon$ is
\begin{enumerate}
\item  $\varepsilon$-\emph{uniformly exponentially stable} ($\varepsilon$-ES, for short) if there exist
$\varepsilon^{\star}>0$, $C>0$, and $\delta>0$ such that
\begin{equation}\label{eq-ES-unif}
\|\Phi^\varepsilon_{M}(t,0)\|\leq C e^{-\delta t},\,\forall t\geq 0, \,\forall M\in \mathcal{S}_{\mathcal{M}}, \forall \varepsilon\in (0,\varepsilon^{\star});
\end{equation}
\item $\varepsilon$-\emph{uniformly exponentially unstable} ($\varepsilon$-EU, for short) if there exist $\delta>0$, $C>0$, and $\varepsilon^{\star}>0$ such that for every $\varepsilon\in (0,\varepsilon^{\star})$ there exist $M\in\mathcal{S}_{\mathcal{M}}$ and $z_0\neq 0$  for which
\begin{equation*}
|\Phi_{M}^\varepsilon(t,0)z_0|\geq C e^{\delta t}|z_0|,\qquad \forall\,t\geq 0.
\end{equation*}
\end{enumerate}
\end{definition}

\medskip

\begin{remark}
The $\varepsilon$-uniform exponential stability of $\Sigma$ is stronger than the property of global uniform asymptotic stability  introduced in \cite{Hachemi2011}, since it not only guarantees that $\lambda(\Sigma_\varepsilon)<0$ for every $\varepsilon$ small enough, but also that the stability is uniform with respect to $\varepsilon$, i.e.,   $\lambda(\Sigma_\varepsilon)$ is smaller than the negative number $-\delta$, independent of $\varepsilon$ small, and the constant $C$ appearing in \eqref{eq-ES-unif} is independent of $\varepsilon$ small.
Similar considerations concern {the} $\varepsilon$-uniform exponential instability.
\end{remark}

\subsection{Auxiliary switching systems and main result}

Our stability analysis of $\Sigma$ relies on the comparison with
several auxiliary systems having a single time scale.

Let us first discuss the system $\Sigma_D$ corresponding to the evolution of the fast variable $y$ with $x=0$. More precisely,
let
\[\mathcal{M}_D=\{D\mid  (\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})
\in \mathcal{M}\}\]
and consider the linear switching system
\[
    \Sigma_{D}:\quad \dot y(t)=D(t)y(t),\qquad D\in \mathcal{S}_{\mathcal{M}_D}.\]
We first provide a useful result for the asymptotic behavior of
$\Sigma$.\\

\begin{proposition}\label{prop:toto1}
With the notations above, it holds that
\[\lim_{\varepsilon\to 0}\varepsilon\lambda(\Sigma_\varepsilon)=\max\{
0,\lambda(\Sigma_D)
\
.\]
\end{proposition}
\begin{proof}Consider the family $(\Sigma_{\mathcal{N}_\varepsilon})_{\varepsilon\geq 0}$ of linear switching  systems, where, for every $\varepsilon\geq 0$, the compact set of matrices $\mathcal{N}_\varepsilon\subset M_{n+m}(\mathbb{R})$ is defined by
\[ \mathcal{N}_\varepsilon = \left\{\begin{pmatrix}\varepsilon A &\varepsilon B\\
C &  D
\end{pmatrix}\,|\,  \begin{pmatrix} A &B\\
C &  D
\end{pmatrix}\in \mathcal{M}\right\}.
\]
Then one notices that, for every $\varepsilon>0$, the time rescaling $t\mapsto \varepsilon t$ yields
\[\
\phi_{N_\varepsilon}(t,0)=\phi^{\varepsilon}_{M}(\varepsilon t,0),
\]
where $M$ is an arbitrary signal in $\mathcal{M}$ and the signal $N_\varepsilon\in\mathcal{N}_\varepsilon$ is defined as
\[N_\varepsilon(t)=\begin{pmatrix}
\varepsilon A(t)&\varepsilon B(t)\\
C(t) &  D(t)
\end{pmatrix},
\quad
t\geq 0.
\]This implies at once that, for every $\varepsilon$,
$\varepsilon\lambda(\Sigma_\varepsilon)=\lambda(\Sigma_{\mathcal{N}_\varepsilon})$.
Next notice that the
compact sets
\mathcal{N}_\
$ converge
to $\mathcal{N}_0$
as $\varepsilon$ goes to zero
for the Hausdorff topology defined on the family of compact subsets of $M_{n+m}(\mathbb{R})$.
One gets the conclusion
using
Lemma 3.5 and Equation~(19)
from \cit
{WIRTH200217}.
\end{proof}
The above result yields, in the particular case where $\lambda(\Sigma_D)>0$, that $\lambda(\Sigma_\varepsilon)\ge
\mu/\varepsilon$
for $\mu\in (0,\lambda(\Sigma_D))$ and $\varepsilon>0$ small enough, hence
$\Sigma_\varepsilon$ is unstable with trajectories diverging at an arbitrarily large exponential rate  for $\varepsilon$ small.
This motivates the following working assumption for the rest of the paper.\\

\begin{assumption}\label{UGES-fast}
The switching system $\Sigma_D$ is ES, that is, $\lambda(\Sigma_D)<0$.
\end{assumption}
 In particular, all matrices in $\mathcal{M}_D$ are Hurwitz (and then invertible).
We also  introduce the  set
\begin{equation*
\bar{\mathcal{M}}:=\{A-BD^{-1}C\mid
(\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})
\in\mathcal{M}\}\subset M_{n}(\mathbb{R}),
\end{equation*}
which collects the modes of the switching system
\begin{align*
\bar\Sigma:  \dot {\bar x}(t)&=\left(A(t)-B(t)D(t)^{-1}C(t)\right)\bar x(t),\qquad (\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in \mathcal{S}_{\mathcal{M}}.
\end{align*}

As described in the introduction, we also consider as auxiliary system another switching system, denoted by $\check{\Sigma}$,
with a larger set of modes than $\bar \Sigma$, and which
could be {thought} of as the
slow dynamics corresponding to
signals whose
switching occurs at rate exactly $1/\varepsilon$.
In order to define $\check{\Sigma}$,  let us associate
with every $T>0$ and every $\sigma=(\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in \mathcal{S}_{\mathcal{M}}$  the matrices
\begin{align*
\Lambda_0(T,\sigma)&=\int_{0}^{T}\Phi_{D}(T,s)C(s)\ ds,\\
 \Lambda_1(T,\sigma)&=\int_{0}^{T}\left(A(s) +B(s)\Lambda_0(s,\sigma)\right)\ ds, \\
\Lambda_2(T,\sigma)&=\int_{0}^{T}B(s)\Phi_{D}(s,0)\ ds,
 \end{align*}
and
\[\Lambda(T,\sigma)=\frac{\Lambda_1(T,\sigma)+\Lambda_2(T,\sigma)\left(I_m-\Phi_D(T,0)\right)^{-1}\Lambda_0(T,\sigma)}T.\]
As proved in Lemma~\ref{paolo-lemma} given in Section~\ref{czech}, the set
\[\mathcal{\check M}:=\{\Lambda(T,\sigma)\mid T>0,\;\sigma\in\mathcal{S}_\mathcal{M}\}\]
is bounded in $M_n(\mathbb{R})$.
The
switching system $\check{\Sigma}$
is then defined as
\begin{align}\label{new limit system}
\check\Sigma: \quad \dot {\check x}(t)=M(t)
\check x(t), \quad M\in \mathcal{S}_{\check \mathcal{M}}.
\end{align}

In the case of autonomous systems, i.e., in the absence of switching,
singular perturbation theory~\cite{KOKOTOVIC1976123}
guarantees that
the exponential stability of $\Sigma$
is completely characterized by the associated reduced dynamics~$\bar\Sigma$.
In the switching case, however,
it is well known that the stability of $\bar\Sigma$ is not sufficient to deduce the stability of the perturbed switching system $\Sigma_\varepsilon$ for $\varepsilon\sim 0$~\cite{Malloci2009CDC}.
The main difference
is that, fixed $x\in \mathbb{R}^n$, in the switching case  the $\omega$-limit set $K(x)$
of the \emph{fast dynamics}
 \begin{align}\label{fast}
\Sigma_x: \quad \dot y(t)&= D(t)y(t)+C(t) x,\qquad (\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in \mathcal{S}_{\mathcal{M}},
\end{align}
is in general not reduced to
a single equilibrium point \cite{DellaRossa2022,Nilsson2013}.
The precise definition of the set $K(x)$ is the closure of the union of the $\omega$-limit sets of all trajectories of $\Sigma_x$ starting from the initial condition $y(0)=0$.
We will prove in Proposition~\ref{forward invariance} and Lemma~\ref{lem:KLiphom}
that $K(x)$ is compact and the set-valued map $x\mapsto K(x)$  is  homogeneous of degree one and globally Lipschitz continuous for the Hausdorff distance.

Let us use the set-valued map $K$ to introduce a last  auxiliary system, which is not necessarily a switching system.
Namely, let us set
\begin{equation*
\hat{\mathcal{M}}(x):=\{Ax+B y\mid
(\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in\mathcal{M},\;y\in K(x)\},\quad x\in \mathbb{R}^n,
\end{equation*}
and consider the differential inclusion
 \begin{align*
\hat\Sigma: \quad \dot x(t)&\in \hat{\mathcal{M}}(x(t)). \end{align*}
Notice that $\hat{\mathcal{M}}(x)$ is compact for every $x\in\mathbb{R}^n$, since both $\mathcal{M}$ and $K(x)$ are compact.
We recall that a solution to $\hat\Sigma$  (in the sense of Filippov) is an absolutely continuous map $\mathbb{R}_+\ni t\mapsto x(t)\in \mathbb{R}^n$ such that $\dot x(t)\in \hat{M}(x(t))$ for almost all $t$. The existence of solutions to $\hat{\Sigma}$ for every initial condition is a consequence of
classical results on differential inclusions
(see, e.g., \cite[Chapter 2]{aubin-cellina}).

In analogy {with}
Definition~\ref{def:LSS-lyapunov}, we say that
$\hat\Sigma$ is ES (for \emph{exponentially stable}) if there exist $C>0$ and $\delta>0$ such that every trajectory $x(\cdot)$ of $\hat\Sigma$ satisfies
\[|x(t)|\le C e^{-\delta t}|x(0)|,\qquad \forall t\ge 0.\]
The notion of exponential instability is completely analogous to
the one given for linear switching systems. As for the Lyapunov exponent of $\hat\Sigma$, it can be defined as
\begin{equation*}
   \lambda(\hat\Sigma):=\limsup_{t\to+\infty}\frac{1}{t}\sup\log|x(t)|,
\end{equation*}
where the sup is taken over all trajectories $x(\cdot)$  of~$\hat\Sigma$  with $|x(0)|=1$.
Notice that the restriction to initial conditions with unit norm is justified by the fact that $x\mapsto K(x)$, and hence $x\mapsto {\hat {\mathcal{M}}(x)}$, is homogeneous of degree one.

The Lyapunov exponent $\lambda(\hat\Sigma)$ satisfies the following property, which generalizes the corresponding one for switching systems recalled in
Remark~\ref{rem:pseudo-fenichel}. \\

\begin{lemma}
\label{lem:ES-DI}
System $\hat\Sigma$ is ES if and only if $\lambda(\hat\Sigma)<0$.
\end{lemma}
\begin{proof}
The direct implication is trivial.
On the other hand, by definition of $\lambda(\hat\Sigma)$, if the latter is negative then for every $\delta\in (0,|\lambda(\hat\Sigma)|)$ there exists $T>0$ such that $|\hat x(t)|\leq e^{-\delta t}|\hat x(0)|$ for every solution $\hat x(\cdot)$ of the differential inclusion and $t\geq T$. Furthermore, $|\dot{\hat x}(t)|\leq \hat C|\hat x(t)|$ for some $\hat C>0$, because of the compactness, continuity, and homogeneity of the
map $\hat{\mathcal{M}}$.
Hence $|\hat x(t)|\leq e^{\hat C t}|\hat x(0)|\leq \hat K e^{-\delta t}|\hat x(0)|$ on $[0,T]$, where $\hat K = \max_{t\in [0,T]}e^{(\hat C+\delta) t}=e^{(\hat C+\delta) T}$. We conclude that $|\hat x(t)|\leq \hat Ke^{-\delta t}|\hat x(0)|$ for every $t>0$.
\end{proof}

The following theorem summarizes the main results obtained in this paper.\\

\begin{theorem}\label{thm:main}
Suppose that Assumption~\ref{UGES-fast} holds. Then
\begin{equation}\label{eigenvalues}
\lambda(\bar\Sigma) \leq\lambda(\check\Sigma) \leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_{\varepsilon})\leq \limsup_{\varepsilon\to 0^+}\lambda(\Sigma_{\varepsilon})\leq \lambda(\hat\Sigma).
\end{equation}
Moreover,
\begin{enumerate}
\item if system~$\check\Sigma$ is EU then system $\Sigma$ is $\varepsilon$-EU;
\item If system~$\hat\Sigma$ is ES then system~$\Sigma$ is $\varepsilon$-ES.
\end{enumerate}
\end{theorem}

\section{Comparison between the Lyapunov exponents of $\Sigma$ and $\bar\Sigma$}\label{sec:bar}

We prove in this section that $\lambda(\bar\Sigma) \leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_\varepsilon)$ (Proposition~\ref{epsbehavior}).
Strictly speaking, in view of Proposition~\ref{check} given below, this is not necessary for the proof of Theorem~\ref{thm:main}, but we prefer to provide the proof of this result since it illustrates, in a simplified framework, some of the
ideas used in the proof of the inequality $\lambda(\check\Sigma) \leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_\varepsilon)$.

