# Asymptotic stability conditions for linear coupled impulsive systems with
time-invariant subsystems

Vitalii Slynko<EMAIL_ADDRESS>Sergey Dashkovskiy
<EMAIL_ADDRESS>Ivan Atamas
<EMAIL_ADDRESS>97074 Emil-Fischer-Str. 40, Würzburg

###### Abstract

This article proposes an approach to construct a Lyapunov function for a
linear coupled impulsive system consisting of two time-invariant subsystems.
In contrast to various variants of small-gain stability conditions for coupled
systems, the asymptotic stability property of independent subsystems is not
assumed. To analyze the asymptotic stability of a coupled system, the direct
Lyapunov method is used in combination with the discretization method. The
periodic case and the case when the Floquet theory is not applicable are
considered separately. The main results are illustrated with examples.

###### keywords:

Linear impulsive systems , Lyapunov functions, Lyapunov stability , coupled
systems , time-variant systems.

††journal: Journal of LaTeX Templatesmytitlenotemytitlenotefootnotetext: This
research was supported by the German Research Foundation (DFG) via grants No.
DA 767/12-1 and SL 343/1-1, and by the German academic exchange service
(DAAD), personal ref. no.: 91775148.

##  1 Introduction

Impulsive differential equations can model mechanical systems subjected to
shocks. Instantaneous change in holonomic or nonholonomic constraints imposed
on the system and changes in parameters of the system in time lead to study
impulsive systems with variable coefficients. In this case, it is important to
obtain stability conditions that are robust with respect to variations in the
sequence of moments of impulse action.

The theory of impulsive differential equations emerged as an independent
direction in the modern differential equations theory and system theory,
starting with the classic book [1]. The main methods for studying stability
for systems of differential equations with impulsive action are laid down in
[1, 2]. In modern control theory, impulsive systems are often considered as an
important subclass of hybrid systems [3, 4]. The stability of linear impulsive
systems of differential equations with constant parameters has been a subject
of research in many works. In contrast to linear time-invariant systems of
ordinary differential equations, where the stability problem is exhaustively
solved by the classical Routh-Hurwitz theorem, the stability question is open
in the general case. This is due to the fact that the dynamics of impulsive
systems is determined not only by the parameters of the system, but also by
the sequence of moments of impulse action. It should also be taken into
account that an impulsive system can be asymptotically stable even in the case
when the continuous and discrete dynamics are both unstable. Therefore, the
stability conditions for linear impulsive systems should cover this case as
well, see [5, 6, 7, 8, 9, 10, 11]. In [10, 11], linear impulsive differential
equations in Banach spaces were considered under the assumption that the
moments of impulse action satisfy the ADT condition, assuming that the
continuous and discrete dynamics of the system are both unstable. Using the
identities of the commutator calculus, conditions for asymptotic stability are
obtained. In [7, 8, 12], the dwell-time conditions which guarantee the
asymptotic stability of linear impulsive systems with constant parameters are
obtained. In this case, a construction of a Lyapunov function is reduced to an
approximate solution of a two-point boundary value problem with boundary
conditions in the form of matrix inequalities. A generalization of these
results for some classes of linear impulsive systems with variable
coefficients is given in [8]. In [6], the dwell-times estimates that guarantee
the asymptotic stability of linear impulsive systems are obtained based on the
second Lyapunov method using the derivatives of the second [9] and higher
orders [6] of a Lyapunov function.

The papers [5, 9, 13, 14, 15] are devoted to the study of stability of large-
scale impulsive systems. In [5, 9, 14], Lyapunov vector functions are used to
study the asymptotic stability of the equilibrium of nonlinear impulsive
systems. In [13], the problem of stability of critical equilibria of nonlinear
large-scale impulsive systems is considered. In [15], the input to state
stability theory (ISS) is developed for nonlinear impulsive systems and
sufficient conditions for the global asymptotic stability of nonlinear coupled
impulsive systems are established (small-gain theorems). It should be noted
that all of these results use the assumption of asymptotic stability of
independent subsystems. Hence, it is of interest to find stability conditions
for large-scale (coupled) systems which do not assume a priori the presence of
the asymptotic stability property of independent subsystems. One of the
possible approaches to solve this problem is to use the construction of a
matrix-valued Lyapunov function [16, mart1]. The question of the method of
construction is decisive for the practical application of matrix-valued
Lyapunov functions in stability problems. In [17], for linear time-variant
coupled systems with time-invariant subsystems, a method of construction of a
matrix-valued Lyapunov function by some heuristic simplifications of a matrix
differential equation written in the block form is proposed. Although, this
method allows in some cases to obtain a conclusion about the stability of
linear coupled systems with unstable subsystems, the question about the degree
of conservatism of the obtained sufficient stability conditions is open. For
linear large-scale impulsive systems with variable coefficients, the problem
of choosing the elements of a matrix-valued Lyapunov function has not been
studied.

Note that for time-discrete coupled systems, some approaches to study Lyapunov
stability and ISS in case when some of the independent subsystems do not have
the exponential stability property or, respectively, the ISS property are
presented in [18, 19, 20]. For continuous-time and impulsive coupled systems
with possibly unstable subsystems, the problem of construction of a Lyapunov
function remains open even in the linear case of time-variant systems.

The development of computer calculations has opened up new possibilities to
solve the problem of construction of Lyapunov functions for various classes of
dynamical systems. In recent years, the use of the discretization method to
construct approximate solutions of a Lyapunov matrix differential equations
(or their modifications) has led to significant advances in the theory of
stability of linear hybrid systems with constant parameters [21], systems with
delay [22] and others. The application of the discretization method of
construction of Lyapunov functions makes it possible to obtain with high
accuracy estimates of the dwell-times that guarantee the stability of linear
hybrid systems or the conditions of robust stability. In [23], the
discretization method is used to synthesize robust control of a nonlinear
affine system. The asymptotic stability of a large-scale system, using the
direct Lyapunov method in combination with the discretization method and
identities of the commutator calculus were considered in [24]. Here the
independent subsystems may not have the asymptotic stability property. In
contrast to [24], we consider a coupled impulsive system by substantially
modifying the choice of a candidate of a Lyapunov function using the time-
invariance of (disconnected) independent subsystems.

In our paper, for the first time, it is proposed to apply the discretization
method in order to construct matrix-valued Lyapunov functions for linear
impulsive systems with periodic coefficients. In this case, it is assumed that
the independent subsystems are time-invariant, and for the dwell-times, two
possible cases are considered: they are constant or subject to two-sided
estimates. The elements of a matrix-valued Lyapunov function are constructed
in the bilinear forms with time-variable matrices. The proposed algorithm of
construction of Lyapunov functions admits a simple numerical implementation.

Contributions of this manuscript are as follows. First of all, a new method
for construction of a Lyapunov function for a linear time-variant system
consisting of two coupled time-invariant subsystems is proposed. Conditions
for the asymptotic stability of a linear time-variant system are obtained
under various assumptions about the dwell-times and dynamic properties of
independent subsystems. We show that the proposed approach is applicable in
the case when one of the subsystems is not stable and the classical methods
for studying of coupled systems, which are based on the ideas of Lyapunov
vector functions or small-gain theorems, obtained on the basis of the ISS
concept, are not applicable. In the case when subsystems are asymptotically
stable, but the small-gain conditions are not satisfy, the proposed approach
can still work and leads to less conservative stability conditions. The
proposed results are new not only in the context of the theory of impulsive
systems, but also for coupled periodic ODEs.

We introduce examples where the obtained sufficient stability conditions are
applicable under different assumptions regarding the continuous and discrete
dynamics of the system, in particular, when both dynamics are simultaneously
unstable and the application of known results of the theory of stability of
impulsive systems is impossible or very difficult.

Next section describes the problem statement. Section 3 is devoted to an
informal discussion of the proposed method of construction of a matrix-valued
Lyapunov function. In the fourth section, this method is rigorously justified
for the case of linear impulsive periodic systems. Sufficient conditions for
asymptotic stability are obtained. Section 5 is devoted to the substantiation
of the proposed algorithm of construction of a matrix-valued Lyapunov function
in the case when the dwell-times are not constant (in this case, the Floquet
theory is not applicable). In Section 6 we consider particular cases of
rapidly changing interaction between subsystems. We compare our results with
known small-gain theorems. Section 7 provides numerical examples that are
discussed in Section 8.

Notation. Let $\mathbb{R}^{n}$ be the Euclidian space with standard dot
product and $\mathbb{R}^{n\times m}$ be the linear space of $n\times m$
matrices. For $A\in\mathbb{R}^{n\times n}$, $\sigma(A)$ denotes its spectrum,
$r_{\sigma}(A)$ denotes its spectral radius and norm
$\|A\|=\lambda_{\max}^{1/2}(A^{\operatorname{T\,}}A)$. If
$\sigma(A)\subset\mathbb{R}$, then $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$
are its smallest and largest eigenvalues respectively,
$\lambda_{\max}^{+}(A)=\max(\lambda_{\max}(A),0)$. For any symmetric matrices
$P$ and $Q$ notation $P\geq Q$ means that $P-Q$ is a positive semidefinite
matrix and $P\succ Q$ means that $P-Q$ is a positive definite matrix. We will
also use the Cauchy–Bunyakovsky inequality
$|x^{\operatorname{T\,}}y|\leq\|x\|\|y\|$ for $x,y\in\mathbb{R}^{n}$.

##  2 Statement of the problem

Consider a coupled linear system of impulsive differential equations

$\begin{gathered}\dot{x}_{1}(t)=A_{11}x_{1}(t)+A_{12}(t)x_{2}(t),\quad
t\neq\tau_{k}\\\ \dot{x}_{2}(t)=A_{21}(t)x_{1}(t)+A_{22}x_{2}(t),\quad
t\neq\tau_{k},\\\ x_{1}(t^{+})=B_{11}x_{1}(t)+B_{12}x_{2}(t),\quad
t=\tau_{k},\\\ x_{2}(t^{+})=B_{21}x_{1}(t)+B_{22}x_{2}(t),\quad
t=\tau_{k}\end{gathered}$ (1)

where $x_{i}\in\mathbb{R}^{n_{i}}$, $i=1,2$,
$A_{ij}\,:\mathbb{R}\to\mathbb{R}^{n_{i}\times n_{j}}$ are piece-wise
continuous functions, $i,j=1,2$. Suppose that $A_{ii}$ are constant,
$A_{ij}(t)$, $i\neq j$ are $\theta$-periodic, i.e.,
$A_{ij}(t+\theta)=A_{ij}(t)$ for all $t\in\mathbb{R}$,
$\\{\tau_{k}\\}_{k=0}^{\infty}$ is a sequence of moments of impulse action,
and $B_{ij}\in\mathbb{R}^{n_{i}\times n_{j}}$ are constant. We denote
$x=(x_{1}^{\operatorname{T\,}},x_{2}^{\operatorname{T\,}})^{\operatorname{T\,}}$
and $n=n_{1}+n_{2}$. For $\\{\tau_{k}\\}_{k=0}^{\infty}$, we assume that there
are positive constants $\theta_{1}$ and $\theta_{2}$ such that the dwell time
$T_{k}=\tau_{k}-\tau_{k-1}$, $k\geq 1$ satisfy $\theta_{1}\leq
T_{k}\leq\theta_{2}$.

Coupled systems of the form (1) naturally arise in the process of mathematical
modeling of the dynamics of coupled impulsive time-independent systems that
exchange information with each other. We consider the case when the coupling
functions between subsystems change in time, and independent subsystems are
not subject to parametric disturbances and are described by linear systems
with constant parameters. Classical approaches to the study of the problem of
stability of coupled systems based on a Lyapunov vector function or the
concept of ISS a priori assume the property of asymptotic stability of
independent subsystems. Therefore, it is of interest to get rid of this a
priori assumption, which is caused not by the essence of the problem, but by
the restrictions of existing methods for studying stability.

The results presented below can be extended easily to the case of an arbitrary
number of independent subsystems. Here we restrict ourselves to the case of
two subsystems in order to make the presentation more accessible without
overshadowing it with technical details. We study a class of linear non-
autonomous systems that allow decomposition into subsystems with time-
invariant independent subsystems. The construction of a Lyapunov function for
this class can be significantly simplified due to decomposition, in comparison
with the direct study of the system (1) without resorting to decomposition.