The following lemma recalls a rather classical relation between Lyapunov exponents of linear switching systems that will be useful in the sequel. We provide its proof for completeness. \\

\begin{lemma}\label{shift}
Consider a linear switching system~$\Sigma_{\mathcal{N}}$ as in \eqref{general}.
Then, for every $\mu\in\mathbb{R}$, $\lambda(\Sigma_{{\mathcal{N}}+\mu I_n})=\mu +\lambda(\Sigma_{\mathcal{N}})$.
\end{lemma}

\begin{proof}
Let $\mu\in\mathbb{R}$. Observe that $t\mapsto x(t)$ is a trajectory of~$\Sigma_{\mathcal{N}}$ if and only if $t\mapsto e^{\mu t} x(t)$ is a trajectory of
$\Sigma_{{\mathcal N}+\mu I_n}$.
By consequence,
\begin{align*}
    \lambda(\Sigma_{{\mathcal N}+\mu I_n})&=\limsup_{t\to+\infty}\frac{1}{t}\sup_{N\in \mathcal{S}_{\mathcal{N}+\mu I_n}}\log\|\Phi_{N}(t,0)\|\\
    &=\limsup_{t\to+\infty}\frac{1}{t}\sup_{N\in \mathcal{S}_{\mathcal{N}}}\log\|e^{\mu t}\Phi_{N}(t,0)\|\\
    &=\mu +\lambda(\Sigma_{\mathcal N}),
\end{align*}
concluding the proof.
\end{proof}

\begin{proposition}\label{epsbehavior}
Suppose that Assumption~\ref{UGES-fast} holds. Then $\lambda(\bar\Sigma)\leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_\varepsilon)$.
\end{proposition}

\begin{proof}
For $\mu\in\mathbb{R}$ and $\varepsilon>0$ consider the switching systems
\begin{equation}\label{xypdot}
\Sigma_{\varepsilon,\mu}:\quad \begin{array}{rll}
\dot x(t)&=&A_{\mu}(t)x(t)+B(t)y(t), \\
\varepsilon \dot y(t)&=&C(t)x(t)+D_{\mu}(t)y(t),
\end{array
\end{equation}
and
\begin{align*
\bar\Sigma_{\mu}:  \dot {\bar x}(t)&=\left(A_{\mu}(t)-B(t)D^{-1}(t)C(t)\right)\bar x(t),
\end{align*}
where {$(\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in \mathcal{S}_\mathcal{M}$,}
\[A_{\mu}(t)=A(t)+\mu I_{n}, \quad \mbox{and} \quad D_{\mu}(t)=D(t)+\varepsilon\mu I_{m}.\]
It follows from Lemma~\ref{shift} that
\begin{equation*}
    \lambda(\Sigma_{\varepsilon,\mu})= \lambda(\Sigma_{\varepsilon})+\mu \quad {\rm and } \quad \lambda(\bar \Sigma_{\mu})= \lambda(\bar\Sigma)+\mu.
\end{equation*}
Fix, for now, $\mu\in \mathbb{R}$, $T>0$, and a piecewise-constant switching signal $\sigma=(\begin{smallmatrix}A&B\\mathcal{C}&D\end{smallmatrix})\in \mathcal{S}_\mathcal{M}$.
Denote by $t_1<\dots<t_N$ the switching instants of $\sigma$ within the interval $[0,T]$ and set $t_0=0$, $t_{N+1}=T$.
Following a classical approach (see e.g. \cite{kokotovic1975riccati}), we introduce the variable
\begin{equation}\label{trans}
z(t)=y(t)+D_{\mu}^{-1}(t)C(t)x(t)+\varepsilon P_{\varepsilon}(t) x(t),
\end{equation}
where $P_{\varepsilon}$ is
constant on each interval $[t_k,t_{k+1})$, $k=0,\dots,N$, and is
chosen in such a way
that $\Sigma_{\varepsilon,\mu}$ is equivalently represented on each interval $[t_k,t_{k+1})$ as
\begin{align}
\dot x(t)&=\left(M_{\mu}(t)-\varepsilon B(t)P_{\varepsilon}(t)\right)x(t)+B(t)z(t),\label{xeqdot}\\
\varepsilon \dot z(t)&=\left(D_{\mu}(t)+\varepsilon Q_{\varepsilon}(t)B(t)\right)z(t), \label{yeqdot}
\end{align}
where $M_{\mu}=A_{\mu}-BD_{\mu}^{-1}C$, $Q_{\varepsilon}=D_{\mu}^{-1}C+\varepsilon P_{\varepsilon}$, and $\|P_\varepsilon(t)\|$ is upper bounded uniformly with respect to $t\in [0,T]$ and $\varepsilon$ small enough.
 System~\eqref{xeqdot}-\eqref{yeqdot} is discontinuous at the instants of switching, since the variable $z$ depends on $\sigma$.
Recall that $\Phi_{\varepsilon^{-1}{D_\mu}}$ denotes
the flow associated with $\frac{1}{\varepsilon}D_{\mu}$. Thanks to Assumption~\ref{UGES-fast},  there exist $c,\alpha>0$ such that for $\varepsilon$ small enough and for every $0<s<t$
\begin{equation}\label{phieps}
    \|\Phi_{\varepsilon^{-1}{D_\mu}
(t,s)\|\leq ce^{-\frac{\alpha}{\varepsilon} (t-s)}.
\end{equation}
Observe that there exists a positive constant $K$
independent of $\varepsilon$ (but possibly depending on $T,\mathcal{M},\mu$)
such that for every $(x_0,y_0)\in\mathbb{R}^n\times\mathbb{R}^m$ and  every $\varepsilon>0$ small enough
one has
\begin{equation}\label{eq:wiK}
|(x(t),z(t))|\leq K|(x_0,y_0)|,\qquad t\in [0,T],
\end{equation}
where $(x(\cdot),y(\cdot))$ is the trajectory of $\Sigma_{\varepsilon,\mu}$
associated with $\sigma$ and the initial condition $(x_0,y_0)$ and $z(\cdot)$ is given by \eqref{trans}.
 By a slight abuse of notation, in what follows we still use $K$ to denote possibly larger constants
independent of $\varepsilon$.
For $t\in [t_{k},t_{k+1})$, by applying the variation of constant formula to \eqref{yeqdot} and using~\eqref{trans} for $t=t_k$,  we have
\begin{align*}
    z(t)={}
    \Phi_{\varepsilon^{-1}{D_\mu}}(t,t_{k})z(t_k)
    +\int_{t_{k}}^{t}
    \Phi_{\varepsilon^{-1}{D_\mu}}(t,s)Q_{\varepsilon}(s)B(s)z(s)}ds.

\end{align*}
As an immediate consequence of~\eqref{phieps} and \eqref{eq:wiK}, one gets that
\begin{equation}\label{estimate1}
|z(t)|\leq (e^{-\frac{\alpha}{\varepsilon}(t-t_k)}+\varepsilon) K |(x_0,y_0)|, \,\forall t\in [t_k,t_{k+1}).
\end{equation}
By applying the variation of constant formula to~\eqref{xeqdot}, one deduces that,
for every $t\in[0,T]$,
\begin{align*}
x(t)-\bar x(t)
&=\int_{0}^{t}\Phi_{\bar M_{\mu}}(t,s)B(s)
\left(D^{-1}(s)-D_{\mu}^{-1}(s)\right)C(s)x(s)ds
-\varepsilon\int_{0}^{t}\Phi_{\bar M_\mu}(t,s)B(s) P_{\varepsilon}(s)x(s)ds\\
&+\int_{0}^{t}\Phi_{\bar M_\mu}(t,s)B(s)z(s)ds,
\end{align*}
where
$\bar M_{\mu}:=A_{\mu}-BD^{-1}C$.
By the definition of $D_{\mu}$ we have that
\begin{equation}
    \label{Dmu}
    \|D^{-1}(t)-D_{\mu}^{-1}(t)\|\le \varepsilon K, \qquad \forall\, t\in [0,T].
\end{equation}
Hence, using estimates ~\eqref{eq:wiK} and \eqref{estimate1}, one deduces that
\begin{equation}\label{estimate2}
|x(t)-\bar x(t)
|\leq \varepsilon K|(x_0,y_0)|.
\end{equation}

From~\eqref{trans} together with  \eqref{estimate1}, \eqref{Dmu}, and \eqref{estimate2} we obtain that
\begin{align}
|y(T)+D^{-1}_{N}C_{N}\bar x(T)|\nonumber
&\leq
|z(T)|+K|x(T)-\bar x(T)|+
\varepsilon K |(x_0,y_0)|\leq
(e^{-\frac{\alpha}{\varepsilon}(T-t_N)}+\varepsilon) K |(x_0,y_0)|\nonumber\\
&\le \varepsilon  K |(x_0,y_0)|\label{estimate4}
\end{align}
for $\varepsilon$ sufficiently small, where $ (\begin{smallmatrix} A_N&B_N\\mathcal{C}_N&D_N\end{smallmatrix})$ is the value of $\sigma$ on $[t_N,T]$.

If $\Phi^\varepsilon_{\sigma,\mu}$ denotes the flow of~\eqref{xypdot} associated with $\sigma$,  one deduces from~\eqref{estimate2} and~\eqref{estimate4} that
\begin{equation*
\left\|\Phi^\varepsilon_{\sigma,\mu}(T,0)-\left(\begin{array}{lll}\Phi_{\bar M_\mu} (T,0) & 0\vspace{3pt}\\  -D_N^{-1}C_{N}\Phi_{\bar M_\mu} (T,0)&0\end{array}\right)\right\|\leq \varepsilon K.
\end{equation*}

Let $\mu$ be any constant such that
$\lambda(\bar\Sigma)+\mu=\lambda(\bar\Sigma_\mu)>0$, and choose the time
$T>0$ and the piecewise-constant switching signal $\sigma$
so that the spectral radius of $\Phi_{\bar M_\mu}(T,0)$ is larger than one (this is possible thanks to \eqref{eq:JSR}).
By continuity of the spectral radius,
it follows that
$\rho(\Phi^\varepsilon_{\sigma,\mu}(T,0))$ is also larger than one for $\varepsilon$ small enough.
Then one deduces from \eqref{eq:gelfand} that $\lambda(\Sigma_\varepsilon)+\mu=\lambda(\Sigma_{\varepsilon,\mu})> 0$.
The conclusion follows by arbitrariness of $\mu>-\lambda(\bar \Sigma)$.
\end{proof}

\section{Comparison between the Lyapunov exponents of $\Sigma$ and $\check\Sigma$}\label{czech}

We have the following preliminary results.

\begin{lemma}\label{paolo-lemma}
Under Assumption~\ref{UGES-fast} the set $\mathcal{\check M}$ is bounded.
\end{lemma}
\begin{proof}
By Assumption~\ref{UGES-fast} we have that
\begin{equation}\label{eq-flow-bound}
\|\Phi_D(t_2,t_1)\|\leq ce^{-\alpha (t_2-t_1)}
\end{equation}
for every $t_2\geq t_1$, for some constants $\alpha>0$ and $c\geq 1$ independent of the switching law $D\in \mathcal{S}_{\mathcal{M}_D}$. As a consequence of \eqref{eq-flow-bound} and the boundedness of $\mathcal{M}$, we have
$\|\Lambda_0(T,\sigma)\|\leq C_1 \min\{1,T\}$ and $\|\Lambda_2(T,\sigma)\|\leq C_1 \min\{1,T\}$ for some $C_1>0$, and $\|\Lambda_1(T,\sigma)\|\leq C_2 T$ for some $C_2>0$. Moreover
since $\rho(\Phi_D(T,0))<1$ for every $T>0$ and $D\in \mathcal{S}_{\mathcal{M}_D}$ according to \eqref{eq:gelfand}, $\left(I_m-\Phi_D(T,0)\right)^{-1}$ is well defined and expandable as an absolutely convergent power series in $\Phi_D(T,0)$. Letting $\bar T = \frac{\log(2 c)}{\alpha}$ so that $\|\Phi_D(\tau,0)\|\leq\frac12$ for every $\tau\geq \bar T$, we can write
\begin{align*}
(I_m-&\Phi_D(T,0))^{-1} =\sum_{k\geq 0} \Phi_D(T,0)^k \\
& =\sum_{h=0}^{\floor{\frac{\bar T}{T}}} \Phi_D(T,0)^h + \Phi_D(T,0)^{\floor{\frac{\bar T}{T}}+1} \left(\sum_{h=0}^{\floor{\frac{\bar T}{T}}} \Phi_D(T,0)^h\right) \\
&+ \Phi_D(T,0)^{2({\floor{\frac{\bar T}{T}}+1})} \left(\sum_{h=0}^{\floor{\frac{\bar T}{T}}} \Phi_D(T,0)^h\right)+\dots\\
&= \left(\sum_{k\geq 0} \Phi_D(T,0)^{(\floor{\frac{\bar T}{T}}+1)k}\right)\left(\sum_{h=0}^{\floor{\frac{\bar T}{T}}} \Phi_D(T,0)^h\right).
\end{align*}
As $\Phi_D(T,0)^h$ corresponds to the flow of $\Sigma_D$
for a $T$-periodic signal at time $hT$ we have $\|\Phi_D(T,0)^h\|\leq c$ by \eqref{eq-flow-bound}. Hence
\[
\left\|\sum_{h=0}^{\floor{\frac{\bar T}{T}}} \Phi_D(T,0)^h\right\|\leq \left(1+\floor{\frac{\bar T}{T}}\right)c\leq c + \frac{c\bar T}{T}.
\]
Moreover, as $(\floor{\frac{\bar T}{T}}+1)T\geq \bar T$,
\begin{align*}
\left\|\sum_{k\geq 0} \Phi_D(T,0)^{(\floor{\frac{\bar T}{T}}+1)k}\right\|&\leq \sum_{k\geq 0}\|\Phi_D(T,0)^{(\floor{\frac{\bar T}{T}}+1)}\|^k\leq \sum_{k\geq 0}\frac1{2^k}=2.
\end{align*}
Summing up
\begin{align*}
\|\Lambda(T,\sigma)\|&\leq C_2+\frac{C_1^2\min\{1,T\}^2 (2c+2c\bar{T}/T)}{T}\\
&=C_2+2cC_1^2\min\left\{\frac1T+\frac{\bar T}{T^2},T+\bar T\right\}\\
&\leq C_2+2cC_1^2(1+\bar T),
\end{align*}
concluding the proof of the lemma.
\end{proof}

\begin{lemma}\label{lem:smile}
Let $T>0$, $\sigma = (\begin{smallmatrix}
A&B\\
C&D
\end{smallmatrix})\in \mathcal{S}_\mathcal{M}$, and $\mu\in\mathbb{R}$.
For every $\varepsilon>0$ denote by $\mathscr{M}(\varepsilon)$ the flow at time $\varepsilon T$ of $\Sigma_{\varepsilon,\mu}$ defined in \eqref{xypdot} and  corresponding to the signal $\sigma(\cdot/\varepsilon)$.
Then there exists $P(\varepsilon)$ of the form
\begin{equation}
\label{eq:Peps}
P(\varepsilon)=\begin{pmatrix}I_n & 0\\
Q(\varepsilon) & I_m \end{pmatrix}
\end{equation}
such that
\begin{align}
P(\varepsilon)^{-1}\mathscr{M}(\varepsilon)P(\varepsilon)=
&\begin{pmatrix} I_n+\varepsilon T (\Lambda(T,\sigma)+\mu I_n)+O(\varepsilon^2)&O(\varepsilon)\\
0&\Phi_D(T,0)+O(\varepsilon) \end{pmatrix},\label{eq: ptrans}
\end{align}
where
\begin{equation}\label{Qe}
Q(\varepsilon)=\Big(I_m-\Phi_D(T,0)\Big)^{-1}\Lambda_0(T,\sigma)+O(\varepsilon)
\end{equation}
and the functions $O(\varepsilon^k)$ are such that $\|O(\varepsilon^k)\|\le C\varepsilon^k$ for $\varepsilon$ small, for some positive constant $C$ independent of $\varepsilon$ and $\sigma$.
\end{lemma}
\begin{proof}
We first prove that $\varepsilon\mapsto \mathscr{M}(\varepsilon)$ admits a first order expansion
\begin{equation}\label{DL}
\mathscr{M}(\varepsilon)=\mathscr{M}_{0}+\varepsilon \mathscr{M}_{1}+O(\varepsilon^2),
\end{equation}
for some matrices $\mathscr{M}_{0},\mathscr{M}_{1}$ to be computed.
To see that, we first apply the time rescaling $\tau=t/\varepsilon$ and
we have that $\mathscr{M}(\varepsilon)$ is equal to $\Phi_{N_{0}+\varepsilon N_{1}}(T,0)$,
where the signals $N_0$ and $N_1$ are defined as
\begin{equation*}
N_{0}(\tau)=
\begin{pmatrix}
0 & 0\\
C(\tau) & D(\tau)
\end{pmatrix},
\,
N_{1}(\tau)=
\begin{pmatrix}
 A(\tau)+\mu I_n &  B(\tau)\\
0 & \mu I_m
\end{pmatrix}.
\end{equation*}
By the variation of constant formula, one has
\begin{align*}
\Phi_{{N}_{0}+\varepsilon N_1}(T,0)=
 \Phi_{{N}_{0}}(T,0) +\varepsilon \int_0^T \Phi_{{N}_{0}}(T,\tau)N_1(\tau)\Phi_{{N}_{0}+\varepsilon N_1}(\tau,0)\;d\tau.
\end{align*}
One deduces that
\eqref{DL} holds true with
\begin{align*
\mathscr{M}_{0}=\Phi_{N_{0}}(T,0) \quad \mbox{and} \quad \mathscr{M}_{1}=\int_0^T\Phi_{{N}_{0}}(T,\tau)N_{1}(\tau)
\Phi_{{N}_{0}}(\tau,0)\;d\tau.
\end{align*}
It is easy to get that
\begin{align}
\mathscr{M}_{0}=\begin{pmatrix}I_n&0\\ \Lambda_0(T,\sigma)&
\Phi_D(T,0)\end{pmatrix},\quad \mbox{and} \quad
 \mathscr{M}_{1}=\begin{pmatrix}\Lambda_1(T,\sigma)+T\mu I_n&\Lambda_2(T,\sigma)\\ \Lambda_3(T,\sigma)&
\Lambda_4(T,\sigma)\end{pmatrix},\label{eq:M1}
    \end{align}
where
\begin{align*}\Lambda_3(T,\sigma)&=\Lambda_0(T,\sigma)\left(\Lambda_1(T,\sigma)+T\mu\right)
-\int_0^T \Phi_D(T,\tau) \Lambda_0(\tau,\sigma)(A(\tau)+B(\tau)\Lambda_0(\tau,\sigma))d\tau
\end{align*}
and
\begin{align*}
\Lambda_4(T,\sigma)&=\mu T\Phi_D(T,0)+\Lambda_0(T,\sigma)\Lambda_2(T,\sigma)
-\int_0^T \Phi_D(T,\tau) \Lambda_0(\tau,\sigma)B(\tau)\Phi_D(\tau,0)d\tau.
\end{align*}
According to~\eqref{DL} and~\eqref{eq:M1}, it
follows
that~\eqref{eq: ptrans} holds true with
$P(\varepsilon)$ and $Q(\varepsilon)$ as in \eqref{eq:Peps} and \eqref{Qe}.
This concludes the proof of the lemma.
\end{proof}

We can now prove the main result of this section.

\begin{proposition}
\label{check}
Suppose that Assumption~\ref{UGES-fast} holds. Then $\lambda(\bar\Sigma)\leq \lambda(\check\Sigma)\leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_\varepsilon)$. Moreover, if~$\check\Sigma$ is EU then~$\Sigma$ is $\varepsilon$-EU.
\end{proposition}

\begin{proof}
We first prove the  inequality $\lambda(\bar\Sigma)\leq \lambda(\check\Sigma)$ by showing  that  $
\bar{\mathcal{M}}\subset \mathcal{\check M}$.
To see that, let us check that $A-BD^{-1}C$ belongs to
$\mathcal{\check M}$ for every  $M=(\begin{smallmatrix} A&B\\mathcal{C}&D\end{smallmatrix})$ in $\mathcal{M}$. Indeed, letting
$T>0$ and  $\bar{\sigma}\in\mathcal{S}_\mathcal{M}$ constantly equal to $M$ on $[0,T]$, it holds
\begin{equation*}
\Lambda(T,\bar{\sigma})=A-BD^{-1}C.
\end{equation*}

Assume that $\mu$ is chosen so that
\[
\lambda(\check\Sigma)+\mu>0.
\]
Then, according to \eqref{eq:JSR}, there exist $\ell\in\mathbb{N}$, $\ell$ matrices $\Lambda(T_1,\sigma_1),\dots,\Lambda(T_\ell,\sigma_\ell)\in \check{\mathcal{M}}$,
and $\ell$ positive times $t_1,\dots,t_\ell$ so that
\begin{equation}\label{eq:bigger1}
\rho\Big(e^{t_\ell(\Lambda(T_\ell,\sigma_\ell)+\mu I_n)} \dots e^{t_1(\Lambda(T_1,\sigma_1)+\mu I_n)}\Big)>1.
\end{equation}

Let $\varepsilon$ be sufficiently small so that
$\varepsilon T_k<t_k$
for every $k=1,\dots,\ell$,   and denote by
 $N_k=\floor{\frac{t_k}{\varepsilon T_k}}$  the number of intervals of length $\varepsilon T_k$ contained in $[0,t_k]$.