Note that there exist real constants $\mu_{i}$, $\delta_{i}$, $M_{i}$,
$N_{i}$, $i=1,2$, such that

$\begin{gathered}\|e^{sA_{ii}}\|\leq
M_{i}e^{s\mu_{i}},\quad\|e^{-sA_{ii}}\|\leq N_{i}e^{s\delta_{i}},\quad s\geq
0.\end{gathered}$

Remark 4.1. There are several ways to obtain the estimate $\|e^{tA}\|\leq
Me^{\mu t}$, $t\geq 0$, $A\in\mathbb{R}^{n\times n}$. Here we use a result
from [25], where one can find the estimate

$\begin{gathered}\|e^{tA}\|\leq
e^{\beta_{A}t}\sum\limits_{k=0}^{n-1}\frac{g^{k}_{A}t^{k}}{(k!)^{3/2}},\end{gathered}$
(2)

where $\beta_{A}=\max\\{\operatorname{Re\,}\lambda\,|\lambda\in\sigma(A)\\}$,
$g_{A}=\sqrt{\operatorname{tr\,}(AA^{\operatorname{T\,}})-|\operatorname{tr\,}A^{2}|}$.
From (2), we can derive the estimates we need as follows.

If $g_{A}=0$, $M=1$ and $\mu=\beta_{A}$ then from (2) immediately implies the
estimates we need. Let $g_{A}\neq 0$ and for a given $\epsilon>0$, we denote

$\begin{gathered}M_{\epsilon,A}:=\sup\limits_{t\geq 0}e^{-\epsilon
t}\sum\limits_{k=0}^{n-1}\frac{g^{k}_{A}t^{k}}{(k!)^{3/2}}<\infty,\end{gathered}$

then

$\begin{gathered}\|e^{tA}\|\leq M_{\epsilon,A}e^{(\beta_{A}+\epsilon)t},\quad
t\geq 0.\end{gathered}$

Since the functions $A_{ij}(t)$ are assumed to be bounded and periodic, then
for some positive constants $\gamma_{12}^{(m)}$, $\gamma_{21}^{(m)}$,
$m=0,1,\dots,N-1$ it holds that

$\begin{gathered}\sup\limits_{s\in(mh,(m+1)h]}\|A_{12}(s)\|\leq\gamma_{12}^{(m)},\quad\sup\limits_{s\in(mh,(m+1)h]}\|A_{21}(s)\|\leq\gamma_{21}^{(m)}.\end{gathered}$

For any $m\in\mathbb{Z}$, let $\gamma_{ij}^{(m)}:=\gamma_{ij}^{(\varrho)}$,
where $\varrho$ is the remainder of $m$ devided by $N$. By this we extend the
definition of constants $\gamma_{ij}^{(m)}$ for any $m\in\mathbb{Z}$. Here we
study the asymptotic stability of (1) in the sense of:

Definition 2.1. System of differential equations (1) is called

1) stable if for any $\varepsilon>0$, $t_{0}\in\mathbb{R}$ there exists
$\delta=\delta(\varepsilon,t_{0})>0$ such that the inequality
$\|x_{0}\|<\delta$ implies that $\|x(t,t_{0},x_{0})\|<\varepsilon$ for all
$t\geq t_{0}$;

2) uniformly stable if for any $\varepsilon>0$ there exists
$\delta=\delta(\varepsilon)>0$ such that for all $t_{0}\in\mathbb{R}$ the
inequality $\|x_{0}\|<\delta$ implies $\|x(t,t_{0},x_{0})\|<\varepsilon$ for
$t\geq t_{0}$;

3) asymptotically stable if it is stable and
$\lim\limits_{t\to+\infty}\|x(t,t_{0},x_{0})\|=0$.

Here, $x(t,t_{0},x_{0})$ is the solution of the Cauchy problem (1) with the
initial condition $x(t_{0},t_{0},x_{0})=x_{0}$,
$x_{0}=(x_{10}^{\operatorname{T\,}},x_{20}^{\operatorname{T\,}})^{\operatorname{T\,}}\in\mathbb{R}^{n}$.

The aim of this work is to construct a Lyapunov function for (1) and to prove
sufficient conditions for its stability. The case of the periodic system (1),
when $\theta_{1}=\theta_{2}=\theta$ and the general case, when
$\theta_{1}<\theta_{2}$ are considered separately. If the linear impulsive
system (1) is not periodic, the Floquet theory is not applicable.

##  3 General idea of construction of a matrix-valued Lyapunov function

Here we consider the case $\theta_{1}=\theta_{2}$ and present the general
idea, how we will derive a Lyapunov function for (1). Without any restriction
we assume that $t_{0}=0$. Let
$\mathfrak{V}(t,x)=(v_{ij}(t,\cdot,\cdot))_{i,j=1,2}$ be a matrix-valued
Lyapunov function (MFL) [16, 26]. We choose
$v_{ij}(t,x_{i},x_{j})=x_{i}^{\operatorname{T\,}}P_{ij}(t)x_{j}$, where
$P_{ij}\,:\mathbb{R}\to\mathbb{R}^{n_{i}\times n_{j}}$ are continuous on the
left $\theta$-periodic maps, $P_{ij}(t)=P_{ji}^{\operatorname{T\,}}(t)$. It is
enough to define $P_{ij}(t)$, $i,j=1,2$ on the period $(0,\theta]$. For the
practical application of this rather general design, we provide a method of
construction of $P_{ij}(t)$.

From $\mathfrak{V}$ we proceed to the following scalar Lyapunov function

$\begin{gathered}v(t,x_{1},x_{2})=(1,1)\mathfrak{V}(t,x_{1},x_{2})(1,1)^{\operatorname{T\,}}=v_{11}(t,x_{1})+2v_{12}(t,x_{1},x_{2})+v_{22}(t,x_{2})\\\
=(x_{1}^{\operatorname{T\,}},x_{2}^{\operatorname{T\,}})P(t)(x_{1}^{\operatorname{T\,}},x_{2}^{\operatorname{T\,}})^{\operatorname{T\,}},\end{gathered}$

where $P(t)=(P_{ij}(t))_{i,j=1,2}$ is a block matrix
$P_{ij}(t)=P_{ji}^{\operatorname{T\,}}(t)$ to be designed.

Let matrix $P(t)$ satisfy the condition

$\begin{gathered}\dot{P}(t)+A^{\operatorname{T\,}}(t)P(t)+P(t)A(t)=0,\quad
t\in(k\theta,(k+1)\theta),\quad k\in\mathbb{Z}_{+},\end{gathered}$ (3)

with the block matrix $A(t)=(A_{ij}(t))_{i,j=1,2}$. It is easy to show that
for $t=k\theta$ the following estimate holds

$\begin{gathered}v(k\theta+0,x_{1}(k\theta+0),x_{2}(k\theta+0))=x^{\operatorname{T\,}}(k\theta+0)P(k\theta+0)x(k\theta+0)\\\
=(Bx(k\theta))^{\operatorname{T\,}}P_{0}Bx(k\theta)=x^{\operatorname{T\,}}(k\theta)B^{\operatorname{T\,}}P_{0}Bx(k\theta)\leq\lambda
v(k\theta,x_{1}(k\theta),x_{2}(k\theta)),\end{gathered}$

where $\lambda=\lambda_{\max}(B^{\operatorname{T\,}}P_{0}B(P(\theta))^{-1})$,
$B=(B_{ij})_{i,j=1,2}$ and $P_{0}=P(0+0)$ is a symmetric positive definite
matrix. From (3) it follows that $v(t,x_{1},x_{2})$ satisfies the following
system of impulsive differential inequalities

$\begin{gathered}\dot{v}(t,x_{1}(t),x_{2}(t))=0,\quad t\neq k\theta,\\\
v(t+0,x_{1}(t+0),x_{2}(t+0))\leq\lambda v(t,x_{1}(t),x_{2}(t)),\quad
t=k\theta.\end{gathered}$

By means of the comparison principle [2] the stability investigation of (1)
reduces to the stability question of

$\begin{gathered}\dot{u}(t)=0,\quad t\neq k\theta,\\\ u(t+0)=\lambda
u(t),\quad t=k\theta.\end{gathered}$ (4)

The asymptotic stability of (4) implies the same property for (1). The
condition $\lambda<1$ (which restricts $P_{0}\succ 0$) is necessary and
sufficient for the asymptotic stability of (4) and is equivalent to the
necessary and sufficient condition for the asymptotic stability of (1), as can
be seen from the Floquet-Lyapunov theorem. Resolving (3) on the interval
$(k\theta,(k+1)\theta)$ is equivalent to the calculation of the monodromy
matrix $\Phi$ for (1), i.e., the problem is comparable in complexity to the
integration of (1). The main idea of construction of the matrix-valued
Lyapunov function is to construct an approximate solution to (3) written in
the block form:

$\begin{gathered}\dot{P}_{11}(t)+A_{11}^{\operatorname{T\,}}P_{11}(t)+P_{11}(t)A_{11}=-(P_{12}(t)A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}(t)),\\\
\dot{P}_{22}(t)+A_{22}^{\operatorname{T\,}}P_{22}(t)+P_{22}(t)A_{22}=-(A_{12}^{\operatorname{T\,}}(t)P_{12}(t)+P_{21}(t)A_{12}(t)),\\\
\dot{P}_{12}(t)+A_{11}^{\operatorname{T\,}}P_{12}(t)+P_{12}(t)A_{22}=-(P_{11}(t)A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}(t)).\end{gathered}$
(5)

Since $P(t)$ is $\theta$-periodic, we need to construct a solution to (5) on
the interval $(0,\theta]$. Let $P_{ij}(0+0)=P_{ij}^{(0)}$, $i,j=1,2$,
$P_{ij}^{(0)}=(P_{ji}^{(0)})^{\operatorname{T\,}}$, where
$(P_{ij}^{(0)})_{i,j=1,2}$ is a positive definite block matrix. We use the
discretization method to resolve (5). For $N\in\mathbb{N}$ let
$h=\frac{\theta}{N}$ be the discretization step. We derive an approximate
solution to (5) step by step on the intervals $(mh,(m+1)h]$, $m=0,\dots,N-1$.
By the Cauchy formula applied to the interval $(mh,(m+1)h]$ we obtain integral
representations for the solutions of (5)

$\begin{gathered}P_{11}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-mh)}P_{11}^{(m)}e^{-A_{11}(t-mh)}\\\
-\int\limits_{mh}^{t}e^{-A_{11}^{\operatorname{T\,}}(t-s)}(P_{12}(s)A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}(s))e^{-A_{11}(t-s)}\,ds,\\\
\end{gathered}$
$\begin{gathered}P_{22}(t)=e^{-A_{22}^{\operatorname{T\,}}(t-mh)}P_{22}^{(m)}e^{-A_{22}(t-mh)}\\\
-\int\limits_{mh}^{t}e^{-A_{22}^{\operatorname{T\,}}(t-s)}(A_{12}^{\operatorname{T\,}}(s)P_{12}(s)+P_{21}(s)A_{12}(s))e^{-A_{22}(t-s)}\,ds,\\\
\end{gathered}$
$\begin{gathered}P_{12}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-mh)}P_{12}^{(m)}e^{-A_{22}(t-mh)}\\\
-\int\limits_{mh}^{t}e^{-A_{11}^{\operatorname{T\,}}(t-s)}(P_{11}(s)A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}(s))e^{-A_{22}^{\operatorname{T\,}}(t-s)}\,ds.\end{gathered}$

Here, $P_{ij}^{(m)}=P_{ij}(mh-0)$, $m=1,\dots,N-1$, $i,j=1,2$. For $h$ small
enough we use the approximation $P_{ij}(s)\approx P_{ij}^{(m)}$ in the
integrals which leads to the approximate solutions to (5) for
$t\in(mh,(m+1)h]$ as follows

$\begin{gathered}P_{11}(t)\approx
e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}^{(m)}-\int\limits_{mh}^{t}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,ds)e^{-A_{11}(t-mh)},\end{gathered}$
$\begin{gathered}P_{22}(t)\approx
e^{-A_{22}^{\operatorname{T\,}}(t-mh)}(P_{22}^{(m)}-\int\limits_{mh}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,ds)e^{-A_{22}(t-mh)},\end{gathered}$
$\begin{gathered}P_{12}(t)\approx
e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}^{(m)}-\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds)e^{-A_{22}(t-mh)}.\end{gathered}$

Note that for $h\to 0+$ these approximations converge to the true solutions to
(5). Since the asymptotic stability conditions derived with help of
$\mathfrak{V}(t,x_{1},x_{2})$ are necessary and sufficient, the matrix-valued
Lyapunov function, whose elements are given by these approximations leads to
sufficient asymptotic stability conditions for (1) arbitrarily close to the
necessary ones.