Then, for every $k=1,\dots, \ell$, consider the flow
$\mathscr{M}_k(\varepsilon)$
of system $\Sigma_{\varepsilon,\mu}$ (cf.~\eqref{xypdot}) corresponding to the signal $\sigma_k(\cdot/\varepsilon)$, evaluated at time
$\varepsilon{T_k}$.
Thanks to Lemma~\ref{lem:smile}, there exists $P_k(\varepsilon)$ given by
$P_k(\varepsilon)=\begin{pmatrix}I_n & 0\\
Q_k(\varepsilon) & I_m \end{pmatrix}$
such that
\begin{align*
\mathcal{T}_k(\varepsilon)=P_k(\varepsilon)^{-1}\mathscr{M}_k(\varepsilon)P_k(\varepsilon)
\end{align*}
satisfies
\begin{align*}
\mathcal{T}_k(\varepsilon)=
\begin{pmatrix} I_n+\varepsilon T_k (\Lambda(T_k,\sigma_k)+\mu I_n)+O(\varepsilon^2)&O(\varepsilon)\\
0&\Phi_D(T_k,0)+O(\varepsilon) \end{pmatrix}.
\end{align*}
We repeat $N_k$ times $\sigma_k(\cdot/\varepsilon)$ to get a signal on $[0,\varepsilon T_k N_k]$ and the corresponding flow of $\Sigma_{\varepsilon,\mu}$ at time $\varepsilon T_k N_k$ is given by
\begin{equation}\label{eq:Mk}
\mathscr{M}_k(\varepsilon)^{N_k}=P_k(\varepsilon)\mathcal{T}_k(\varepsilon)^{N_k}P_k(\varepsilon)^{-1}.
\end{equation}
We claim that
\begin{equation}\label{eq:puissanceN1}
\mathcal{T}_k(\varepsilon)^{N_k}=
\begin{pmatrix}e^{ t_k (\Lambda(T_k,\sigma_k)+\mu I_n)}+O(\varepsilon)&O(\varepsilon)\\ 0&O(\varepsilon)
\end{pmatrix}.
\end{equation}
This follows from the general formula
\[\begin{pmatrix}A_{11}&A_{12}\\0&A_{22}\end{pmatrix}^N=\begin{pmatrix}A_{11}^N&W\\0&A_{22}^N\end{pmatrix},\,
W=\sum_{j=0}^{N-1} A_{11}^jA_{12}A_{22}^{N-j-1},\]
applied to
 $N=N_k$,
 \[A_{11}=I+B_{11}/N_k+O(\varepsilon^2)=e^{\frac{B_{11}}{N_k}}+O(\varepsilon^2)\] with $B_{11}=t_k (\Lambda(T_k,\sigma_k)+\mu I_n)$, and
 \[A_{22}=\Phi_D(T_k,0)+O(\varepsilon).\] Then, for $\varepsilon$ small enough one gets that
\begin{eqnarray}
\|A_{11}^j\|&\le& \Big(1+\frac{2\|B_{11}\|}j\Big)^j\le e^{2\|B_{11}\|},\quad 1\le j\le N_k,\label{eq:A11-1}\\
A_{11}^{N_k}&=&e^{B_{11}}+O(\varepsilon).\nonumber
\end{eqnarray}
Moreover, since $\rho(A_{22})<1$ for $\varepsilon$ small, it follows that
\begin{equation}\label{eq:A22}
\|A_{22}^j\|\leq K\lambda^j,\quad 1\leq j\leq N_k,
\end{equation}
for some $K>0$ and $\lambda\in (0,1)$ independent of $\varepsilon$ small enough and $k\in\{1,\dots\ell\}$. In particular, \[\|A_{22}^{N_k}\|\leq K\lambda^{\frac{t_k}{\varepsilon T_k}-1}=O(\varepsilon).\]
Using now \eqref{eq:A11-1}, \eqref{eq:A22}, and recalling that $K$ can be taken so that $\|A_{12}\|\leq K\varepsilon$, one deduces that
\begin{align*}
\|W\|\le \sum_{j=0}^{N_k-1}\|A_{11}^j\|\|A_{12}\| \|A_{22}^{N_k-j-1}\|
\le K^2e^{2\|B_{11}\|}\varepsilon \sum_{j=0}^{N_k-1} \lambda^{N_k-1-j}=C\varepsilon,
\end{align*}
for some $C>0$ independent of $\varepsilon$ small enough and $k\in\{1,\dots\ell\}$. This concludes the proof of \eqref{eq:puissanceN1}.

We next use \eqref{Qe} and \eqref{eq:puissanceN1} in \eqref{eq:Mk} to deduce that
\begin{equation}\label{eq:good-Mk}
\mathscr{M}_k(\varepsilon)^{N_k}=\begin{pmatrix}e^{ t_k (\Lambda(T_k,\sigma_k)+\mu I_n)}&0\\
r_k & 0\end{pmatrix}+O(\varepsilon),
\end{equation}
where \[r_k=\Big(I_m-\Phi_{D_{\sigma_k}}(T_k,0)\Big)^{-1}\Lambda_0(T_k,\sigma_k)e^{t_k(\Lambda(T_k,\sigma_k)+\mu I_n)}.\]

Set $t_{\varepsilon}=\varepsilon\sum_{k=1}^\ell N_kT_k$
 and notice that $t_{\varepsilon}$ tends to $t_1+\dots+t_\ell$ as $\varepsilon$ tends to zero.
 We concatenate the $N_k$ times repetitions of $\sigma_k(\cdot/\varepsilon)$ for $k=1,\dots,\ell$ to get a signal on $[0,t_{\varepsilon}]$ and the corresponding flow of $\Sigma_{\varepsilon,\mu}$ at time $t_{\varepsilon}$ is given by the matrix product
\[\Upsilon_{\varepsilon}=\mathscr{M}_\ell(\varepsilon)^{N_\ell}\mathscr{M}_{\ell-1}(\varepsilon)^{N_{\ell-1}}\cdots \mathscr{M}_{1}(\varepsilon)^{N_{1}}.\]
Using \eqref{eq:good-Mk}, one deduces that
\[
\Upsilon_{\varepsilon}=\begin{pmatrix}
e^{t_\ell(\Lambda(T_\ell,\sigma_\ell)+\mu I_n)} \cdots e^{t_1(\Lambda(T_1,\sigma_1)+\mu I_n)}&0\\
r & 0\end{pmatrix}+O(\varepsilon),
\]
where the matrix $r$ does not depend on $\varepsilon$.

Using \eqref{eq:bigger1}, one gets that  $\rho(\Upsilon_{\varepsilon})>1$ for $\varepsilon$ small enough, yielding that
\[\lambda(\Sigma_\varepsilon)+\mu=\lambda(\Sigma_{\varepsilon,\mu})>0.
\]
 In particular $\liminf_{\varepsilon\to 0^+}\lambda(\Sigma_\varepsilon)+\mu\ge 0$. By letting $\mu$ tend to $-\lambda(\check\Sigma)$ from above, one concludes that $\lambda(\check\Sigma) \leq \liminf_{\varepsilon\to 0^+} \lambda(\Sigma_\varepsilon)$ as desired.

We are left to show that if~$\check\Sigma$ is EU then~$\Sigma$ is $\varepsilon$-EU.
For this purpose, we take $\mu=0$ in the previous calculations and we observe that the flow $\Upsilon_{\varepsilon,t}$ of $\Sigma_\varepsilon$ at time $t\in [0,t_{\varepsilon}]$ corresponding to the signal defined above satisfies
\begin{align}
\Upsilon_{\varepsilon,t}=
\begin{pmatrix}
e^{(t-\sum_{h=1}^{k-1}t_h)\Lambda(T_k,\sigma_k)} \prod_{\ell=1}^{k-1}e^{t_{\ell}\Lambda(T_{\ell},\sigma_{\ell})}+O(\varepsilon) & O(\varepsilon)\\
r(\varepsilon,t) & q(\varepsilon,t)\end{pmatrix},\label{ups!}
\end{align}
whenever  $t\in [\sum_{h=1}^{k-1}t_h,\sum_{h=1}^{k}t_h]$, for some matrix functions $r,q$,
where the terms $O(\varepsilon)$ are uniform with respect to $t\in [0,t_{\varepsilon}]$.
The matrix $\Upsilon_{\varepsilon,t_\varepsilon}$ converges, as $\varepsilon$ goes to zero, to
\[\bar\Upsilon = \begin{pmatrix} e^{t_\ell \Lambda(T_\ell,\sigma_\ell)} \cdots e^{t_1 \Lambda(T_1,\sigma_1)} & 0\\ \bar r & 0\end{pmatrix},\]
for some matrix $\bar r$.
For $\varepsilon$ small enough we construct a trajectory $z_\varepsilon(t) = (x_\varepsilon(t),y_\varepsilon(t))$ of $\Sigma_\varepsilon$ satisfying $|z_\varepsilon(t)| \geq \hat C e^{\hat\lambda t}|z_\varepsilon(0)|>0$ for every $t\geq 0$, with $\hat\lambda \in (0,\lambda(\check\Sigma))$
and $\hat C>0$ independent of $\varepsilon$. Let {$\pi$} be the sum of the projectors on the generalized eigenspaces associated with the eigenvalues of  $\bar\Upsilon$ of modulus $\rho(e^{t_\ell \Lambda(T_\ell,\sigma_\ell)} \cdots e^{t_1 \Lambda(T_1,\sigma_1)})>1$.
Since $\Upsilon_{\varepsilon,t_\varepsilon}$ converges to $\bar\Upsilon$ as $\varepsilon$ goes to zero, by classical results (see~\cite[Theorem 5.1, Chapter II]{kato}) there exists $\pi_\varepsilon$, a sum of projectors on generalized eigenspaces of $\Upsilon_{\varepsilon,t_\varepsilon}$, satisfying $\lim_{\varepsilon\to 0}\pi_\varepsilon = \pi$, and the corresponding eigenvalues also converge. Let $v_\varepsilon$ be a possibly complex eigenvector of the restriction of $\Upsilon_{\varepsilon,t_\varepsilon}$ to the image of $\pi_\varepsilon$, associated with an eigenvalue $\alpha_\varepsilon$. If $\alpha_\varepsilon$ is real then $v_\varepsilon$ can be taken real as well, otherwise we assume without loss of generality that $v_\varepsilon$
satisfies $|\mathrm{Re}(v_\varepsilon)| = \min_{\theta\in\mathbb{R}}|\mathrm{Re}(e^{i\theta}v_\varepsilon)|$.
In particular $|\mathrm{Re}(\beta v_\varepsilon)|\geq |\beta| |\mathrm{Re}(v_\varepsilon)|$ for every $\beta\in\mathbb{C}$. Note that, for every positive integer $k$, one has that $(\Upsilon_{\varepsilon,t_\varepsilon})^k v_\varepsilon = \alpha_\varepsilon^k v_\varepsilon$ and $(\Upsilon_{\varepsilon,t_\varepsilon})^k\bar{v}_\varepsilon =\bar{\alpha}_\varepsilon^k \bar{v}_\varepsilon$ which implies $(\Upsilon_{\varepsilon,t_\varepsilon})^k \mathrm{Re}(v_\varepsilon) = \mathrm{Re}(\alpha_\varepsilon^k v_\varepsilon)$.

Consider the trajectory $z_\varepsilon(t) = (x_\varepsilon(t),y_\varepsilon(t))$ of $\Sigma_\varepsilon$ obtained applying the flow $\Upsilon_{\varepsilon,t}$ to the initial condition $z_\varepsilon(0)=\mathrm{Re}(v_\varepsilon)$ and repeating periodically after time $t_\varepsilon$. Letting $\Pi_x$ be the projection of a vector of $\mathbb{R}^{n+m}$ onto its first $n$ components, it is easy to see that $|\Pi_x v|\geq C|v|$ for every $v$ in the image of $\bar\Upsilon$, where
\[C=(1+\|\bar r\|^2\|(e^{t_\ell \Lambda(T_\ell,\sigma_\ell)} \cdots e^{t_1 \Lambda(T_1,\sigma_1)})^{-1}\|^2)^{-1/2}.\]
Hence, for every nonnegative integer $h$ and $\varepsilon$ small enough,
\begin{align}
|x_\varepsilon(ht_\varepsilon)|&= |\Pi_x z_\varepsilon(ht_\varepsilon)|\nonumber\\
&= |\Pi_x \pi_\varepsilon z_\varepsilon(ht_\varepsilon)|\nonumber\\
&\geq |\Pi_x \pi z_\varepsilon(ht_\varepsilon)| -\|\Pi_x\|\|\pi-{\pi}_\varepsilon\|\,|z_\varepsilon(ht_\varepsilon)|\nonumber\\
&\geq |\Pi_x \pi z_\varepsilon(ht_\varepsilon)| -\|\pi-\pi_\varepsilon\|\,|z_\varepsilon(ht_\varepsilon)|\nonumber\\
&\geq C|\pi z_\varepsilon(ht_\varepsilon)|-\|\pi-{\pi}_\varepsilon\|\,|z_\varepsilon(ht_\varepsilon)|\nonumber\\
&\geq C|z_\varepsilon(ht_\varepsilon)|-(1+C)\|{\pi}-{\pi}_\varepsilon\|\,|z_\varepsilon(ht_\varepsilon)|\nonumber\\
&\geq \frac{C}{2}|z_\varepsilon(ht_\varepsilon)|.\label{eq-periodic-1}
\end{align}
As $z_\varepsilon(ht_\varepsilon) = \mathrm{Re}(\alpha_\varepsilon^h v_\varepsilon)$ we also have
\begin{equation}
\label{eq-periodic-2}
|x_\varepsilon(ht_\varepsilon)|\geq \frac{C}{2}|\alpha_\varepsilon|^h |z_\varepsilon(0)|.
\end{equation}
By~\eqref{ups!}, \eqref{eq-periodic-1}, and setting $\kappa = \max_{\Lambda\in \mathcal{\check M}} \|\Lambda\|$ it follows that
\begin{align}\label{eq-periodic-3}
|x_\varepsilon(ht_\varepsilon+\tau)|&\geq |x_\varepsilon(ht_\varepsilon)|e^{-\kappa \tau}-|z_\varepsilon(ht_\varepsilon)|O(\varepsilon)\nonumber\\
&\geq  |x_\varepsilon(ht_\varepsilon)|e^{-\kappa \tau}-\frac{2}{C}|x_\varepsilon(ht_\varepsilon)|O(\varepsilon)\nonumber\\
&\geq \frac{1}{2}|x_\varepsilon(ht_\varepsilon)|e^{-\kappa \tau}
\end{align}
for every $\tau\in [0,t_\varepsilon]$. Take $\varepsilon$ small enough in such a way that $1<\underline\alpha\leq |\alpha_\varepsilon|\leq \bar\alpha$ for some positive constants $\underline\alpha,\bar\alpha$, and $ t_\varepsilon\leq \bar t$, where $\bar t= 2 \sum_{h=1}^\ell t_h$. By~\eqref{eq-periodic-2} and \eqref{eq-periodic-3} we thus get for every $t\geq 0$
\begin{align*}
|z_\varepsilon(t)|\geq |x_\varepsilon(t)|&\geq \frac{C}{4}e^{-\kappa t_\varepsilon}|\alpha_\varepsilon|^{\floor{\frac{t}{t_\varepsilon}}}|z_\varepsilon(0)|
\geq  \frac{Ce^{-\kappa t_\varepsilon}}{4|\alpha_\varepsilon|}e^{\frac{\log|\alpha_\varepsilon|}{t_\varepsilon}t}|z_\varepsilon(0)|\geq \hat Ce^{\hat\lambda t}|z_\varepsilon(0)|,
\end{align*}
where $\hat C=\frac{Ce^{-\kappa\bar t }}{4\bar\alpha}$ and
$\hat\lambda = \frac{\log\underline\alpha}{\bar t}$, which shows that $\Sigma$ is $\varepsilon$-EU.
\end{proof}