##  4 The case when the system is periodic

###  4.1 Construction of the matrix-valued Lyapunov function

We proceed to a rigorous justification of the proposed method of construction
in the case when the dwell-times are constant and equal to the period $\theta$
of $A_{ij}(t)$. We introduce discretization parameters: the number of nodes
$N\in\mathbb{N}$ and the discretization step length $h=\frac{\theta}{N}$. Let
$P_{0}=(P_{ij}^{(0)})_{i,j=1,2}$ be a positive definite symmetric block
matrix, $P_{ij}^{(0)}\in\mathbb{R}^{n_{i}\times n_{j}}$,
$P_{ij}^{(0)}=(P_{ji}^{(0)})^{\operatorname{T\,}}$. We define recursively the
matrices $P_{ij}^{(m)},\quad
P_{ji}^{(m)}=(P_{ij}^{(m)})^{\operatorname{T\,}},\quad i,j=1,2$ as follows

$\begin{gathered}P_{11}^{(m+1)}=e^{-A_{11}^{\operatorname{T\,}}h}(P_{11}^{(m)}-\int\limits_{mh}^{(m+1)h}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,ds)e^{-A_{11}h},\end{gathered}$
(6)
$\begin{gathered}P_{22}^{(m+1)}=e^{-A_{22}^{\operatorname{T\,}}h}(P_{22}^{(m)}-\int\limits_{mh}^{(m+1)h}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,ds)e^{-A_{22}h}\end{gathered}$
(7)
$\begin{gathered}P_{12}^{(m+1)}=e^{-A_{11}^{\operatorname{T\,}}h}(P_{12}^{(m)}-\int\limits_{mh}^{(m+1)h}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds)e^{-A_{22}h}.\end{gathered}$
(8)

Next, define the matrices $P_{ij}(t)$, $i,j=1,2$,
$P_{ij}(t)=P_{ji}^{\operatorname{T\,}}(t)$ on the intervals $(mh,(m+1)h]$ by
setting

$\begin{gathered}P_{11}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}^{(m)}-\int\limits_{mh}^{t}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,ds)e^{-A_{11}(t-mh)},\end{gathered}$
(9)
$\begin{gathered}P_{22}(t)=e^{-A_{22}^{\operatorname{T\,}}(t-mh)}(P_{22}^{(m)}-\int\limits_{mh}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,ds)e^{-A_{22}(t-mh)}\end{gathered}$
(10)
$\begin{gathered}P_{12}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}^{(m)}-\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds)e^{-A_{22}(t-mh)}.\end{gathered}$
(11)

We define $\mathfrak{V}(t,x_{1},x_{2})=(v_{ij}(t,.,.))_{i,j=1,2}$, where
$\,v_{ij}(t,x_{i},x_{j})=x_{i}^{\operatorname{T\,}}P_{ij}(t)x_{j}$,
$P_{ij}(t)=P_{ji}^{\operatorname{T\,}}(t)$, $i,j=1,2$. Using the function
$\mathfrak{V}$, we construct the scalar Lyapunov function [16]

$\begin{gathered}v(t,x_{1},x_{2})=v_{11}(t,x_{1})+2v_{12}(t,x_{1},x_{2})+v_{22}(t,x_{2}).\end{gathered}$
(12)

We establish some auxiliary estimates for the derivatives of the components of
$\mathfrak{V}$ along the solutions of (1) and estimates for the Lyapunov
function $v(t,x_{1},x_{2})$, necessary for construction of a comparison linear
impulsive differential equation and obtaining sufficient conditions for the
asymptotic stability of (1). We will also use the following proposition.

Proposition 4.1 Let $M,\mu>0$ be such that $\|e^{tA}\|\leq Me^{t\mu}$, $t\geq
0$. Then

$\begin{gathered}\|e^{tA}-I\|\leq\frac{\|A\|M}{\mu}(e^{t\mu}-1),\quad t\geq
0.\end{gathered}$ (13)

Proof. Let $X(t)=e^{tA}-I$, then $X(0)=0$ and $\dot{X}(t)=A(X(t)+I)$, applying
the Cauchy formula, we obtain $X(t)=\int\limits_{0}^{t}e^{A(t-s)}A\,ds$, which
implies (13).

The following assertion is necessary to verify the positive-definiteness
conditions for the proposed Lyapunov function $v(t,x_{1},x_{2})$. Since this
function depends explicitly on time $t$, it is impossible to check this
condition pointwise for all $t\in[0,\theta]$. The following Lemma 4.1 reduces
this checking to a finite number of conditions.

Lemma 4.1. Let $z_{1m}=e^{-A_{11}(t-mh)}x_{1}$,
$z_{2m}=e^{-A_{22}(t-mh)}x_{2}$, $t\in(mh,(m+1)h]$. Then,

$\begin{gathered}\lambda_{\min}(\Pi_{m})\|z_{m}\|^{2}\leq
v(t,x_{1},x_{2})\leq\lambda_{\max}(\Xi_{m})\|z_{m}\|^{2},\\\ \text{for
all}\quad t\in(mh,(m+1)h],\quad m=0,\dots,N-1,\end{gathered}$ (14)

where
$z_{m}=(z_{1m}^{\operatorname{T\,}},z_{2m}^{\operatorname{T\,}})^{\operatorname{T\,}}$,
$\|z_{m}\|^{2}=\|z_{1m}\|^{2}+\|z_{2m}\|^{2}$,
$\Pi_{m}=(\pi_{ij}^{(m)})_{i,j=1,2}$, $\Xi_{m}=(\xi_{ij}^{(m)})_{i,j=1,2}$ are
block matrices with the elements

$\begin{gathered}\pi_{11}^{(m)}=P_{11}^{(m)}-h(2\gamma_{21}^{(m)}\|P_{12}^{(m)}\|+(\gamma_{12}^{(m)}\|P_{11}^{(m)}\|+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|))I_{n_{1}},\\\
\pi_{22}^{(m)}=P_{22}^{(m)}-h(2\gamma_{12}^{(m)}\|P_{12}^{(m)}\|+(\gamma_{12}^{(m)}\|P_{11}^{(m)}\|+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|))I_{n_{2}},\\\
\pi_{12}^{(m)}=P_{12}^{(m)},\quad\pi_{21}^{(m)}=P_{21}^{(m)}\end{gathered}$
(15)
$\begin{gathered}\xi_{11}^{(m)}=P_{11}^{(m)}+h(2\gamma_{21}^{(m)}\|P_{12}^{(m)}\|+(\gamma_{12}^{(m)}\|P_{11}^{(m)}\|+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|))I_{n_{1}},\\\
\xi_{22}^{(m)}=P_{22}^{(m)}+h(2\gamma_{12}^{(m)}\|P_{12}^{(m)}\|+(\gamma_{12}^{(m)}\|P_{11}^{(m)}\|+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|))I_{n_{2}},\\\
\xi_{12}^{(m)}=P_{12}^{(m)},\quad\xi_{21}^{(m)}=P_{21}^{(m)}.\end{gathered}$

The proof of this statement is given in the Appendix.

Let

$\begin{gathered}\eta_{11}^{(m)}:=\sqrt{\|P_{11}^{(m)}\|^{2}+\|P_{12}^{(m)}\|^{2}}+\|P_{12}^{(m)}\|,\\\
\eta_{22}^{(m)}:=\sqrt{\|P_{22}^{(m)}\|^{2}+\|P_{12}^{(m)}\|^{2}}+\|P_{12}^{(m)}\|\end{gathered}$
$\begin{gathered}\eta_{12}^{(m)}:=\frac{1}{2}(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)}\\\
+\sqrt{(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})^{2}+16(\gamma_{21}^{(m)})^{2}\|P_{12}^{(m)}\|^{2}}).\end{gathered}$
$\begin{gathered}\eta_{21}^{(m)}:=\frac{1}{2}(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)}\\\
+\sqrt{(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})^{2}+16(\gamma_{12}^{(m)})^{2}\|P_{12}^{(m)}\|^{2}}).\end{gathered}$

We denote

$\begin{gathered}\alpha_{11}^{(m)}:=\gamma_{12}^{(m)}\|A_{22}\|N_{1}M_{2}\eta_{11}^{(m)},\quad\alpha_{12}^{(m)}:=\gamma_{12}^{(m)}\|A_{11}\|N_{1}\eta_{11}^{(m)},\end{gathered}$
$\begin{gathered}\alpha_{21}^{(m)}:=\gamma_{21}^{(m)}\|A_{11}\|N_{2}M_{1}\eta_{22}^{(m)},\quad\alpha_{22}^{(m)}:=\gamma_{21}^{(m)}\|A_{22}\|N_{2}\eta_{22}^{(m)},\end{gathered}$

$\begin{gathered}\Theta_{m}(h):=\frac{\alpha_{11}^{(m)}}{\mu_{2}}\Big{(}\frac{e^{(\mu_{2}+\delta_{1})h}-1}{\mu_{2}+\delta_{1}}-\frac{e^{\delta_{1}h}-1}{\delta_{1}}\Big{)}+\frac{\alpha_{12}^{(m)}}{\delta_{1}}\Big{(}\frac{e^{h\delta_{1}}-1}{\delta_{1}}-h\Big{)}\\\
+\frac{\alpha_{21}^{(m)}}{\mu_{1}}\Big{(}\frac{e^{(\mu_{1}+\delta_{2})h}-1}{\mu_{1}+\delta_{2}}-\frac{e^{\delta_{2}h}-1}{\delta_{2}}\Big{)}+\frac{\alpha_{22}^{(m)}}{\delta_{2}}\Big{(}\frac{e^{h\delta_{2}}-1}{\delta_{2}}-h\Big{)}+\\\
+2\Big{(}\gamma_{12}^{(m)}\eta_{12}^{(m)}N_{1}M_{2}\Big{(}\frac{he^{(\mu_{2}+\delta_{1})h}}{\mu_{2}+\delta_{1}}-\frac{e^{(\mu_{2}+\delta_{1})h}-1}{(\mu_{2}+\delta_{1})^{2}}\Big{)}\\\
+\gamma_{21}^{(m)}\eta_{21}^{(m)}N_{2}M_{1}\Big{(}\frac{he^{(\mu_{1}+\delta_{2})h}}{\mu_{1}+\delta_{2}}-\frac{e^{(\mu_{1}+\delta_{2})h}-1}{(\mu_{1}+\delta_{2})^{2}}\Big{)}\Big{)}.\end{gathered}$
(16)

The following lemma establishes estimates for the change of the Lyapunov
function on each of the discretization intervals $(mh,(m+1)h]$.

Lemma 4.2 If $\Pi_{m}$ are positive definite, for all $m=0,\dots,N-1$, then

$\begin{gathered}v((m+1)h,x_{1}((m+1)h),x_{2}((m+1)h))\\\ \leq
e^{\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}}v(mh+0,x_{1}(mh+0),x_{2}(mh+0)).\end{gathered}$
(17)

Proof. Let $t\in(mh,(m+1)h)$. It can be established by direct calculations
that

$\begin{gathered}\dot{v}(t,x_{1}(t),x_{2}(t))=x_{1}^{\operatorname{T\,}}(t)(P_{12}(t)A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}(t)\\\
-e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}^{(m)}A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)})e^{-A_{11}(t-mh)})x_{1}(t)\\\
+2x_{1}^{\operatorname{T\,}}(t)(P_{11}(t)A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}(t)\\\
-e^{-A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}^{(m)}A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)})e^{-A_{22}(t-mh)})x_{2}(t)\\\
+x_{2}^{\operatorname{T\,}}(t)((A_{12}^{\operatorname{T\,}}(t)P_{12}(t)+P_{21}(t)A_{12}(t))\\\
-e^{-A_{22}^{\operatorname{T\,}}(t-mh)}(A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(t))e^{-A_{22}(t-mh)})x_{2}(t)\\\
=z_{1m}^{\operatorname{T\,}}(t)(e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}(t)A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}(t))e^{A_{11}(t-mh)}\\\
-(P_{12}^{(m)}A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)}))z_{1m}(t)\\\
+2z_{1m}^{\operatorname{T\,}}(t)(e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}(t)A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}(t))e^{A_{22}(t-mh)}\\\
-(P_{11}^{(m)}A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)}))z_{2m}(t)\\\
+z_{2m}^{\operatorname{T\,}}(t)(e^{A_{22}^{\operatorname{T\,}}(t-mh)}(A_{12}^{\operatorname{T\,}}(t)P_{12}(t)+P_{21}(t)A_{12}(t))e^{A_{22}(t-mh)}\\\
-(A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(t)))z_{2m}(t)\end{gathered}$
(18)

Let us consider separately

$\begin{gathered}e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}(t)A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}(t))e^{A_{11}(t-mh)}-(P_{12}^{(m)}A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)})\\\
=e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{12}(t)A_{21}(t)e^{A_{11}(t-mh)}-P_{12}^{(m)}A_{21}(t)\\\
+e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)P_{21}(t)e^{A_{11}(t-mh)}-A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)}\end{gathered}$

Taking into account the explicit expressions for $P_{12}(t)$ from (11) we get

$\begin{gathered}e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{12}(t)A_{21}(t)e^{A_{11}(t-mh)}-P_{12}^{(m)}A_{21}(t)\\\
=(P_{12}^{(m)}-\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds)e^{-A_{22}(t-mh)}A_{21}(t)e^{A_{11}(t-mh)}\\\
-P_{12}^{(m)}A_{21}(t))=P_{12}^{(m)}(e^{-A_{22}(t-mh)}A_{21}(t)e^{A_{11}(t-mh)}-A_{21}(t))\\\
-\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds)e^{-A_{22}(t-mh)}A_{21}(t)e^{A_{11}(t-mh)}\end{gathered}$

Consequently, using (13) we obtain

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{12}(t)A_{21}(t)e^{A_{11}(t-mh)}-P_{12}^{(m)}A_{21}(t)\|\\\
\leq\gamma_{21}^{(m)}\Big{(}\|P_{12}^{(m)}\|(\frac{\|A_{11}\|M_{1}N_{2}}{\mu_{1}}e^{\delta_{2}(t-mh)}(e^{\mu_{1}(t-mh)}-1)+\frac{N_{2}\|A_{22}\|}{\delta_{2}}(e^{\delta_{2}(t-mh)}-1))\\\
+(t-mh)(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|)M_{1}N_{2}e^{(t-mh)(\mu_{1}+\delta_{2})}\Big{)}\end{gathered}$

and

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)P_{21}(t)e^{-A_{11}(t-mh)}-A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)}\|\\\
\leq\gamma_{21}^{(m)}\Big{(}\|P_{12}^{(m)}\|(\frac{\|A_{11}\|M_{1}N_{2}}{\mu_{1}}e^{\delta_{2}(t-mh)}(e^{\mu_{1}(t-mh)}-1)+\frac{N_{2}\|A_{22}\|}{\delta_{2}}(e^{\delta_{2}(t-mh)}-1))\\\
+(t-mh)(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|)M_{1}N_{2}e^{(t-mh)(\mu_{1}+\delta_{2})}\Big{)}\end{gathered}$

Hence, applying the triangle inequality we obtain

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{12}(t)A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}(t))e^{A_{11}(t-mh)}-(P_{12}^{(m)}A_{21}(t)+A_{21}^{\operatorname{T\,}}(t)P_{21}^{(m)})\|\\\
\leq
2\gamma_{21}^{(m)}\Big{(}\|P_{12}^{(m)}\|(\frac{\|A_{11}\|M_{1}N_{2}}{\mu_{1}}e^{\delta_{2}(t-mh)}(e^{\mu_{1}(t-mh)}-1)+\frac{N_{2}\|A_{22}\|}{\delta_{2}}(e^{\delta_{2}(t-mh)}-1))\\\
+(t-mh)(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\gamma_{21}^{(m)}\|P_{22}^{(m)}\|)M_{1}N_{2}e^{(t-mh)(\mu_{1}+\delta_{2})}\Big{)}:=\psi_{11}^{(m)}(t).\end{gathered}$

Let us transform the following expression

$\begin{gathered}e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}(t)A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}(t))e^{A_{22}(t-mh)}-(P_{11}^{(m)}A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)})\\\
=e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{11}(t)A_{12}(t)e^{A_{22}(t-mh)}-P_{11}^{(m)}A_{12}(t)\\\
+e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)P_{22}(t)e^{A_{22}(t-mh)}-A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)}\end{gathered}$

Taking into account (9), consider separately

$\begin{gathered}e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{11}(t)A_{12}(t)e^{A_{22}(t-mh)}-P_{11}^{(m)}A_{12}(t)\\\
=P_{11}^{(m)}(e^{-A_{11}(t-mh)}A_{12}(t)e^{A_{22}(t-mh)}-A_{12}(t))\\\
-\int\limits_{mh}^{t}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,dse^{-A_{11}(t-mh)}A_{12}(t)e^{A_{22}(t-mh)}\end{gathered}$

Consequently, using (13) we get

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}P_{11}(t)A_{12}(t)e^{A_{22}(t-mh)}-P_{11}^{(m)}A_{12}(t)\|\\\
\leq\gamma_{12}^{(m)}\Big{(}\|P_{11}^{(m)}\|(\frac{\|A_{22}\|M_{2}N_{1}}{\mu_{2}}e^{\delta_{1}(t-mh)}(e^{\mu_{2}(t-mh)}-1)+\frac{N_{1}\|A_{11}\|}{\delta_{1}}(e^{\delta_{1}(t-mh)}-1))\\\
+2(t-mh)\|P_{12}^{(m)}\|\gamma_{21}^{(m)}N_{1}M_{2}e^{(t-mh)(\delta_{1}+\mu_{2})}\Big{)}.\end{gathered}$

Taking into account (10) we obtain

$\begin{gathered}e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)P_{22}(t)e^{A_{22}(t-mh)}-A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)}\\\
=(e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)e^{-A_{22}^{\operatorname{T\,}}(t-mh)}-A_{21}^{\operatorname{T\,}}(t))P_{22}^{(m)}\\\
-e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)e^{-A_{22}^{\operatorname{T\,}}(t-mh)}\int\limits_{mh}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,ds\end{gathered}$

Thus, taking into account (13) we have

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}A_{21}^{\operatorname{T\,}}(t)P_{22}(t)e^{A_{22}(t-mh)}-A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)}\|\\\
\leq\gamma_{21}^{(m)}\Big{(}\|P_{22}^{(m)}\|(\frac{\|A_{11}\|M_{1}N_{2}}{\mu_{1}}e^{\delta_{2}(t-mh)}(e^{\mu_{1}(t-mh)}-1)+\frac{N_{2}\|A_{22}\|}{\delta_{2}}(e^{\delta_{2}(t-mh)}-1))\\\
+2(t-mh)\|P_{12}^{(m)}\|\gamma_{12}^{(m)}N_{2}M_{1}e^{(t-mh)(\delta_{2}+\mu_{1})}\Big{)}\end{gathered}$

Thereby, finally we find the following estimate

$\begin{gathered}\|e^{A_{11}^{\operatorname{T\,}}(t-mh)}(P_{11}(t)A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}(t))e^{A_{22}(t-mh)}-(P_{11}^{(m)}A_{12}(t)+A_{21}^{\operatorname{T\,}}(t)P_{22}^{(m)})\|\\\
\leq\gamma_{12}^{(m)}\Big{(}\|P_{11}^{(m)}\|(\frac{\|A_{22}\|M_{2}N_{1}}{\mu_{2}}e^{\delta_{1}(t-mh)}(e^{\mu_{2}(t-mh)}-1)+\frac{N_{1}\|A_{11}\|}{\delta_{1}}(e^{\delta_{1}(t-mh)}-1))\\\
+2(t-mh)\|P_{12}^{(m)}\|\gamma_{21}^{(m)}N_{1}M_{2}e^{(t-mh)(\delta_{1}+\mu_{2})}\Big{)}\\\
+\gamma_{21}^{(m)}\Big{(}\|P_{22}^{(m)}\|(\frac{\|A_{11}\|M_{1}N_{2}}{\mu_{1}}e^{\delta_{2}(t-mh)}(e^{\mu_{1}(t-mh)}-1)+\frac{N_{2}\|A_{22}\|}{\delta_{2}}(e^{\delta_{2}(t-mh)}-1))\\\
+2(t-mh)\|P_{12}^{(m)}\|\gamma_{12}^{(m)}N_{2}M_{1}e^{(t-mh)(\delta_{2}+\mu_{1})}\Big{)}:=\psi_{12}^{(m)}(t).\end{gathered}$

Further, we consider

$\begin{gathered}e^{A_{22}^{\operatorname{T\,}}(t-mh)}(A_{12}^{\operatorname{T\,}}(t)P_{12}(t)+P_{21}(t)A_{12}(t))e^{A_{22}(t-mh)}-(A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(t))\\\
=e^{A_{22}^{\operatorname{T\,}}(t-mh)}A_{12}^{\operatorname{T\,}}(t)P_{12}(t)e^{A_{22}(t-mh)}-A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}\\\
+e^{A_{22}^{\operatorname{T\,}}(t-mh)}P_{21}(t)A_{12}(t)e^{A_{22}(t-mh)}-P_{21}^{(m)}A_{12}(t)\end{gathered}$

Taking into account (11) we obtain

$\begin{gathered}e^{A_{22}^{\operatorname{T\,}}(t-mh)}A_{12}^{\operatorname{T\,}}(t)P_{12}(t)e^{A_{22}(t-mh)}-A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}\\\
=(e^{A_{22}^{\operatorname{T\,}}(t-mh)}A_{12}^{\operatorname{T\,}}(t)e^{-A_{11}^{\operatorname{T\,}}(t-mh)}-A_{12}^{\operatorname{T\,}}(t))P_{12}^{(m)}\\\
-e^{A_{22}^{\operatorname{T\,}}(t-mh)}A_{12}^{\operatorname{T\,}}(t)e^{-A_{11}^{\operatorname{T\,}}(t-mh)}\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds\end{gathered}$

Consequently, applying (13) we obtain

$\begin{gathered}\|e^{A_{22}^{\operatorname{T\,}}(t-mh)}A_{12}^{\operatorname{T\,}}(t)P_{12}(t)e^{A_{22}(t-mh)}-A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}\|\\\
\leq\gamma_{12}^{(m)}\Big{(}\|P_{12}^{(m)}\|(\frac{\|A_{22}\|M_{2}N_{1}}{\mu_{2}}e^{\delta_{1}(t-mh)}(e^{\mu_{2}(t-mh)}-1)+\frac{N_{1}\|A_{11}\|}{\delta_{1}}(e^{\delta_{1}(t-mh)}-1))\\\
+(t-mh)(\|P_{22}^{(m)}\|\gamma_{21}^{(m)}+\gamma_{12}^{(m)}\|P_{11}^{(m)}\|)M_{2}N_{1}e^{(t-mh)(\mu_{2}+\delta_{1})}\Big{)}\end{gathered}$

Thus,

$\begin{gathered}\|e^{A_{22}^{\operatorname{T\,}}(t-mh)}(A_{12}^{\operatorname{T\,}}(t)P_{12}(t)+P_{21}(t)A_{12}(t))e^{A_{22}(t-mh)}\\\
-(A_{12}^{\operatorname{T\,}}(t)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(t))\|\\\ \leq
2\gamma_{12}^{(m)}\Big{(}\|P_{12}^{(m)}\|(\frac{\|A_{22}\|M_{2}N_{1}}{\mu_{2}}e^{\delta_{1}(t-mh)}(e^{\mu_{2}(t-mh)}-1)+\frac{N_{1}\|A_{11}\|}{\delta_{1}}(e^{\delta_{1}(t-mh)}-1))\\\
+(t-mh)(\|P_{22}^{(m)}\|\gamma_{21}^{(m)}+\gamma_{12}^{(m)}\|P_{11}^{(m)}\|)M_{2}N_{1}e^{(t-mh)(\mu_{2}+\delta_{1})}\Big{)}:=\psi_{22}^{(m)}(t)\end{gathered}$

We define matrices $\Psi_{m}(t)=(\psi_{ij}^{(m)}(t))_{i,j=1,2}$, then

$\begin{gathered}\dot{v}(t,x_{1}(t),x_{2}(t))\leq\zeta_{m}^{\operatorname{T\,}}(t)\Psi_{m}(t)\zeta_{m}(t),\end{gathered}$
(19)

where $\zeta_{m}(t)=(\|z_{1m}(t)\|,\|z_{2m}(t)\|)^{\operatorname{T\,}}$. We
represent $\Psi_{m}(t)$ in the following form

$\begin{gathered}\Psi_{m}(t)=\gamma_{12}^{(m)}\Big{(}\|A_{22}\|M_{2}N_{1}e^{\delta_{1}(t-mh)}\frac{e^{\mu_{2}(t-mh)}-1}{\mu_{2}}+N_{1}\|A_{11}\|\frac{e^{\delta_{1}(t-mh)}-1}{\delta_{1}}\Big{)}\Upsilon_{1}\\\
+\gamma_{21}^{(m)}\Big{(}\|A_{11}\|M_{1}N_{2}e^{\delta_{2}(t-mh)}\frac{e^{\mu_{1}(t-mh)}-1}{\mu_{1}}+N_{2}\|A_{22}\|\frac{e^{\delta_{2}(t-mh)}-1}{\delta_{2}}\Big{)}\Upsilon_{2}\\\
+2\gamma_{12}^{(m)}N_{1}M_{2}(t-mh)e^{(t-mh)(\mu_{2}+\delta_{1})}\widetilde{\Upsilon}_{1}+2\gamma_{21}^{(m)}N_{2}M_{1}(t-mh)e^{(t-mh)(\mu_{1}+\delta_{2})}\widetilde{\Upsilon}_{2},\end{gathered}$

where

$\begin{gathered}\Upsilon_{1}=\begin{pmatrix}0&\|P_{11}^{(m)}\|\\\
\|P_{11}^{(m)}\|&2\|P_{12}^{(m)}\|\end{pmatrix},\quad\Upsilon_{2}=\begin{pmatrix}2\|P_{12}^{(m)}\|&\|P_{22}^{(m)}\|\\\
\|P_{22}^{(m)}\|&0\end{pmatrix}\end{gathered}$

$\begin{gathered}\widetilde{\Upsilon}_{1}=\begin{pmatrix}0&2\|P_{12}^{(m)}\|\gamma_{21}^{(m)}\\\
2\|P_{12}^{(m)}\|\gamma_{21}^{(m)}&(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})\end{pmatrix},\quad\end{gathered}$
$\begin{gathered}\widetilde{\Upsilon}_{2}=\begin{pmatrix}(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})&2\|P_{12}^{(m)}\|\gamma_{12}^{(m)}\\\
2\|P_{12}^{(m)}\|\gamma_{12}^{(m)}&0\end{pmatrix},\quad\end{gathered}$

Direct calculations show $\|\Upsilon_{1}\|=\eta_{11}^{(m)}$,
$\|\Upsilon_{2}\|=\eta_{22}^{(m)}$,
$\|\widetilde{\Upsilon}_{1}\|=\eta_{12}^{(m)}$,
$\|\widetilde{\Upsilon}_{2}\|=\eta_{21}^{(m)}$.