\begin{example}[$\check \Sigma$ gives sharper bounds than $\bar \Sigma$]
Consider system $\Sigma$ with
\[
\mathcal{M}=\left\{M_1=\begin{pmatrix}
    -1& 1 \\
     0& -0.1
\end{pmatrix}, M_2=\begin{pmatrix}
    -3& 0 \\
     {2}& -0.1
\end{pmatrix}\right\}.
\]
The stability of singularly perturbed planar switching systems is completely characterized in~\cite[Theorem 2]{Hachemi2011} through some necessary and sufficient conditions. Based on this characterization (cf., in particular, Item (SP5) in \cite[Theorem 2]{Hachemi2011})  the  condition
\begin{align}\label{eq:SP5}
    \Gamma(M_1,M_2)&:=\frac{1}{2}\left(\mathrm{tr}(M_1)\mathrm{tr}(M_2)-\mathrm{tr}(M_1M_2)\right)
   <-\sqrt{\det(M_1)\det(M_2)}
\end{align}
implies that $\Sigma_\varepsilon$ is EU for all $\varepsilon>0$. Condition~\eqref{eq:SP5} is satisfied in the case of this example with $\Gamma(M_1,M_2)=-0.8$ and $\det(M_1M_2)=0.03$.
Look now at systems $\bar\Sigma$ and $\check\Sigma$.
We have $\bar\mathcal{M}=\{-1, -3\}$ and then the associated system $\bar\Sigma$ is ES.
Concerning system $\check\Sigma$, let us consider the switching signal
\[\sigma(t)=\alpha(t)M_1+(1-\alpha(t))M_2
\]
associated with the 2-periodic function
\begin{equation*}
 \alpha(t)=
 \begin{cases}
    1 & t\in [0,1],  \\
     0 & t\in [1,2],
 \end{cases}
\end{equation*}
and take $T=2$. For this choice of $\sigma$ and $T$ one can easily verify that \[\Lambda(T,\sigma)=-2+100(1-e^{-0.2})^{-1}(1-e^{-0.1})^2>0.\] Then $\check\Sigma$ is EU, as illustrated in Figure~\ref{fig1}.

\begin{figure}[!ht]
    \centering
    \includegraphics[scale=0.5]{SF.pdf}
    \caption{Fast and slow variables evolution of system $\Sigma_\varepsilon$ with signal $\sigma$ and $\varepsilon=0.1$, starting from $(x_0,y_0)=(1,1)$. }
    \label{fig1}
\end{figure}

\end{example}

\section{Comparison between the Lyapunov exponents of $\Sigma$ and $\hat \Sigma$}\label{sec:hat}

\subsection{Definition and structural properties of the differential inclusion $\hat\Sigma$}
The following lemma studies the $\omega$-limit set of the dynamics $\Sigma_x$ given by~\eqref{fast}.\\
\begin{lemma}\label{independance}
Let $x\in\mathbb{R}^n $. Then, for every $y_0\in\mathbb{R}^m$ and $\sigma\in\mathcal{S}_{\mathcal{M}}$, the $\omega$-limit set $\omega_{\sigma}^{x}$ of the trajectory of $\Sigma_x$ associated with $\sigma$ and starting at $y_0$ does not depend on the initial condition $y_0$.
\end{lemma}

\begin{proof}
Let $y_0, y_1\in \mathbb{R}^{m}$ and let $y_0(\cdot),y_1(\cdot)$ be the trajectories of $\Sigma_x$ associated with $\sigma$ and starting from $y_0$ and $y_1$, respectively. Setting $z(\cdot)=y_0(\cdot)-y_1(\cdot)$, one has $\dot z(t)=D(t)z(t)$, and thanks to Assumption~\ref{UGES-fast}, there exist $c, \delta>0$ such that
\begin{equation}\label{eq: invariance 1}
|y_0(t)-y_1(t)|\leq ce^{-\delta t}|y_0-y_1|, \quad \forall\, t\geq 0.
\end{equation}
By definition of an $\omega$-limit set, one deduces at once that $\omega_\sigma^x$ does not depend on $y_0$.
 \end{proof}

Let us introduce the set valued-map $K:\mathbb{R}^n\leadsto\mathbb{R}^m$ defined by $K(x)=\overline{\bigcup_{\sigma}\omega_{\sigma}^{x}}$. We have the following proposition.\\

\begin{proposition}\label{forward invariance}
For each $x\in\mathbb{R}^n $ the set $K(x)$ is compact and forward invariant for the dynamics of $\Sigma_x$, and there exist $c, \delta>0$ such that
\begin{equation}\label{eq:invariance}
d(y_0(t),K(x))\leq ce^{-\delta t}d(y_0,K(x)), \quad \forall t\ge 0,\;y_0\in \mathbb{R}^m,
\end{equation}
for all $\sigma\in \mathcal{S}_{\mathcal{M}}$, where $y_0(\cdot)$ is the trajectory of~$\Sigma_x$ associated with $\sigma$ and starting from $y_0$.
\end{proposition}

\begin{proof}
Let $c$ and $\delta$ be as in \eqref{eq: invariance 1}.
The set  $K(x)$ is closed by definition and its boundedness follows from the fact that
for every $\sigma\in \mathcal{S}_\mathcal{M}$ the corresponding solution $y(\cdot)$ of $\Sigma_x$ with initial condition $y(0)=0$ satisfies
\[|y(t)|=\left|\int_0^t \Phi_D(t,s)C(s)x\;ds\right|
\le \frac{c|x|\max_{M\in\mathcal{M}}\|M\|}\delta,\]
for $t\geq0$. Hence $K(x)$ is compact.

Let now $\bar y\in K(x)$, i.e., $\bar y=\lim_{k\to +\infty}y^k$, where $y^k\in \omega_{\sigma_k}^{x}$, $\sigma_k\in \mathcal{S}_\mathcal{M}$. For $\tilde t> 0$ and $\tilde\sigma$,
denote by $\Phi_{x,\tilde\sigma}$ the flow of $\Sigma_x$ associated with the signal $\tilde\sigma$ and
let us prove that $\tilde y:=\Phi_{x,\tilde\sigma}(\tilde t,0)\bar y$ is in $K(x)$. For $k\geq 1$, let $r_k=|\tilde y-\Phi_{x,\tilde\sigma}(\tilde t, 0)y^k)|$ and notice that $\lim_{k\to +\infty} r_k=0$. Moreover, for $k\geq 1$, there exists $\rho_k>0$ such that
$\Phi_{x,\tilde\sigma}(\tilde t,0)B_{\rho_k}(y^k)\subset
B_{2 r_k}(\tilde y)$.
We next define recursively the sequence
$(z_{k})_{k\geq 0}$ with $z_0\in \mathbb{R}^m$ by setting
\begin{equation*}
    z_{k+1}=
    \Phi_{x,\tilde\sigma}(\tilde t,0)\Phi_{x,\sigma_{k+1}}(t_k,0)z_k,\qquad k\ge 0,
\end{equation*}
where the sequence of times $t_k$
is chosen so that $\Phi_{x,\sigma_{k+1}}(t_k,0)z_k
$ is in $B_{\rho_{k+1}}(y^{k+1})$. This is possible since $y^{k+1}\in  \omega_{\sigma_{k+1}}^{x}$ for $k\geq 0$.
By construction, $z_k$ is in $B_{2 r_{k}}(\tilde y)$ for every $k\ge 1$. Moreover, $z_k=
\Phi_{x,\bar \sigma}(\tau_k,0)z_0$
where $\tau_k\to \infty$ and $\bar \sigma$ is constructed by repeatedly concatenating $\sigma_k|_{[0,t_k]}$ and $\tilde\sigma|_{[0,\tilde t]}$.
Then $\tilde y$ is in $\omega^x_{\bar\sigma}\subset K(x)$.

Finally, using~\eqref{eq: invariance 1} with $y_1\in K(x)$ and the forward invariance of $K(x)$ for the dynamics of $\Sigma_x$, we have
\begin{equation*}
d(y_0(t),K(x))\leq |y_0(t)-y_1(t)|\leq ce^{-\delta t}|y_0-y_1|, \quad \forall\, t\geq 0,
\end{equation*}
and one gets \eqref{eq:invariance} by arbitrariness of $y_1\in K(x)$.
 \end{proof}

\begin{remark}
The contents of the above proposition are essentially
contained in the preprint \cite{DellaRossa2022} (Theorem~1 and Proposition~2), where the authors study general switching affine  systems and the role of $K(x)$ is played by the set $\mathcal{K}_\infty$.
\end{remark}

\begin{lemma}\label{lem:KLiphom}
The set-valued map $K$ is globally Lipschitz continuous
for the Hausdorff distance and homogeneous of degree one.
\end{lemma}

\begin{proof}
Notice that, given $x\in \mathbb{R}^n$ and a nonzero $\lambda\in \mathbb{R}$, $y(\cdot)$ is a trajectory of $\Sigma_x$ if and only if $\lambda y(\cdot)$ is a trajectory of $\Sigma_{\lambda x}$. We deduce that $z\in \omega_{\sigma}^x$ if and only if $\lambda z\in \omega_{\sigma}^{\lambda x}$ and hence that $K(\lambda x) = \lambda K(x)$.

As for  the Lipschitz continuity of $K$, let $x_1, x_2\in \mathbb{R}^n$. It is easy to
{deduce from the variation of constant formula} that there exists
$L_K>0$
 independent of $x_1, x_2$ such that
\begin{equation*}
|y_1(t)-y_2(t)|\leq L_K|x_1-x_2|, \qquad \forall\, t\geq 0,\forall\, \sigma\in\mathcal{S}_{\mathcal{M}},
\end{equation*}
where $y_1(\cdot)$ and $y_2(\cdot)$ are the trajectories of $\Sigma_{x_1}$ and $\Sigma_{x_2}$, respectively, associated with $\sigma$ and starting from the same initial condition $y_0$. By consequence, we have
 \begin{equation*}
\max\{d(y_1, \omega_{\sigma}^{x_2}), d(y_2, \omega_{\sigma}^{x_1})\}\leq L_K|x_1-x_2|,
\end{equation*}
for $y_1\in\,\omega_{\sigma}^{x_1}$ and  $y_2\in\,\omega_{\sigma}^{x_2}$, where $\omega_{\sigma}^{x_1}$ and $\omega_{\sigma}^{x_2}$ are the $\omega$-limit sets of $\Sigma_{x_1}$ and $\Sigma_{x_2}$, respectively, which, thanks to Lemma~\ref{independance}, do not depend on the initial condition $y_0$. From the previous inequality  we obtain that,
 \begin{equation*
\max\{d(y_1, \cup_{\sigma}\omega_{\sigma}^{x_2}), d(y_2, \cup_{\sigma}\omega_{\sigma}^{x_1})\}\leq L_K|x_1-x_2|,
\end{equation*}
for $y_1\in\,\cup_{\sigma}\omega_{\sigma}^{x_1}$ and  $y_2\in\,\cup_{\sigma}\omega_{\sigma}^{x_2}$.
By a standard density argument,
 \begin{equation*}\label{eq2:lipschitz}
\max\{d(y_1, K(x_2)
), d(y_2,K(x_1)
)\}
\leq L_K|x_1-x_2|,
\end{equation*}
for $y_1\in K(x_1)$ and $y_2\in K(x_2)$,
from which we obtain, by the arbitrariness of $y_1$ and $y_2$, that
\begin{equation*}
d_{H}(K(x_1), K(x_2))\leq L_K|x_1-x_2|,
\end{equation*}
where
we recall that
$d_{H}$ denotes the Hausdorff distance in $\mathbb{R}^m$.
\end{proof}

\subsection{

Asymptotic estimates by converse Lyapunov arguments
}}
The argument provided below bears similarities with proofs given in \cite{WATBLED2005362}, where more general dynamics are considered.

We consider the
$\mu$-shifted differential inclusion
\[\hat\Sigma_{\mu}: \quad \dot{\hat x}\in
\{A_\mu\hat x+B y\mid M=\left(\begin{smallmatrix}A & B\\mathcal{C} & D\end{smallmatrix}\right)
\in \mathcal{M},\; y\in K(\hat x)\},
\]
where
$A_{\mu}=A+\mu I_{n}$ with $\mu\in\mathbb{R}$.
By homogeneity of $K(\cdot)$ it follows that for every solution $\hat x(\cdot)$ of $\hat\Sigma$ the trajectory $t\mapsto \hat{x}_\mu(t) = e^{\mu t}\hat{x}(t)$ is a solution of $\hat\Sigma_{\mu}$. As a consequence $\lambda(\hat\Sigma_{\mu}) = \lambda(\hat\Sigma)+\mu$.
Hence, recalling that $\lambda(\Sigma_{\varepsilon,\mu}) = \lambda(\Sigma_\varepsilon)+\mu$,
in order to prove the right-hand side of inequality~\eqref{eigenvalues} it is enough to show that, under Assumption~\ref{UGES-fast}, $\lambda(\hat\Sigma_{\mu})<0$ implies $\lambda(\Sigma_{\varepsilon,\mu})<0$ for $\varepsilon$ small enough.
In order to prove the latter statement we will construct a common Lyapunov function for the systems $\Sigma_{\varepsilon,\mu}$ as the sum of Lyapunov functions for $\hat\Sigma_{\mu}$ and for $\Sigma_x$ defined next.
Assume that $\lambda({\hat\Sigma_\mu})<0$. Then, by Lemma~\ref{lem:ES-DI}, $|\hat x(t)|\leq ce^{-\gamma t}|\hat x(0)|$ for some $c\geq 1$ and $\gamma>0$, for every trajectory $\hat x(\cdot)$ of $\hat\Sigma_{\mu}$.
Define
\begin{equation}\label{eq:V1}
    V_1(x) = \sup_{\hat x(\cdot), t\in [0,\hat t\,]}e^{\gamma t}|\hat x(t)|^2,
\end{equation}
where the supremum is computed among all trajectories $\hat x(\cdot)$  of $\hat\Sigma_{\mu}$ starting from $x\in \mathbb{R}^n$, and $\hat t=\frac{\log(c)}{2\gamma}$.

Furthermore, let $\delta\in (0,|\lambda(\Sigma_D)|)$ and $\bar c\ge 1$ be such that $|y(t)|\leq \bar c e^{-\delta t}|y(0)|$ for every trajectory $y(\cdot)$ of $\Sigma_D$. By Proposition~\ref{forward invariance} one has
\[d(y_x(t),K(x))\leq \bar c e^{-\delta t}d(y_x(0),K(x))\] for every trajectory $y_x$ of $\Sigma_x$, for every $x\in\mathbb{R}^n$ and $t>0$. Define
\begin{equation}\label{eq:V2}
    V_2(x,y) = \sup_{y_x(\cdot), t\in [0,\bar t\,]}e^{\delta t}d(y_x(t),K(x))^2,
\end{equation}
where the supremum is computed among all trajectories $y_x(\cdot)$ of $\Sigma_x$ starting from $y\in \mathbb{R}^m$  and $\bar t = \frac{\log(\bar c)}{2\delta}$.

In the next lemma, we summarize the main properties of $V_1$ and $V_2$.
\begin{lemma}\label{lem:V1V2}
The positive definite functions $V_1$ and $V_2$ introduced
in \eqref{eq:V1} and \eqref{eq:V2} are homogeneous of degree two and locally Lipschitz continuous. Moreover, $V_1$ and $V_2$  are nonincreasing along every trajectory of $\hat\Sigma_\mu$ and $\Sigma_x$ respectively and satisfy the following estimates:
\begin{align
V_1(\hat x(t))&\leq e^{-\gamma t} V_1(\hat x(0)),\label{eq:LyapV1}\\
V_2(x,\bar{y}_x( t))&\leq e^{-\delta t} V_2(x,\bar{y}_x(0)),\label{eq:LyapV2}
\end{align}
where $t\geq 0$, $x\in \mathbb{R}^n$, $\hat x(\cdot)$ is an arbitrary trajectory of $\hat\Sigma_\mu$, and $\bar{y}_x(\cdot)$ is an arbitrary trajectory of $\Sigma_x$.
\end{lemma}
\begin{proof}
It is clear that both $V_1$ and $V_2$ are homogeneous of degree two. We give the proof of the remaining properties for $V_2$, the corresponding arguments for $V_1$ being completely analogous.

Let us next show that $V_2$ is locally Lipschitz continuous.
For every bounded set $\mathcal{B}\subset \mathbb{R}^m$ there exists a compact set of $\mathbb{R}^m$ containing every trajectory of $\Sigma_x$ starting from $\mathcal{B}$. Take two points $y_1,y_2$ in $\mathcal{B}$ and consider the trajectories $y^1_x,y^2_{x}$ of $\Sigma_x$ corresponding to the same switching law starting respectively from $y_1$ and $y_2$. Then  $|y^2_x(t)-y^1_x(t)|\leq \bar c e^{-\delta t}|y_2-y_1|$  for every $t\geq 0$. We deduce that the function $y\mapsto e^{\delta t}d(y_x(t),K(x))^2$, where $y_x$ is the trajectory starting from $y\in \mathcal{B}$ corresponding to a fixed switching law $\sigma\in \mathcal{S}_\mathcal{M}$, is Lipschitz continuous, and the Lipschitz constant does not depend on $t$ nor on the switching law. Since the supremum among a family of uniformly Lipschitz continuous functions is Lipschitz continuous, we deduce that $V_2$ is locally Lipschitz continuous in the variable $y$.
Similarly, local Lipschitz continuity  of the map $x\mapsto e^{\delta t}d(y_x(t),K(x))^2$ for a fixed switching law follows from the fact that $y_x(\cdot)$ is affine with respect to the variable $x$ and $K(\cdot)$ is Lipschitz continuous. Furthermore, the corresponding Lipschitz constant (locally) does not depend on $t$ nor on the switching law.

Consider now a trajectory $\bar{y}_x(\cdot)$ of $\Sigma_x$ and let us prove that $V_2(x,\cdot)$ is nonincreasing along it. For $h>0$ one has
\begin{align*}
V_2(x,\bar{y}_x( h)) & =  \sup_{\substack{y_x(\cdot),\ y_x|_{[0, h]}=\bar y_x|_{[0, h]}\\ t\in [ h,\bar t+ h]}}e^{\delta (t- h)}d(y_x(t),K(x))^2\\
& \leq  \sup_{\substack{y_x(\cdot),\ y_x(0)=\bar{y}_x(0)\\t\in [ h,\bar t+ h]}}e^{\delta (t- h)}d(y_x(t),K(x))^2\\
& \leq  \sup_{\substack{y_x(\cdot),\ y_x(0)=\bar{y}_x(0)\\t\in [0,\bar t+ h]}}e^{\delta (t- h)}d(y_x(t),K(x))^2\\
& =  e^{-\delta h}\sup_{\substack{y_x(\cdot),\ y_x(0)=\bar{y}_x(0)\\t\in [0,\bar t]}}e^{\delta t}d(y_x(t),K(x))^2\\
& = e^{-\delta h} V_2(x,\bar{y}_x(0)),
\end{align*}
where  we have used the fact that
\begin{align*}
e^{\delta t}d(y_x(t),K(x))^2
&\leq \bar c^2 e^{-\delta t}d(y_x(0),K(x))^2<d(y_x(0),K(x))^2
\end{align*}
for every $t>\bar t$ along every trajectory of $\Sigma_x$.
\end{proof}

 Based on the above construction of $V_1$ and $V_2$, we  next show the existence of a common Lyapunov function allowing us to prove that  $(\Sigma_{\varepsilon,\mu})_{\varepsilon>0}$ is $\varepsilon$-ES. \\
\begin{proposition}
There exists $\chi>0$, $0 < \alpha_- < \alpha_+$, $\eta>0$, and $\varepsilon_*>0$ such that, setting $V = V_1+\chi V_2$, one has
\begin{align}
 \alpha_-|(x,y)|^2 \leq V(x,y) \leq \alpha_+|(x,y)|^2\label{Lyap-1},\qquad &\forall x\in \mathbb{R}^n,\;\forall y\in \mathbb{R}^m,\\
 V(x(t),y(t))\leq V(x(0),y(0)) e^{-\eta t},\qquad&\forall t\ge 0,\label{Lyap-2}
\end{align}
where~\eqref{Lyap-2} holds true for every solution $(x(\cdot),y(\cdot))$  of $\Sigma_{\varepsilon,\mu}$ for $\varepsilon<\varepsilon_*$. As a consequence $\limsup_{\varepsilon\to 0^+} \lambda(\Sigma_{\varepsilon}) \leq \lambda(\hat\Sigma)$ and if $\lambda(\hat\Sigma)<0$  then $\Sigma$ is $\varepsilon$-ES.
\end{proposition}