For $\Psi_{m}(t)$ the following estimates hold

$\begin{gathered}\|\Psi_{m}(t)\|\leq\alpha_{11}^{(m)}e^{\delta_{1}(t-mh)}\frac{e^{\mu_{2}(t-mh)}-1}{\mu_{2}}+\alpha_{12}^{(m)}\frac{e^{\delta_{1}(t-mh)}-1}{\delta_{1}}\\\
+\alpha_{21}^{(m)}e^{\delta_{2}(t-mh)}\frac{e^{\mu_{1}(t-mh)}-1}{\mu_{1}}+\alpha_{22}^{(m)}\frac{e^{\delta_{2}(t-mh)}-1}{\delta_{2}}\\\
+2(\gamma_{12}^{(m)}\eta_{12}^{(m)}N_{1}M_{2}(t-mh)e^{(t-mh)(\mu_{2}+\delta_{1})}+\gamma_{21}^{(m)}\eta_{21}^{(m)}N_{2}M_{1}(t-mh)e^{(t-mh)(\mu_{1}+\delta_{2})}).\end{gathered}$

Therefore, for $\int_{mh}^{(m+1)h}\|\Psi_{m}(s)\|\,ds$ the following estimate
holds

$\begin{gathered}\int\limits_{mh}^{(m+1)h}\|\Psi_{m}(s)\|\,ds\leq\frac{\alpha_{11}^{(m)}}{\mu_{2}}\Big{(}\frac{e^{(\mu_{2}+\delta_{1})h}-1}{\mu_{2}+\delta_{1}}-\frac{e^{\delta_{1}h}-1}{\delta_{1}}\Big{)}+\frac{\alpha_{12}^{(m)}}{\delta_{1}}\Big{(}\frac{e^{h\delta_{1}}-1}{\delta_{1}}-h\Big{)}\\\
+\frac{\alpha_{21}^{(m)}}{\mu_{1}}\Big{(}\frac{e^{(\mu_{1}+\delta_{2})h}-1}{\mu_{1}+\delta_{2}}-\frac{e^{\delta_{2}h}-1}{\delta_{2}}\Big{)}+\frac{\alpha_{22}^{(m)}}{\delta_{2}}\Big{(}\frac{e^{h\delta_{2}}-1}{\delta_{2}}-h\Big{)}+\\\
+2\Big{(}\gamma_{12}^{(m)}\eta_{12}^{(m)}N_{1}M_{2}\Big{(}\frac{he^{(\mu_{2}+\delta_{1})h}}{\mu_{2}+\delta_{1}}-\frac{e^{(\mu_{2}+\delta_{1})h}-1}{(\mu_{2}+\delta_{1})^{2}}\Big{)}\\\
+\gamma_{21}^{(m)}\eta_{21}^{(m)}N_{2}M_{1}\Big{(}\frac{he^{(\mu_{1}+\delta_{2})h}}{\mu_{1}+\delta_{2}}-\frac{e^{(\mu_{1}+\delta_{2})h}-1}{(\mu_{1}+\delta_{2})^{2}}\Big{)}\Big{)}=\Theta_{m}(h).\end{gathered}$

From (19), taking into account the Lemma 4.1, for $t\in(mh,(m+1)h)$ we obtain

$\begin{gathered}\dot{v}(t,x_{1}(t),x_{2}(t))\leq\zeta_{m}^{\operatorname{T\,}}\Psi_{m}(t)\zeta_{m}(t)\leq\|\Psi_{m}(t)\|(\|z_{1m}(t)\|^{2}+\|z_{2m}(t)\|^{2})\\\
\leq\frac{\|\Psi_{m}(t)\|}{\lambda_{\min}(\Pi_{m})}v(t,x_{1}(t),x_{2}(t)),\end{gathered}$

Integrating this differential inequality we get

$\begin{gathered}v((m+1)h,x_{1}((m+1)h),x_{2}((m+1)h))\\\ \leq
e^{\int\limits_{mh}^{(m+1)h}\frac{\|\Psi_{m}(s)\|}{\lambda_{\min}(\Pi_{m})}\,ds}v(mh+0,x_{1}(mh+0),x_{2}(mh+0))\\\
\leq
e^{\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}}v(mh+0,x_{1}(mh+0),x_{2}(mh+0)),\end{gathered}$

which completes the proof of the Lemma.

###  4.2 Conditions for the asymptotic stability

We establish sufficient conditions for the asymptotic stability of (1) using
$\mathfrak{V}(t,x_{1},x_{2})$ constructed in the previous section and Lemma
4.2. Recall that $P_{0}=(P_{ij}^{(0)})_{i,j=1,2}$,
$P_{N}=(P_{ij}^{(N)})_{i,j=1,2}$ are block matrices, where $P_{ij}$ are
defined by (6)-(8).

Theorem 4.1. Let $N\in\mathbb{N}$ and $P_{0}\succ 0$ be such that for
$m=0,\dots,N-1$ the matrices $\Pi_{m}$ defined by (15) are positive definite.
If

$\begin{gathered}Q:=\sum\limits_{m=0}^{N-1}\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}+\ln\lambda_{\max}(P_{N}^{-1}B^{\operatorname{T\,}}P_{0}B)<0,\end{gathered}$
(20)

where $h=\frac{\theta}{N}$ and $\Theta_{m}(h)$ are defined in (16), then (1)
is asymptotically stable.

Proof. From (17) it follows that for all $m=0,\dots,N-1$

$\begin{gathered}v((m+1)h,x_{1}((m+1)h),x_{2}((m+1)h))\\\
\leq\exp\Big{(}\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}\Big{)}v(mh+0,x_{1}(mh+0),x_{2}(mh+0)).\end{gathered}$

Therefore,

$\begin{gathered}v(\theta,x_{1}(\theta),x_{2}(\theta))\leq\exp\Big{(}\sum\limits_{m=0}^{N-1}\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}\Big{)}v(0+0,x_{1}(0+0),x_{2}(0+0)).\end{gathered}$

For $t=\theta$,

$\begin{gathered}v(\theta+0,x_{1}(\theta+0),x_{2}(\theta+0))=x^{\operatorname{T\,}}(\theta+0)P_{0}x(\theta+0)\\\
=x^{\operatorname{T\,}}(\theta)B^{\operatorname{T\,}}P_{0}Bx(\theta)=(P_{N}^{1/2}x(\theta))^{\operatorname{T\,}}P_{N}^{-1/2}B^{\operatorname{T\,}}P_{0}BP_{N}^{-1/2}P_{N}^{1/2}x(\theta)\\\
\leq\lambda_{\max}(P_{N}^{-1/2}B^{\operatorname{T\,}}P_{0}BP_{N}^{-1/2})\|P_{N}^{1/2}x(\theta)\|^{2}=\lambda_{\max}(P_{N}^{-1}B^{\operatorname{T\,}}P_{0}B)v(\theta,x_{1}(\theta),x_{2}(\theta)).\end{gathered}$

We recall that here
$x(\theta)=(x_{1}^{\operatorname{T\,}}(\theta),x_{2}^{\operatorname{T\,}}(\theta))^{\operatorname{T\,}}$.
Consequently,

$\begin{gathered}v(\theta+0,x_{1}(\theta+0),x_{2}(\theta+0))\\\
\leq\lambda_{\max}(P_{N}^{-1}B^{\operatorname{T\,}}P_{0}B)\exp\Big{(}\sum\limits_{m=0}^{N-1}\frac{\Theta_{m}(h)}{\lambda_{\min}(\Pi_{m})}\Big{)}v(0+0,x_{1}(0+0),x_{2}(0+0))\\\
=e^{Q}v(0+0,x_{1}(0+0),x_{2}(0+0)).\end{gathered}$

Due to the periodicity in $t$ of (1) and $\mathfrak{V}(t,.,.)$ for any
$k\in\mathbb{Z}_{+}$ the following inequality holds

$\begin{gathered}v((k+1)\theta+0,x_{1}((k+1)\theta+0),x_{2}((k+1)\theta+0))\\\
\leq e^{Q}v(k\theta+0,x_{1}(k\theta+0),x_{2}(k\theta+0)).\end{gathered}$

Therefore,

$\begin{gathered}v(k\theta+0,x_{1}(k\theta+0),x_{2}(k\theta+0))\leq
e^{Qk}v(0+0,x_{1}(0+0),x_{2}(0+0)).\end{gathered}$

From Lemma 4.1 it follows that

$\begin{gathered}\lambda_{\min}(\Pi_{0})\|x(k\theta+0)\|^{2}\leq
v(k\theta+0,x_{1}(k\theta+0),x_{2}(k\theta+0))\\\ \leq
e^{Qk}v(0+0,x_{1}(0+0),x_{2}(0+0))\leq
e^{Qk}\lambda_{\max}(\Xi_{0})\|x(0+0)\|^{2},\end{gathered}$

which is equivalent to

$\begin{gathered}\|x(k\theta+0)\|\leq\sqrt{\frac{\lambda_{\max}(\Xi_{0})}{\lambda_{\min}(\Pi_{0})}}e^{Qk/2}\|x(0+0)\|.\end{gathered}$

By conditions of Theorem 4.1, $Q<0$, therefore, $\|x(k\theta+0)\|\to 0$ as
$k\to\infty$ which proves the asymptotic stability of (1). The theorem is
proved.

To check the condition (20), it is necessary to calculate $\Theta_{m}(h)$. In
addition, with the constants $\gamma_{ij}^{(m)}$, $m=0,\dots,N-1$ included in
$\Theta_{m}(h)$, it is necessary to calculate the matrices $P_{ij}^{(m)}$,
$i,j=1,2$, $m=0,\dots,N$ using the recurrent formulas (6)-(8). These formulas
contain $e^{-A_{ii}h}$ and $\int_{mh}^{(m+1)h}A_{ij}(s)\,ds$. Such
calculations are accessible to modern computing tools. It is intuitively clear
that $h$ should be chosen such that
$h\sup_{t\in[0,\theta],i,j=1,2}\|P_{ij}(t)\|\ll 1$ , i.e. so that the change
of $P_{ij}(t)$ is rather small over the discretization period $(mh,(m+1)h]$.

##  5 The case of a non-periodic system

Consider the case of a linear impulsive system (1) when the dwell-times are
subject to the conditions $\theta_{1}\leq T_{k}\leq\theta_{2}$,
$\theta_{1}\neq\theta_{2}$. In this case, the system (1) is not periodic,
since its solutions do not have the property of invariance with respect to the
semigroup $\theta\mathbb{Z}_{+}$ and the Floquet theory is not applicable.

The Lyapunov function $v(t,x_{1},x_{2})$ has a quadratic form (12), we will
construct $P_{ij}(t)$, $i,j=1,2$ step by step on every interval
$[\tau_{k},\tau_{k+1}]$. If $\tau_{k}\notin h\mathbb{Z}$, then on the interval
$(\tau_{k},d_{k}h]$, where $d_{k}h$ is the grid node nearest on the left to
$\tau_{k}$, we choose $P_{ij}(t)$ as a constant on the interval
$(\tau_{k},d_{k}h]$. Between the nodes we construct $P_{ij}(t)$ similarly to
the periodic case. Finally, if $\tau_{k+1}\notin h\mathbb{Z}$, then on the
interval $(\varkappa_{k}h,\tau_{k+1}]$ ($\varkappa_{k}h$ is the grid node
nearest on the right to $\tau_{k+1}$) we choose $P_{ij}(t)$ again as a
constants, i.e. $P_{ij}(t)=P_{ij}(\varkappa_{k}h-0)$, for
$t\in(\varkappa_{k}h,\tau_{k+1}]$. Thus, it becomes necessary to estimate the
derivative of the Lyapunov function $v(t,x_{1},x_{2})$ on the time intervals
$(\tau_{k},d_{k}h]$, $(\varkappa_{k}h,\tau_{k+1}]$. For this we will introduce
Assumption 5.1 and Lemma 5.1.

###  5.1 Construction of the Lyapunov function

To construct a Lyapunov function, the discretization method is also used here.
The discretization parameters are: the number of nodes $N\in\mathbb{N}$,
$N\geq 2$ and $h=\frac{\theta}{N}$. We denote
$N_{3}=-\Big{[}\frac{2h-\theta_{1}}{h}\Big{]}$,
$N_{4}=\Big{[}\frac{\theta_{2}}{h}\Big{]}$. In this case, additional
assumptions regarding the connections between the subsystems are required.

Assumption 5.1. There are positive constants $l_{ij}^{(m)}$, $i,j=1,2$, $i\neq
j$, $m=0,\dots,N-1$ such that

$\begin{gathered}\sup\limits_{t\in(mh,(m+1)h]}\|A_{ij}(t)-A_{ij}(mh)\|\leq
l_{ij}^{(m)}h.\end{gathered}$

For any $m\in\mathbb{Z}$, let $l_{ij}^{(m)}:=l_{ij}^{(\varrho)}$, where
$\varrho$ is the remainder of dividing $m$ by $N$. Let
$\mathfrak{l}_{m}:=\sqrt{(l_{12}^{(m)})^{2}+(l_{21}^{(m)})^{2}}$.

We denote the constant matrices

$\begin{gathered}A_{m}=\begin{pmatrix}A_{11}&A_{12}(mh)\\\
A_{21}(mh)&A_{22}\end{pmatrix},\quad m\in\mathbb{Z}.\end{gathered}$

Let $P_{0}=(P_{ij}^{(0)})_{i,j=1,2}$,
$P_{ij}^{(0)}=(P_{ij}^{(0)})^{\operatorname{T\,}}$ be a positive definite
symmetric block matrix. We define sequences of block matrices
$P_{m}^{(l)}=(P_{ij}^{(m,l)})_{i,j=1,2}$, $l=0,\dots,N-1$, $m=0,\dots,N_{4}-1$
as follows $P_{ij}^{(0,l)}\equiv P_{ij}^{(0)}$

$\begin{gathered}P_{11}^{(m+1,l)}=e^{-A_{11}^{\operatorname{T\,}}h}(P_{11}^{(m,l)}\\\
-\int\limits_{(m+l)h}^{(m+l+1)h}(P_{12}^{(m,l)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m,l)})\,ds)e^{-A_{11}h},\\\
\end{gathered}$ (21)
$\begin{gathered}P_{22}^{(m+1,l)}=e^{-A_{22}^{\operatorname{T\,}}h}(P_{22}^{(m,l)}\\\
-\int\limits_{(m+l)h}^{(m+l+1)h}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m,l)}+P_{21}^{(m,l)}A_{12}(s))\,ds)e^{-A_{22}h},\\\
\\\ \end{gathered}$ (22)
$\begin{gathered}P_{12}^{(m+1,l)}=e^{-A_{11}^{\operatorname{T\,}}h}(P_{12}^{(m,l)}\\\
-\int\limits_{(m+l)h}^{(m+l+1)h}(P_{11}^{(m,l)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m,l)})\,ds)e^{-A_{22}h}.\end{gathered}$
(23)

Next, we define the matrices $P_{ij}(t)$, $i,j=1,2$ sequentially on the
intervals $(\tau_{k},\tau_{k+1}]$. Let
$\widetilde{\tau}_{k}=\tau_{k}-\Big{[}\frac{\tau_{k}}{\theta}\Big{]}\theta\in[0,\theta)$,
$l_{k}:=\Big{[}\frac{\widetilde{\tau}_{k}}{h}\Big{]}+1$,
$d_{k}:=\Big{[}\frac{\tau_{k}}{h}\Big{]}+1$,
$\varkappa_{k}=\Big{[}\frac{\tau_{k+1}}{h}\Big{]}$. It is easy to show that

$\begin{gathered}(\tau_{k},\tau_{k+1}]=(\tau_{k},d_{k}h]\cup\bigcup\limits_{l=d_{k}}^{\varkappa_{k}-1}(lh,(l+1)h]\cup(\varkappa_{k}h,\tau_{k+1}].\end{gathered}$

Let $P_{ij}(t)=P_{ij}^{(0)}$ for $t\in(\tau_{k},d_{k}h]$. On each interval
$((m+d_{k})h,(m+1+d_{k})h]$, $m=0,\dots,\varkappa_{k}-d_{k}-1$, we put

$\begin{gathered}P_{11}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-(m+d_{k})h)}(P_{11}^{(m,l_{k})}\\\
-\int\limits_{(m+d_{k})h}^{t}(P_{12}^{(m,l_{k})}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m,l_{k})})\,ds)e^{-A_{11}(t-(m+d_{k})h)},\end{gathered}$
(24)
$\begin{gathered}P_{22}(t)=e^{-A_{22}^{\operatorname{T\,}}(t-(m+d_{k})h)}(P_{22}^{(m,l_{k})}\\\
-\int\limits_{(m+d_{k})h}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m,l_{k})}+P_{21}^{(m,l_{k})}A_{12}(s))\,ds)e^{-A_{22}(t-(m+d_{k})h)},\end{gathered}$
(25)
$\begin{gathered}P_{12}(t)=e^{-A_{11}^{\operatorname{T\,}}(t-(m+d_{k})h)}(P_{12}^{(m,l_{k})}\\\
-\int\limits_{(m+d_{k})h}^{t}(P_{11}^{(m,l_{k})}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m,l_{k})})\,ds)e^{-A_{22}(t-(m+d_{k})h)}.\end{gathered}$
(26)

For $t\in(\varkappa_{k}h,\tau_{k+1}]$, let
$P_{ij}(t)=P_{ij}^{(\varkappa_{k}-d_{k},l_{k})}$.

Remark. Note that $P(t)$ is continuous on $(\tau_{k},\tau_{k+1})$ and left-
continuous at $t=\tau_{k+1}$. Indeed, due to the $\theta$-periodicity of
$A_{ij}(t)$, the matrices $P_{ij}(t)$ satisfy

$\begin{gathered}P_{ij}((m+d_{k})h+0)=P_{ij}^{(m,l_{k})},\quad
m=0,\dots,\varkappa_{k}-d_{k}-1,\end{gathered}$ (27)
$\begin{gathered}P_{ij}((m+d_{k})h-0)=P_{ij}^{(m,l_{k})},\quad
m=0,\dots,\varkappa_{k}-d_{k}-1.\end{gathered}$ (28)

Let us prove (27) and (28). We restrict ourselves to the case $(i,j)=(1,1)$,
since other cases one can consider in a similar way. In case $m=0$ the
formulas (27) and (28) are obvious. From the equality (24) it is easy to show
that (27) is true for all $m=1,\dots,\varkappa_{k}-d_{k}-1$.

Next we prove (28). From (24) we have

$\begin{gathered}P_{11}((m+d_{k})h-0)=e^{-A_{11}^{\operatorname{T\,}}h}(P_{11}^{(m-1,l_{k})}\\\
-\int\limits_{(m+d_{k}-1)h}^{(m+d_{k})h}(P_{12}^{(m-1,l_{k})}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m-1,l_{k})})\,ds)e^{-A_{11}h},\end{gathered}$
(29)

In the integral (29) make the change of variables
$\widetilde{s}:=s+(l_{k}-d_{k})h$, then

$\begin{gathered}P_{11}((m+d_{k})h-0)=e^{-A_{11}^{\operatorname{T\,}}h}(P_{11}^{(m-1,l_{k})}\\\
-\int\limits_{(m+l_{k}-1)h}^{(m+l_{k})h}(P_{12}^{(m-1,l_{k})}A_{21}(\widetilde{s}+(l_{k}-d_{k})h)\\\
+A_{21}^{\operatorname{T\,}}(\widetilde{s}+(l_{k}-d_{k})h)P_{21}^{(m-1,l_{k})})\,ds)e^{-A_{11}h},\end{gathered}$
(30)

Note that $(d_{k}-l_{k})h\in\theta\mathbb{Z}$. Indeed, the definition of the
function $x\mapsto[x]$ implies the inequalities

$\begin{gathered}\Big{[}\frac{\tau_{k}}{h}\Big{]}\leq\frac{\tau_{k}}{h}<\Big{[}\frac{\tau_{k}}{h}\Big{]}+1,\quad\Big{[}\frac{\widetilde{\tau}_{k}}{h}\Big{]}\leq\frac{\widetilde{\tau}_{k}}{h}<\Big{[}\frac{\widetilde{\tau}_{k}}{h}\Big{]}+1.\end{gathered}$
(31)

Therefore, from (31) it follows that

$\begin{gathered}\frac{\tau_{k}-\widetilde{\tau}_{k}}{h}-1<\Big{[}\frac{\tau_{k}}{h}\Big{]}-\Big{[}\frac{\widetilde{\tau}_{k}}{h}\Big{]}<\frac{\tau_{k}-\widetilde{\tau}_{k}}{h}+1.\end{gathered}$

From the definition of $\widetilde{\tau}_{k}$ follows
$\frac{\tau_{k}-\widetilde{\tau}_{k}}{h}=[\frac{\tau_{k}}{\theta}]\frac{\theta}{h}=[\frac{\tau_{k}}{\theta}]N\in\mathbb{Z}$,
hence

$\begin{gathered}\Big{[}\frac{\tau_{k}}{\theta}\Big{]}N-1<\Big{[}\frac{\tau_{k}}{h}\Big{]}-\Big{[}\frac{\widetilde{\tau}_{k}}{h}\Big{]}<\Big{[}\frac{\widetilde{\tau}_{k}}{\theta}\Big{]}N+1.\end{gathered}$
(32)

Since $[\frac{\tau_{k}}{h}]-[\frac{\widetilde{\tau}_{k}}{h}]\in\mathbb{Z}$,
then
$[\frac{\tau_{k}}{h}]-[\frac{\widetilde{\tau}_{k}}{h}]=[\frac{\tau_{k}}{\theta}]N$
and
$h([\frac{\tau_{k}}{h}]-[\frac{\widetilde{\tau}_{k}}{h}])=[\frac{\tau_{k}}{\theta}]Nh=[\frac{\tau_{k}}{\theta}]\theta\in\theta\mathbb{Z}$,
which means $(d_{k}-l_{k})h\in\theta\mathbb{Z}$. In turn, $A_{12}(s)$ and
$A_{21}(s)$ are periodic functions, therefore
$A_{12}(s+(d_{k}-l_{k})h)=A_{12}(s)$ and $A_{21}(s+(d_{k}-l_{k})h)=A_{21}(s)$,
then for equality (30), taking into account (21),
$P_{11}((m+d_{k})h-0)=P_{11}^{(m,l)}$.

We choose the elements $\mathfrak{V}(t,x)$ in the form
$v_{ij}(t,x_{i},x_{j})=x_{i}^{\operatorname{T\,}}P_{ij}(t)x_{j}$, $i,j=1,2$.

The following lemma uses Assumption 5.1 and is needed to estimate the
derivative of the Lyapunov function $v(t,x_{1},x_{2})$ on $[\tau_{k},d_{k}h)$
and $(\varkappa_{k}h,\tau_{k+1}]$, where $P_{ij}(t)$ is constant.