\begin{proof}
The right inequality in~\eqref{Lyap-1} follows from the bounds
\begin{align*}
V_1(x)\leq c^2 |x|^2
\end{align*}
and
\begin{align*}
V_2(x,y)\leq \bar c^2\, d(y,K(x))^2&\leq \bar c^2\, (|y|+d(0,K(x)))^2
\leq \bar c^2(|y|+L_K|x|)^2,
\end{align*}
where $L_K$ is the Lipschitz constant for $K(\cdot)$. Concerning the left inequality in~\eqref{Lyap-1}, note that $d(y,K(x))\geq |y|-L_K|x|$, where $L_K$ is the Lipschitz constant for $K(\cdot)$. Then, either $|y|> 2 L_K|x|$, in which case $d(y,K(x))> \frac12 |y|$ and
\[V(x,y)\geq |x|^2+\chi d(y,K(x))^2>|x|^2+\frac{\chi}4|y|^2, \]
or $|y|\leq 2 L_K|x|$, in which case
\[V(x,y)\geq |x|^2\geq \frac{1}{1+4L_K^2} |(x,y)|^2.\]
The desired inequality holds true with \[\alpha_-=\min\left\{\frac{1}{1+4L_K^2},\frac{\chi}4\right\}.\]
Consider now a trajectory $(x(\cdot),y(\cdot))$  of $\Sigma_{\varepsilon,\mu}$ corresponding to the switching law $\sigma$. In order to prove~\eqref{Lyap-2} we first estimate the difference $V(x(\varepsilon h),y(\varepsilon h))-V(x(0),y(0))$ for small $h$.
Note that
\begin{align*}
&|x(\varepsilon h)-x(0)|=\varepsilon|(x(0),y(0))|\, O(h),\\
&|y(\varepsilon h)-y(0)|=|(x(0),y(0))|\,O(h),
\end{align*}
where $O(h)$ denotes a function bounded in absolute value by $Ch$,
where the constant $C$ does not depend on $x(0)$, $y(0)$, $h$, nor $\varepsilon$, for $\varepsilon,h$ in a small right-neighborhood of zero.
Knowing that \[t\mapstod(\dot x(t),A_{\mu}(t)x(t)+B(t)K(x(t)))\leq \|B\|d(y(t),K(x(t)))\] is integrable, we deduce from
Theorem~10.4.1 in \cit
{Aubin-Frankowska} that
there exists a solution $\tilde x(\cdot)$ of $\hat\Sigma_{\mu}$
such that $\tilde x(0)=x(0)$ and
\begin{align*}\label{eq:paolo}
|x(\varepsilon h)-\tilde x(\varepsilon h)|
&\leq \|B\|e^{(\|A_{\mu}\|+L_K\|B\|)\varepsilon h}\int_0^{\varepsilon h}{d(y(s),K(x(s)))ds}\\
& \leq C_1\varepsilon h \,d(y(0),K(x(0)))+\varepsilon|(x(0),y(0))| \,O(h^2),
\nonumber
\end{align*}
for some $C_1>0$, where we have used the fact that
\[d(p,P)\leq |p-q| + d(q,Q) + d_H(Q,P), \]
for $p,q\in \mathbb{R}^m$ and $P,Q\subset \mathbb{R}^m$.
Hence
\begin{align*}
V_1(x(\varepsilon h)) - V_1(\tilde x( h))\leq& \varepsilon |(x(0),y(0))|\,d(y(0),K(x(0)))\,O( h)
+\varepsilon |(x(0),y(0))|^2\,O( h^2),
\end{align*}
as it follows from the fact that the Lipschitz constant of $V_1$ on a ball of radius $r$ is of order $r$.
By using~\eqref{eq:LyapV1},
\begin{align*}
V_1(x(\varepsilon h)) - V_1(x(0))
&=V_1(x(\varepsilon h)) - V_1(\tilde x(\varepsilon h)) + V_1(\tilde x(\varepsilon h)) -  V_1(x(0))\\
&\leq V_1(x(0))(e^{-\gamma \varepsilon h}-1)+\varepsilon|(x(0),y(0))|^2\,O( h^2) +\varepsilon |(x(0),y(0))|\,d(y(0),K(x(0)))\,O( h).
\end{align*}
Moreover,
\begin{align*}
V_2(x(\varepsilon h),y(\varepsilon h))-V_2(x(0),y(0))
&\leq  |V_2(x(\varepsilon h),y(\varepsilon h))- V_2(x(0),y(\varepsilon h))|
+ |V_2(x(0),y(\varepsilon h))-V_2(x(0),\tilde y(\varepsilon h))|\\
&+ V_2(x(0),\tilde y(\varepsilon h)) -V_2(x(0),y(0)),
\end{align*}
where $\tilde y({\cdot}/{\varepsilon})$ is the solution of $\Sigma_{x(0)}$ starting at $y(0)$
and corresponding to the signal $\sigma({\cdot}/{\varepsilon})$.
The first term is of order  $\varepsilon |(x(0),y(0))|^2 O(h)$, while, by~\eqref{eq:LyapV2}, \[V_2(x(0),\tilde y(\varepsilon h)) -V_2(x(0),y(0))\leq (e^{-\delta h}-1)V_2(x(0),y(0)).\] Furthermore
\begin{align*}
\frac{d}{dt}(y(t)-\tilde y(t)) &= \frac1{\varepsilon}D(t)(y(t)-\tilde y(t))
+ \frac1{\varepsilon}C(t)(x(t)-x(0))+\mu y(t),
\end{align*}
from which one gets that \[|y(\varepsilon h)-\tilde y(\varepsilon h)| =\varepsilon |(x(0),y(0))|\,O( h),\] so that
\[
|V_2(x(0),y(\varepsilon h))-V_2(x(0),\tilde y(\varepsilon h))|=\varepsilon|(x(0),y(0))|^2\,O( h).
\]
Summing up,
\begin{align*}
&V_2(x(\varepsilon h),y(\varepsilon h))-V_2(x(0),y(0))
\leq (e^{-\delta h}-1)V_2(x(0),y(0)) + \varepsilon|(x(0),y(0))|^2\,O(h)
\end{align*}
and
\begin{align*}
V(x(\varepsilon h),y(\varepsilon h))  -V(x(0),y(0))
&\leq  (e^{-\gamma \varepsilon h}-1)V_1(x(0)) + \chi(e^{-\delta h}-1)V_2(x(0),y(0))\\
&+ C_2 \varepsilon h|(x(0),y(0))|\,d(y(0),K(x(0)))  \\
&+ C_3 \chi \varepsilon h|(x(0),y(0))|^2+ C_4 \varepsilon h^2|(x(0),y(0))|^2,
\end{align*}
where  $C_2,C_3,C_4$ do not depend on $x(0)$, $y(0)$, $h$, nor $\varepsilon$, for $h,\varepsilon$ in a small right-neighborhood of zero.
We have
\begin{align*}
\limsup_{h\to 0^+}\frac{V(x(\varepsilon h),y(\varepsilon h))  -V(x(0),y(0))}{h}
&\leq -\gamma\varepsilon V_1(x(0)) -  \delta\chi V_2(x(0),y(0))
+ C_2 \varepsilon |(x(0),y(0))|\,d(y(0),K(x(0)))\\
& + C_3 \chi \varepsilon |(x(0),y(0))|^2.
\end{align*}
We write the right-hand side as the sum of the three terms

\begin{align*}
W_1 &= -\frac{\gamma\varepsilon}2 (V_1(x(0)) + \chi V_2(x(0),y(0)))
=-\frac{\gamma\varepsilon}2 V(x(0),y(0)),\\
W_2 &= -\frac{\chi}4 (2\delta-\gamma\varepsilon)V_2(x(0),y(0)) - C_3  \chi \varepsilon |(x(0),y(0))|^2 -C_2\varepsilon |(x(0),y(0))|\,d(y(0),K(x(0))) \\
&\leq  -\frac{\chi}4 (2\delta-\gamma\varepsilon)d(y(0),K(x(0))^2 - C_3  \chi \varepsilon |(x(0),y(0))|^2 -C_2\varepsilon |(x(0),y(0))|\,d(y(0),K(x(0))), \\
W_3 &= -\frac{\gamma\varepsilon}2 V(x(0)) -\frac{\chi}4 (2\delta-\gamma\varepsilon)V_2(x(0),y(0)) + 2C_3  \chi \varepsilon |(x(0),y(0))|^2\\
&\leq -\frac{\gamma\varepsilon}2 |x(0)|^2 -\frac{\chi}4 (2\delta-\gamma\varepsilon)d(y(0),K(x(0))^2 + 2C_3  \chi \varepsilon |(x(0),y(0))|^2,
\end{align*}
where the inequalities in $W_2,W_3$ are obtained assuming $\varepsilon\leq \frac{2\delta}{\gamma}$.
For any given $\chi>0$, it is easy to see that $W_2\leq 0$ if $\varepsilon$ is small enough.
Since \[|y(0)|\leq L_K|x(0)|+d(y(0),K(x(0)))\] we get
\[|(x(0),y(0))|^2 \leq (1+2L_K^2)|x(0)|^2 + 2d(y(0),K(x(0)))^2\] so that $W_3\leq 0$, provided that $\chi$ is chosen so that
\[-\frac{\gamma}2 + 2C_3  \chi (1+2L_K^2) \leq 0\]
and $\varepsilon$ is small enough.
By a time-shift we obtain, for $t\geq 0$,
\begin{align*}
    \limsup_{\tau\to 0^+}&\frac{V(x(t+\tau),y(t+\tau))  -V(x(t),y(t))}{\tau}
\leq -\frac{\gamma}2 V(x(t),y(t)).
\end{align*}
Since $V(x(\cdot),y(\cdot))$ is absolutely continuous we deduce that
\[\frac{d}{dt}V(x(t),y(t)) \leq -\frac{\gamma}{2} V(x(t),y(t)), \quad \hbox{a.e. }t\geq 0,\]
and
\[V(x(t),y(t))\leq V(x(0),y(0)) e^{-\frac{\gamma}{2}t}, \quad \forall\, t\geq 0,\] concluding the proof.
\end{proof}

redpajama_set_name: RedPajamaArXiv


The Minnesota Department of Health and Hennepin County Public Health are investigating an outbreak of measles in Hennepin County children. For current information on the number of cases, go to the MDH Measles website (www.health.state.mn.us/divs/idepc/diseases/measles/).
All confirmed cases are currently Somali Minnesotan children living in Hennepin County. For current information on who is affected, go to the MDH Measles website (www.health.state.mn.us/divs/idepc/diseases/measles/).
Is this problem specific to the Somali community?
This outbreak is about unvaccinated children, not specific communities or ethnic groups. Measles is a concern anywhere you have significant numbers of unvaccinated children, such as ECFE groups, schools, etc. Unfortunately, Minnesota's Somali communities have been targeted with misinformation about vaccine risks and have subsequently struggled with low rates of MMR immunization.
Measles is a highly contagious disease caused by a virus. It can lead to hospitalization and in rare cases death. Measles spreads through the air by coughing or sneezing. You can get measles by just being in the same room as someone who has measles.
Symptoms include a high fever, cough, runny nose and watery eyes, followed by a rash that typically spreads from head to the rest of the body. A person with measles can pass it to others from four days before their rash appears to four days after it appears.
Most people in Minnesota can't get sick from measles because they have been vaccinated or have had measles before. However, in recent years MMR vaccination rates have declined in some communities and groups – often due to fears related to misinformation about vaccine risks. This decline in vaccination rates increases the risk – not only for those who choose not to be vaccinated, but also for those who cannot be vaccinated because of their health status (for example, a weakened immune system) or because they are too young.
Measles can spread very easily among unvaccinated people, so the best way to protect yourself and your community is to make sure everyone who is able receives the measles, mumps and rubella (MMR) vaccine.
How can I check that my children and I have been vaccinated?
Contact your health care provider to confirm that you and your children have received the MMR vaccine. You can also request vaccination records by calling 651-201-3980 or visiting MIIC Immunization Records Requests (www.health.state.mn.us/divs/idepc/immunize/registry/immrecords.html).
How common is measles in Minnesota and the U.S.?
Minnesota typically sees a handful of measles cases every year. Cases that show up are typically linked to international travel.
What should I do if I think someone I know has measles?
People at highest risk of getting measles are those who have not been vaccinated. If a family member or loved one has symptoms of measles, call your doctor or clinic. Keep the ill person at home and avoid having visitors until you have talked with your doctor or clinic. The doctor or clinic will tell you if you should come in for a visit.
What should I do if my family member or I have been exposed to measles?
State and county health departments are working to identify all places where people could have been exposed to measles. They are contacting parents of children who were exposed and are at risk of getting measles to provide them with instructions and steps to take to lower their chances of getting sick. People at highest risk for getting measles and with a higher risk for complications may be able to get immune globulin, which is a type of medicine given as a shot that has antibodies that can fight the measles virus.
How did this outbreak start?
Minnesota Department of Health disease investigators are still looking into the origin of this outbreak. Measles outbreaks typically start with a person becoming infected during international travel. When they return they can spread it to others who are not vaccinated or who have not had measles in the past.
What are you doing to help the community?
Stopping an outbreak means preventing sick people from passing their infection to others who are at risk. We're contacting people who may have been exposed to measles to get them treatment and keep them from passing the virus along to others.
We're also working to make sure as many people as possible are vaccinated. This includes informing people about the measles risk and partnering with Somali community leaders and health care providers to counteract vaccine misinformation.
What is your advice to parents who are concerned about measles?
All Minnesota children 12 months and older who have not received a measles, mumps and rubella (MMR) vaccine should get it now. Babies younger than 12 months may have some protection from their mothers if their mothers have been vaccinated or have had measles.
MMR vaccine is given to children in two doses. The first dose offers good protection, and the second dose provides extra security.
Due to the higher risk of measles infection for Somali Minnesotan children in the current measles outbreak, MDH recommends that all Somali Minnesotan children statewide who have already received their first dose of MMR vaccine get their second dose now. This special vaccine schedule is commonly recommended during outbreaks. Parents of Somali Minnesotan children should contact their child's health care provider and specifically tell them the child needs the MMR vaccine. This may help avoid a longer wait associated with scheduling a routine appointment.
Children in Hennepin County can also get an early second dose of the MMR vaccine.
Vaccine recommendations may expand if the measles outbreak spreads to more communities.
What about parents who are concerned about vaccine costs?
For parents concerned about the cost of immunizations, the Minnesota Vaccines for Children Program provides free or low-cost vaccines for eligible children through age 18. More information is available atCan My Child Get Free or Low Cost Shots? (www.health.state.mn.us/divs/idepc/immunize/howpay.html).
For more information on measles and for updates as the investigation continues, please visit the Minnesota Department of Health website at www.health.state.mn.us.


text:
Biografia
Specialista delle prove tecniche originario di Modane, Blanc vinse il titolo nazionale francese nella combinata nel 1979 e nel 1981 e gareggiò in Coppa del Mondo almeno fino al 1983, senza ottenere risultati di rilievo; non prese parte a rassegne olimpiche né ottenne piazzamenti ai Campionati mondiali.