Lemma 5.1. Let
$\mathfrak{l}_{m}:=\sqrt{(l_{12}^{(m)})^{2}+(l_{21}^{(m)})^{2}}$, then

$\begin{gathered}\dot{v}(t,x(t))\leq\lambda_{\max}(P_{0}^{-1}(A_{l_{k}-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l_{k}-1}+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|I_{n}))v(t,x(t)),\quad
t\in(\tau_{k},d_{k}h],\end{gathered}$
$\begin{gathered}\dot{v}(t,x(t))\leq\lambda_{\max}((P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1}(A_{l_{k}+\varkappa_{k}-d_{k}}^{\operatorname{T\,}}P_{\varkappa_{k}-d_{k}}^{(l_{k})}+P_{\varkappa_{k}-d_{k}}^{(l_{k})}A_{l_{k}+\varkappa_{k}-d_{k}}\\\
+2h\mathfrak{l}_{l_{k}+\varkappa_{k}-d_{k}}\|P_{\varkappa_{k}-d_{k}}^{(l_{k})}\|I_{n}))v(t,x(t)),\quad
t\in(\varkappa_{k}h,\tau_{k+1}].\end{gathered}$

Proof. Let $t\in(\tau_{k},d_{k}h]$, then, taking into account that
$(l_{k}-d_{k})h\in\theta\mathbb{Z}$

$\begin{gathered}\dot{v}(t,x(t))=x^{\operatorname{T\,}}(t)(A^{\operatorname{T\,}}(t)P(t)+P(t)A(t))x(t)=x^{\operatorname{T\,}}(t)(A^{\operatorname{T\,}}(t)P_{0}+P_{0}A(t))x(t)\\\
=x^{\operatorname{T\,}}(t)(A^{\operatorname{T\,}}((d_{k}-1)h)P_{0}+P_{0}A((d_{k}-1)h))x(t)\\\
+x^{\operatorname{T\,}}(t)((A(t)-A((d_{k}-1)h))^{\operatorname{T\,}}P_{0}+P_{0}(A(t)-A((d_{k}-1)h)))x(t)\\\
\leq
x^{\operatorname{T\,}}(t)(A^{\operatorname{T\,}}((l_{k}-1)h)P_{0}+P_{0}A((l_{k}-1)h))x(t)+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|x^{\operatorname{T\,}}(t)x(t)\\\
\leq(P_{0}^{1/2}x(t))^{\operatorname{T\,}}P_{0}^{-1/2}(A^{\operatorname{T\,}}((l_{k}-1)h)P_{0}+P_{0}A((l_{k}-1)h)+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|I_{n})P_{0}^{-1/2}(P_{0}^{1/2}x(t))\\\
\leq\lambda_{\max}(P_{0}^{-1}(A_{l_{k}-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l_{k}-1}+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|I_{n}))v(t,x(t)).\end{gathered}$
(33)

The second part of the lemma can be proved similarly. The lemma is proved.

Lemma 5.2 is similar to Lemma 4.1 and reduces the verification of the positive
definiteness of the Lyapunov function to a finite number of inequalities.

Lemma 5.2. If $z_{im}=e^{-A_{ii}(t-mh)}x_{i}$ for $t\in(mh,(m+1)h],$ $i=1,2$,
then

$\begin{gathered}\lambda_{\min}(\Pi_{m}^{(l)})\|z_{m}\|^{2}\leq
v(t,x_{1},x_{2})\leq\lambda_{\max}(\Xi_{m}^{(l)})\|z_{m}\|^{2},\\\ \text{for
all}\quad t\in(mh,(m+1)h],\quad m=0,\dots,N-1,\end{gathered}$ (34)

where
$z_{m}=(z_{1m}^{\operatorname{T\,}},z_{2m}^{\operatorname{T\,}})^{\operatorname{T\,}}$,
$\|z_{m}\|^{2}=\|z_{1m}\|^{2}+\|z_{2m}\|^{2}$,
$\Pi_{m}^{(l)}=(\pi_{ij}^{(m,l)})_{i,j=1,2}$,
$\Xi_{m}^{(l)}=(\xi_{ij}^{(m,l)})_{i,j=1,2}$ are block matrices with the
elements

$\begin{gathered}\pi_{11}^{(m,l)}=P_{11}^{(m,l)}-h(2\gamma_{21}^{(m+l)}\|P_{12}^{(m,l)}\|+(\gamma_{12}^{(m+l)}\|P_{11}^{(m,l)}\|+\gamma_{21}^{(m+l)}\|P_{22}^{(m,l)}\|))I_{n_{1}},\\\
\pi_{22}^{(m)}=P_{22}^{(m,l)}-h(2\gamma_{12}^{(m+l)}\|P_{12}^{(m,l)}\|+(\gamma_{12}^{(m+l)}\|P_{11}^{(m,l)}\|+\gamma_{21}^{(m+l)}\|P_{22}^{(m,l)}\|))I_{n_{2}},\\\
\pi_{12}^{(m)}=P_{12}^{(m,l)},\quad\pi_{21}^{(m,l)}=P_{21}^{(m,l)}\end{gathered}$
(35)
$\begin{gathered}\xi_{11}^{(m,l)}=P_{11}^{(m,l)}+h(2\gamma_{21}^{(m+l)}\|P_{12}^{(m,l)}\|+(\gamma_{12}^{(m+l)}\|P_{11}^{(m,l)}\|+\gamma_{21}^{(m+l)}\|P_{22}^{(m,l)}\|))I_{n_{1}},\\\
\xi_{22}^{(m,l)}=P_{22}^{(m,l)}+h(2\gamma_{12}^{(m+l)}\|P_{12}^{(m,l)}\|+(\gamma_{12}^{(m+l)}\|P_{11}^{(m,l)}\|+\gamma_{21}^{(m+l)}\|P_{22}^{(m,l)}\|))I_{n_{2}},\\\
\xi_{12}^{(m,l)}=P_{12}^{(m,l)},\quad\xi_{21}^{(m,l)}=P_{21}^{(m,l)}.\end{gathered}$

The proof is similar to the proof of Lemma 4.1.

We denote

$\begin{gathered}\eta_{11}^{(m,l)}:=\sqrt{\|P_{11}^{(m,l)}\|^{2}+\|P_{12}^{(m,l)}\|^{2}}+\|P_{12}^{(m,l)}\|,\\\
\eta_{22}^{(m,l)}:=\sqrt{\|P_{22}^{(m,l)}\|^{2}+\|P_{12}^{(m,l)}\|^{2}}+\|P_{12}^{(m,l)}\|\end{gathered}$
$\begin{gathered}\eta_{12}^{(m,l)}:=\frac{1}{2}(\|P_{11}^{(m,l)}\|\gamma_{12}^{(m+l)}+\|P_{22}^{(m,l)}\|\gamma_{21}^{(m+l)}\\\
+\sqrt{(\|P_{11}^{(m,l)}\|\gamma_{12}^{(m+l)}+\|P_{22}^{(m,l)}\|\gamma_{21}^{(m+l)})^{2}+16(\gamma_{21}^{(m,l)})^{2}\|P_{12}^{(m,l)}\|^{2}}).\end{gathered}$
$\begin{gathered}\eta_{21}^{(m,l)}:=\frac{1}{2}(\|P_{11}^{(m,l)}\|\gamma_{12}^{(m+l)}+\|P_{22}^{(m,l)}\|\gamma_{21}^{(m+l)}\\\
+\sqrt{(\|P_{11}^{(m,l)}\|\gamma_{12}^{(m+l)}+\|P_{22}^{(m,l)}\|\gamma_{21}^{(m+l)})^{2}+16(\gamma_{12}^{(m+l)})^{2}\|P_{12}^{(m,l)}\|^{2}}).\end{gathered}$
$\begin{gathered}\alpha_{11}^{(m,l)}:=\gamma_{12}^{(m+l)}\|A_{22}\|N_{1}M_{2}\eta_{11}^{(m,l)},\quad\alpha_{12}^{(m,l)}:=\gamma_{12}^{(m+l)}\|A_{11}\|N_{1}\eta_{11}^{(m,l)},\end{gathered}$
$\begin{gathered}\alpha_{21}^{(m,l)}:=\gamma_{21}^{(m+l)}\|A_{11}\|N_{2}M_{1}\eta_{22}^{(m,l)},\quad\alpha_{22}^{(m,l)}:=\gamma_{21}^{(m+l)}\|A_{22}\|N_{2}\eta_{22}^{(m,l)},\end{gathered}$

Let

$\begin{gathered}\Theta_{m}^{(l)}(h):=\frac{\alpha_{11}^{(m,l)}}{\mu_{2}}\Big{(}\frac{e^{(\mu_{2}+\delta_{1})h}-1}{\mu_{2}+\delta_{1}}-\frac{e^{\delta_{1}h}-1}{\delta_{1}}\Big{)}+\frac{\alpha_{12}^{(m,l)}}{\delta_{1}}\Big{(}\frac{e^{h\delta_{1}}-1}{\delta_{1}}-h\Big{)}\\\
+\frac{\alpha_{21}^{(m,l)}}{\mu_{1}}\Big{(}\frac{e^{(\mu_{1}+\delta_{2})h}-1}{\mu_{1}+\delta_{2}}-\frac{e^{\delta_{2}h}-1}{\delta_{2}}\Big{)}+\frac{\alpha_{22}^{(m,l)}}{\delta_{2}}\Big{(}\frac{e^{h\delta_{2}}-1}{\delta_{2}}-h\Big{)}+\\\
+2\Big{(}\gamma_{12}^{(m+l)}\eta_{12}^{(m,l)}N_{1}M_{2}\Big{(}\frac{he^{(\mu_{2}+\delta_{1})h}}{\mu_{2}+\delta_{1}}-\frac{e^{(\mu_{2}+\delta_{1})h}-1}{(\mu_{2}+\delta_{1})^{2}}\Big{)}\\\
+\gamma_{21}^{(m+l)}\eta_{21}^{(m,l)}N_{2}M_{1}\Big{(}\frac{he^{(\mu_{1}+\delta_{2})h}}{\mu_{1}+\delta_{2}}-\frac{e^{(\mu_{1}+\delta_{2})h}-1}{(\mu_{1}+\delta_{2})^{2}}\Big{)}\Big{)}.\end{gathered}$
(36)

Lemma 5.3 is similar to Lemma 4.2 and allows us to estimate the variations of
the Lyapunov function $v(t,x_{1},x_{2})$ on each discritization interval
$((m+d_{k})h,(m+d_{k}+1)h)$.

Lemma 5.3 Let $\Pi_{m}^{(l)}$ be positive definite for
$m=0,\dots,\varkappa_{k}-d_{k}-1$, $l=0,\dots,N-1$, then for
$t\in((m+d_{k})h,(m+d_{k}+1)h)$, we have

$\begin{gathered}v((m+d_{k}+1)h,x_{1}((m+d_{k}+1)h),x_{2}((m+d_{k}+1)h))\\\
\leq
e^{\frac{\Theta_{m}^{(l)}(h)}{\lambda_{\min}(\Pi_{m}^{(l)})}}v((m+d_{k})h+0,x_{1}((m+d_{k})h+0),x_{2}((m+d_{k})h+0))\end{gathered}$
(37)

The proof of Lemma 5.3 is similar to the proof of Lemma 4.2.

###  5.2 Conditions for the asymptotic stability

The Lyapunov function $v(t,x_{1},x_{2})$ constructed in the previous
subsection and and Lemmas 5.1-5.3 allows us to formulate sufficient conditions
for the asymptotic stability of (1) when $\theta_{1}\neq\theta_{2}$.

Theorem 5.1. Let $N\in\mathbb{N}$ and $P_{0}\succ 0$ be such that for
$m=0,\dots,N_{4}$, $l=0,\dots,N-1$, the matrices $\Pi_{m}^{(l)}$ defined by
(35) are positive definite and for all $M\in\mathbb{N}$, such that $N_{3}\leq
M\leq N_{4}-1$ the following inequality hold

$\begin{gathered}h\lambda_{\max}^{+}(P_{0}^{-1}(A_{l-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l-1}+2h\mathfrak{l}_{l-1}\|P_{0}\|I_{n}))\\\
+h\lambda_{\max}^{+}((P_{M}^{(l)})^{-1}(A_{l+M}^{\operatorname{T\,}}P_{M}^{(l)}+P_{M}^{(l)}A_{l+M}+2h\mathfrak{l}_{l+M}\|P_{M}^{(l)}\|I_{n}))\\\
+\sum\limits_{m=0}^{M-1}\frac{\Theta_{m}^{(l)}(h)}{\lambda_{\min}(\Pi_{m}^{(l)})}+\ln\lambda_{\max}((P_{M}^{(l)})^{-1}B^{\operatorname{T\,}}P_{0}B)<0,\end{gathered}$
(38)

where $h=\frac{\theta}{N}$, $P_{ij}^{(m,l)}$ are defined in (21)-(23) and
$\Theta_{m}^{(l)}(h)$ in (36). Then (1) is asymptotically stable, if
$T_{k}\in[\theta_{1},\theta_{2}]$.