Palmarès

Campionati francesi

Note

redpajama_set_name: RedPajamaWikipedia


text:
Diamond, now part of the Agilico group, teed off and raised more than £1,500 for two local charities Tyneside and Northumberland Mind and Chrysalis Tynedale.
On Friday 16th July, a total of 52 golfers, including a combination of staff from Diamond and other local businesses took to the sunny greens at Wearside Golf Club for their fourth charity golf day supporting mental health and dementia support across the region.
There was a full programme for the day, starting with bacon rolls to ward off the hunger pains at 8.30am. In total there were 13 teams competing in the tournament which had a positive and competitive atmosphere. The day culminated in a celebratory two course lunch at the clubhouse where prizes were presented for the best score, along with a fundraising raffle, Mulligan and Blind Betting.
John Burns, Managing Director, Diamond said: "We are always keen to do whatever we can to support charities, especially given the difficulties of the last 18 months. As the coronavirus pandemic continues to have an impact on all aspects of our lives, particularly on our mental health and wellbeing. We were delighted to be able to support Tyneside and Northumberland MIND and Chrysalis Tynedale at this year's event".
Alex Wilson, CEO, Tyneside and Northumberland Mind said: "I would like to thank John and the team for this amazing effort and all the teams and individuals who supported the day. Without help like this we cannot deliver our work in mental health which is needed now more so than ever across our region.
Rosie Robson, Manager, Chrysalis Tynedale said: "We are very grateful to John for giving Chrysalis the opportunity to promote our charity and for the generous donations. As a small local charity, we do not have a big marketing budget and bearing in mind the negative impact lockdown has had on people with dementia and their carers, we are particularly grateful at this time."
Diamond should be congratulated for organising such a fantastic event and for raising such a significant amount for two well-deserved charities that do great work across our region.

redpajama_set_name: RedPajamaCommonCrawl


text:
TURKISH MILITARY BRACES FOR NEW BATTLE OVER CONSTITUTION
Publication: Eurasia Daily Monitor Volume: 4 Issue: 162
By: Gareth Jenkins
September 4, 2007 04:00 AM Age: 15 years
Following its failure to prevent the election of Abdullah Gul as Turkey's president, the Turkish military is now regrouping and preparing for the next battle in what is likely to be a long campaign to preserve the ideological legacy of Mustafa Kemal Ataturk, who founded the Turkish Republic in 1923 and whose core principles, such as secularism and territorial integrity, are enshrined in the current Turkish constitution.
The Turkish parliament finally elected Gul as president on August 28, four months after the military intervened to block his first attempt and force the government to call an early general election. In the third round of voting Gul secured the support of 339 of the 550 members of parliament, well past the 267 required. However, in a sign of their displeasure at his appointment, all of the Turkish high command boycotted his swearing-in ceremony (Milliyet, Zaman, August 29). When the military hosted a reception in Ankara on August 30 to mark Victory Day in the Turkish War of Independence, it refused to invited Gul's headscarfed wife. General Yasar Buyukanit, chief of the Turkish General Staff (TGS), broke with tradition by failing to accompany Gul to his car at the end of the reception (Hurriyet, Radikal, August 31). Perhaps more significantly, in speeches at ceremonies at other military facilities across the country, several officers warned that they were committed to protecting secularism — at the cost of their lives, if necessary (Milliyet, August 31).
There is an awareness in the TGS that Gul's election as president was a defeat and that the military's failure to prevent it has damaged its public prestige. But the military's attention is now likely to shift away from the presidency and back to the Justice and Development Party (AKP) government. The TGS is bracing itself to respond if it believes Ataturk's values are under threat.
On September 2, Prime Minister Recep Tayyip Erdogan read out his new government's program in parliament. But the program contained a series of vague declarations of intent rather than any specific proposals (Radikal, September 3). Once the government has received its vote of confidence on September 5, parliament will go into recess until the end of the month, further reducing the possibility of an immediate confrontation between the AKP and the TGS.
Even though it has lost the battle over the presidency, the TGS will now focus on the much more important issue of the AKP's plans for a new constitution to replace the one introduced in 1982 during a period of military rule. For the last few months a group of AKP deputies and legal experts have been working on the draft of the new constitution. In late July, one of the group's members, a professor of constitutional law and AKP member of parliament called Zafer Uskul, provoked a furious reaction from Turkish secularists when he announced that he was opposed to including a commitment to Ataturk's principles in the constitution (Milliyet, July 30).
Uskul's statements were later disowned by the AKP, which described them as merely an expression of personal opinion (NTV, July 30). However, the text of the new constitution has yet to be made public, amid numerous press reports of serious disagreements among the experts responsible for drafting it.
On September 3, AKP Deputy Chairman Dengir Mir Mehmet Firat held a press conference to announce that the group had completed an initial draft, which he described as a "civilian constitution prepared by the people" (CNNTurk, September 3). But he refused to share the draft text with the Turkish people, merely promising that it would be made public within the next week.
One of the main issues of contention is an article in the current constitution that makes religious education compulsory in all schools. The article was introduced by the military junta that drafted the constitution in an attempt to create an ideological bulwark against communism. In practice, even though Turkey is theoretically a secular state, the provisions in the constitution have been used to teach Sunni Islam, which has infuriated Turkey's 10 million-strong Alevi community.
Few in the TGS expect the AKP to issue a direct challenge by removing the references to Ataturk from the new draft constitution. However, there is concern that hardliners within the AKP may try to present themselves as liberals by omitting the article on compulsory religious education while including provisions that would ease restrictions on the teaching of religion outside the school system. This could weaken the state's control over how religion is taught and could result in the inculcation of a hard-line interpretation of Islam.
SCO SUMMIT FOCUSES ON ENERGY COOPERATIONNext
SHORTFALLS IN RUSSIAN OIL DELIVERIES TO ..


>> Google loves Social Bookmarks coming from domains that belong to high level of authority.
>> Imagine the amount of Social Bookmarks juice from authority sites and your site will be highly benefit from this.
>> We all know how important of Social Bookmarks are to everyone's website.
>> If you are looking to gain relevant and trusted Social Bookmarks with high authority then you should definitively buy this.100% natural Penguin, panda and hummingbird safe SEO Social Bookmarks.
>> Your can provide your own title(up to 3) and description(up to 3) also.
>> Full Links Reports will be Provided(without login).
20+ High Authority PR4-9 Social Bookmarks only is ranked 5 out of 5. Based on 269 user reviews.


The voting process is easy. Simply submit your yes or no response below and leave a comment if you'd like. It is vitally important that we get as many votes as possible. Voting for the Board of Directors is your right and responsibility of professional membership. As such, please note that only votes from professional members in good standing will be counted. Votes will be accepted until Saturday April 29. We look forward to seeing you at the ASNY Annual Meeting for those who can attend, and as always, we are happy to hear from you!


Fleet Feet Sports Sun Prairie is about living an active, healthy lifestyle. We will provide the highest quality customer service and products to our customers. Whether you stand all day at work or are training for a marathon, we offer a variety of shoes, inserts, apparel and training programs to enhance your experience.


It's great news that you get high quality offers from your advertisers, but how to promote them to your affiliates can be a headache. We are glad to announce that all this can be cured by Offerslook Email Marketing feature.
Click Email Marketing->>Mailbox->>Start New Email to begin a new email campaign.
There are three types of recipients, Affiliates,Advertiser, Employees. And they are further divided into more categories.
You can also customize the recipients as you want.
You can customize the sender name, subject, banner and mail content. In the mail content, some short codes are provided so that your recipients can receive personalized emails instead of bulk emails.
To promote specific offers, you can also add them to the email at the bottom page.
After finishing your email content, you can choose to preview or send the email to recipients directly.
You can also contact your Offerslook account manager for more details.


text:
Q: Call a class method in an after_save That is may be a stupid point, but I don't find the solution.
I have a simple model with a class method update_menu and I want that it is called after each save of instance.
Class Category
  attr_accessible :name, :content

  def self.menu
     @@menu ||= update_menu
  end

  def self.update_menu
     @@menu = Category.all
  end
end

So what is the correct syntax to get the after_filter call update_menu?
I tried:
after_save :update_menu

But it looks for the method on the instance (which does not exist) and not on the class.
Thanks for your answers.

A: after_save :update_menu

def updated_menu
  self.class.update_menu
end

this will call the class update_menu method

A: Make it an instance method by removing self.
# now an instance method
def update_menu
   @@menu = Category.all
end

It doesn't make much sense to have an after_save callback on a class method. Classes aren't saved, instances are. For example:
# I'm assuming the code you typed in has typos since
# it should inherit from ActiveRecord::Base
class Category < ActiveRecord::Base
  attr_accessible :name
end

category_one = Category.new(:name => 'category one')
category_one.save  # saving an instance

Category.save # this wont work

A: after_save 'self.class.update_menu'

Rails will evaluate a symbol as an instance method.  In order to call a class method, you have to pass a string which will be evaluated in the correct context.
NOTE:  This only works with rails 4.  See Erez answer for Rails 5.

redpajama_set_name: RedPajamaStackExchange


If you're looking for a seasoned attorney with a knack for litigating complex cases, Bill is the man to see. Since 1975, he has worked tirelessly to provide his clients with the aggressive, effective representation they need — and his dedication has not gone unnoticed. Today, Bill is highly respected within the Idaho legal community.
Joe received his Juris Doctor degree in 2006 but was working in law firms long before that time. Today, he focuses his practice primarily on getting compensation for victims of injury, standing up for those who have been wrongfully terminated, and representing people facing criminal charges.
Having practiced law since 1988, you'd be hard-pressed to find another attorney who can handle complex business law matters as effectively and efficiently as Boyd. From contract and license agreements to intellectual property disputes (and everything in between), Boyd is a go-to resource for employers and employees.
The newest member of our team, Johannes has more than 14 years of experience as an attorney. His primary focuses are criminal defense, family law, environmental law and government procurement. He is an accomplished advocate for his clients both in and out of trial.


Android Smartphones are really amazing but might be annoying sometimes as a simple swipe might wipe out vital information on your the device like the WhatsApp messages and history. Today we will see how to recover deleted WhatsApp messages.
Whatsapp makes backups automatically each day around 2am in the morning to the phones internal memory. This allows users to retrieve old messages that are deleted, i.e messages sent or received before the backup time. This is an advantage on recovering deleted WhatsApp messages and we don't need to worry about losing WhatsApp chat histories.
In addition to backing up data offline to internal memory, data backups can be set to be stored online to the Google Drive platform automatically. Similar to offline backup, we can restore deleted messages, but in an easier way with this backup method.
To do this, we just need to re-install WhatsApp to restore data stored in Google Drive or on the device internal storage. On startup of WhatsApp after reinstalling, it will automatically offer an option to recover the old data stored on Google's Drive or search for backup on internal storage. Apart from these two ways of automatically making message Backups. WhatsApp users can also create WhatsApp backups manually, without the need for backup.
To access the WhatsApp backup option, friends can open the WhatsApp app, then click the three dot icon (the other menu icon) at the top right corner, click on Settings, select Chats, and then go to the Chats Backup menu. We can set up a data backup from the settings menu.
If you mistakenly deleted a WhatsApp message you can retrieve it using data recovery using Google Drive if it's set. But if not, the following below reviews how to read or return messages that have been deleted in the application WhatsApp manually.
3. WhatsApp data backup in that folder has format msgstore-YYYY-MM-DD.1.db.crypt ** with YYYY is year, MM is the month, and DD is the date. Copy the data you want to restore according to the backup date to another folder.
After a reinstalling successfully, you will be directed to recover the backup data already stored. Tap Recover to restore data.
Viola, our message has been recovered. But, if the message does not return, it can happen because we choose the wrong data. In addition, you also have to be careful because after reinstall it could be the data we mean even come missing.


This 90 town car headlight wiring diagram image have been published. You are able to obtain this amazing image to your laptop, netbook or desktop computer. You also can easily save these pages to you favourite bookmarking sites. Ways to acquire this 90 town car headlight wiring diagram image? It is simple, you can utilize the save link or place your cursor to the graphic and right click then choose save as.
90 town car headlight wiring diagram is among the most photos we discovered on the internet from reliable sources. We tend to talk about this 90 town car headlight wiring diagram image on this page because based on information coming from Google engine, It is one of many best queries keyword on the internet. And we also think you came here were searching for this information, are not You? From many choices online we are sure this photo might be a best guide for you, and we sincerely hope you are delighted by what we present.
Were very thankful if you leave a comment or reviews about this 90 town car headlight wiring diagram article. We are going to apply it for much better future articles. We thank you for your visit to our website. Make sure you get the information you are looking for. Do not forget to share and love our reference to help further develop our website.