Proof. Consider the variation of the Lyapunov function on the interval
$(\tau_{k},\tau_{k+1}]$. As a consequence of Lemma 5.1, we have

$\begin{gathered}v(hd_{k},x(hd_{k}))\\\
\leq\exp\Big{(}\lambda_{\max}(P_{0}^{-1}(A_{l_{k}-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l_{k}-1}+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|I_{n}))(hd_{k}-\tau_{k})\Big{)}v(\tau_{k}+0,x(\tau_{k}+0))\\\
\leq\exp\Big{(}\lambda_{\max}^{+}(P_{0}^{-1}(A_{l_{k}-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l_{k}-1}+2h\mathfrak{l}_{l_{k}-1}\|P_{0}\|I_{n}))h\Big{)}v(\tau_{k}+0,x(\tau_{k}+0)),\end{gathered}$
(39) $\begin{gathered}v(\tau_{k+1},x(\tau_{k+1}))\\\
\leq\exp\Big{(}\lambda_{\max}((P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1}(A_{l_{k}+\varkappa_{k}-d_{k}}^{\operatorname{T\,}}P_{\varkappa_{k}-d_{k}}^{(l_{k})}+P_{\varkappa_{k}-d_{k}}^{(l_{k})}A_{l_{k}+\varkappa_{k}-d_{k}}\\\
+2h\mathfrak{l}_{l_{k}+\varkappa_{k}-d_{k}}\|P_{\varkappa_{k}-d_{k}}^{(l_{k})}\|I_{n}))(\tau_{k+1}-h\varkappa_{k})\Big{)}v(\tau_{k}+0,x(\tau_{k}+0))\\\
\leq\exp\Big{(}\lambda_{\max}^{+}((P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1}(A_{l_{k}+\varkappa_{k}-d_{k}}^{\operatorname{T\,}}P_{\varkappa_{k}-d_{k}}^{(l_{k})}+P_{\varkappa_{k}-d_{k}}^{(l_{k})}A_{l_{k}+\varkappa_{k}-d_{k}}\\\
+2h\mathfrak{l}_{l_{k}+\varkappa_{k}-d_{k}}\|P_{\varkappa_{k}-d_{k}}^{(l_{k})}\|I_{n}))h\Big{)}v(\varkappa_{k}h+0,x(\varkappa_{k}h+0)).\end{gathered}$
(40)

Lemma 5.3 implies the inequalities

$\begin{gathered}v((m+1+d_{k})h,x((m+1+d_{k})h))\\\
\leq\exp\Big{(}\frac{\Theta_{m}^{(l)}(h)}{\lambda_{\min}(\Pi_{m}^{(l_{k})})}\Big{)}v((m+d_{k})h+0,x((m+d_{k})h+0))\end{gathered}$
(41)

for all $m=0,\dots,\varkappa_{k}-d_{k}-1$.

At the moment of impulse action $t=\tau_{k+1}$, we get

$\begin{gathered}v(\tau_{k+1}+0,x(\tau_{k+1}+0))\\\
=x^{\operatorname{T\,}}(\tau_{k+1}+0)P_{0}x(\tau_{k+1}+0)=x^{\operatorname{T\,}}(\tau_{k+1})B^{\operatorname{T\,}}P_{0}Bx(\tau_{k+1})\\\
=((P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{1/2}x(\tau_{k+1}))^{\operatorname{T\,}}(P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1/2}B^{\operatorname{T\,}}P_{0}B(P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1/2}(P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{1/2}x(\tau_{k+1})\\\
\leq\lambda_{\max}((P_{\varkappa_{k}-d_{k}}^{(l_{k})})^{-1}B^{\operatorname{T\,}}P_{0}B)v(\tau_{k+1},x(\tau_{k+1}))\end{gathered}$
(42)

Comparing the inequalities (39)—(42), we obtain

$\begin{gathered}v(\tau_{k+1}+0,x(\tau_{k+1}+0))\leq
e^{Q_{\varkappa_{k}-d_{k}}^{(l_{k})}}v(\tau_{k}+0,x(\tau_{k}+0)),\end{gathered}$
(43)

where

$\begin{gathered}Q_{s}^{(l)}=h\lambda_{\max}^{+}(P_{0}^{-1}(A_{l-1}^{\operatorname{T\,}}P_{0}+P_{0}A_{l-1}+2h\mathfrak{l}_{l-1}\|P_{0}\|I_{n}))\\\
+h\lambda_{\max}^{+}((P_{s}^{(l)})^{-1}(A_{l+s}^{\operatorname{T\,}}P_{s}^{(l)}+P_{s}^{(l)}A_{l+s}+2h\mathfrak{l}_{l+s}\|P_{s}^{(l)}\|I_{n}))\\\
+\sum\limits_{m=0}^{s-1}\frac{\Theta_{m}^{(l)}(h)}{\lambda_{\min}(\Pi_{m}^{(l)})}+\ln\lambda_{\max}((P_{s}^{(l)})^{-1}B^{\operatorname{T\,}}P_{0}B)\end{gathered}$

Note that from the condition $\theta_{1}\leq\theta_{2}$ it follows that
$N_{3}\leq N_{4}-1$. By assumption about dwell-times, $\theta_{1}\leq
T_{k}\leq\theta_{2}$. Therefore,

$\begin{gathered}h(\varkappa_{k}-d_{k})<T_{k}\leq\theta_{2},\end{gathered}$

and since $\varkappa_{k}-d_{k}\in\mathbb{Z}_{+}$, we get
$\varkappa_{k}-d_{k}\leq N_{4}-1$. On the other hand,

$\begin{gathered}\theta_{1}\leq
T_{k}=\tau_{k+1}-\varkappa_{k}h+h(\varkappa_{k}-d_{k})+d_{k}h-\tau_{k}\leq
h(\varkappa_{k}-d_{k})+2h\end{gathered}$

Consequently, $\varkappa_{k}-d_{k}\geq\frac{\theta_{1}}{h}-2$, and since
$\varkappa_{k}-d_{k}\in\mathbb{Z}_{+}$, we have $\varkappa_{k}-d_{k}\geq
N_{3}$. From the conditions of the theorem, it follows that

$\begin{gathered}Q_{\max}=\max\limits_{l=0,\dots,N-1,N_{3}\leq s\leq
N_{4}-1}Q_{s}^{(l)}<0.\end{gathered}$

It follows from the inequality (43) that for all $k\in\mathbb{Z}_{+}$,

$\begin{gathered}v(\tau_{k+1}+0,x_{1}(\tau_{k+1}+0),x_{2}(\tau_{k+1}+0))\leq
e^{Q_{\max}}v(\tau_{k}+0,x_{1}(\tau_{k}+0),x_{2}(\tau_{k}+0)).\end{gathered}$
(44)

From the inequalities (44), we find

$\begin{gathered}v(\tau_{k}+0,x(\tau_{k}+0))\leq
e^{kQ_{\max}}v(\tau_{0}+0,x(\tau_{0}+0)).\end{gathered}$ (45)

Hence,

$\begin{gathered}\lambda_{\min}(P_{0})\|x(\tau_{k}+0)\|^{2}\leq
v(\tau_{k}+0,x(\tau_{k}+0))\\\ \leq
e^{kQ_{\max}}v(\tau_{0}+0,x(\tau_{0}+0))\leq
e^{kQ_{\max}}\lambda_{\max}(P_{0})\|x(\tau_{0}+0)\|^{2}.\end{gathered}$

and

$\begin{gathered}\|x(\tau_{k}+0)\|\leq
e^{kQ_{\max}/2}\sqrt{\frac{\lambda_{\max}(P_{0})}{\lambda_{\min}(P_{0})}}\|x(\tau_{0}+0)\|.\end{gathered}$

Without loss of generality, we assume that $t_{0}<\tau_{0}$. Using the
Gronwall-Bellman inequality, one can show that there exists a positive
constant $C_{t_{0}}$ such that $\|x(\tau_{0}+0)\|\leq C_{t_{0}}\|x_{0}\|$. It
follows from the inequality $T_{k}<\theta_{2}$ that there exists a positive
constant $C_{2}$ such that for all $t\in(\tau_{k},\tau_{k+1}]$, the inequality
$\|x(t)\|\leq C_{2}\|x(\tau_{k}+0)\|$ holds. Let $t\in(\tau_{k},\tau_{k+1}]$.
Then, the inequality

$\begin{gathered}t-t_{0}=t-\tau_{k}+\sum\limits_{s=1}^{k}T_{s}+\tau_{0}-t_{0}\leq
k\theta_{2}+\theta_{2}+\tau_{0}-t_{0}\end{gathered}$

implies that

$\begin{gathered}kQ_{\max}\leq\frac{Q_{\max}}{\theta_{2}}(t-t_{0})-\frac{(\theta_{2}+\tau_{0}-t_{0})Q_{\max}}{\theta_{2}}.\end{gathered}$

Thus,

$\begin{gathered}\|x(t)\|\leq Ce^{-\beta(t-t_{0})}\|x_{0}\|,\quad t\geq
t_{0},\end{gathered}$ (46)

where

$\begin{gathered}C=C_{t_{0}}C_{2}\exp\Big{(}-\frac{(\theta_{2}+\tau_{0}-t_{0})Q_{\max}}{2\theta_{2}}\Big{)}\sqrt{\frac{\lambda_{\max}(P_{0})}{\lambda_{\min}(P_{0})}},\quad\beta=-\frac{Q_{\max}}{2\theta_{2}}>0.\end{gathered}$

The estimate (46) implies the exponential stability of (1). The theorem is
proved.

To verify the conditions of Theorem 5.1, it is necessary to calculate
$\Theta_{m}^{(l)}(h)$. In addition, with the constants $\gamma_{ij}^{(m+l)}$,
$m=0,\dots,N-1$, $l=0,\dots,N-1$ included in $\Theta_{m}^{(l)}(h)$, it is
necessary to calculate the matrices $P_{ij}^{(m,l)}$, $i,j=1,2$, using the
recurrent formulas (6)-(8) and check a finite number $N_{4}-N_{3}$ of
inequalities (38).

##  6 Comparison of results

Here we compare the results obtained in the Section 4 with the known stability
conditions for coupled time-variant systems. We restrict ourselves to a
special case of high-frequency periodic functions, assuming dwell-time
$T_{k}=\theta$ are constant and $\theta$ is a sufficiently small parameter.
The value of this parameter we obtain from Theorem 4.1 applied to the case
$N=1$. We will compare our results with small-gain conditions obtained on the
basis of the ISS theory and the Lyapunov vector function. We will also
consider the case when one of the independent subsystems is not stable, and
the known approaches from the theory of stability of coupled systems are not
applicable.

Consider (1) with
$\widehat{A}_{ij}=\frac{1}{\theta}\int_{0}^{\theta}A_{ij}(t)\,dt$ for $i\neq
j$. Let $B=(B_{ij})_{i,j=1,2}$,
$\widehat{A}=\begin{pmatrix}0&\widehat{A}_{12}\\\
\widehat{A}_{21}&0\end{pmatrix}$ are block matrices. Consider the system of
linear matrix inequalities

$\begin{gathered}\operatorname{diag\,}\\{e^{\theta
A_{11}^{\operatorname{T\,}}},e^{\theta
A_{22}^{\operatorname{T\,}}}\\}B^{\operatorname{T\,}}P_{0}B\operatorname{diag\,}\\{e^{\theta
A_{11}},e^{\theta A_{22}}\\}\prec
P_{0}-\theta(\widehat{A}^{\operatorname{T\,}}P_{0}+P_{0}\widehat{A}).\end{gathered}$
(47)

Suppose it has a solution $P_{0}=(P_{ij}^{(0)})_{i,j=1,2}$ as a symmetric
positive-definite matrix, i.e.
$P_{ij}^{(0)}=(P_{ji}^{(0)})^{\operatorname{T\,}}$. Let us define matrices

$\begin{gathered}\pi_{11}^{(0)}=P_{11}^{(0)}-\theta(2\gamma_{21}^{(0)}\|P_{12}^{(0)}\|+(\gamma_{12}^{(0)}\|P_{11}^{(0)}\|+\gamma_{21}^{(0)}\|P_{22}^{(0)}\|))I_{n_{1}},\\\
\pi_{22}^{(0)}=P_{22}^{(0)}-\theta(2\gamma_{12}^{(0)}\|P_{12}^{(0)}\|+(\gamma_{12}^{(0)}\|P_{11}^{(0)}\|+\gamma_{21}^{(0)}\|P_{22}^{(0)}\|))I_{n_{2}},\\\
\pi_{12}^{(0)}=P_{12}^{(0)},\quad\pi_{21}^{(0)}=P_{21}^{(0)}\end{gathered}$

and block matrix $P_{1}=(P_{ij}^{(1)})_{i,j=1,2}$,
$(P_{ij}^{(1)})^{\operatorname{T\,}}=P_{ji}^{(1)}$ with the blocks

$\begin{gathered}P_{11}^{(1)}=e^{-A_{11}^{\operatorname{T\,}}\theta}(P_{11}^{(0)}-\theta(P_{12}^{(0)}\widehat{A}_{21}+\widehat{A}_{21}^{\operatorname{T\,}}P_{21}^{(0)}))e^{-A_{11}\theta},\end{gathered}$
(48)
$\begin{gathered}P_{22}^{(1)}=e^{-A_{22}^{\operatorname{T\,}}\theta}(P_{22}^{(0)}-\theta(\widehat{A}_{12}^{\operatorname{T\,}}P_{12}^{(0)}+P_{21}^{(0)}\widehat{A}_{12}))e^{-A_{22}\theta}\end{gathered}$
(49)
$\begin{gathered}P_{12}^{(1)}=e^{-A_{11}^{\operatorname{T\,}}\theta}(P_{12}^{(0)}-\theta(P_{11}^{(0)}\widehat{A}_{12}+\widehat{A}_{21}^{\operatorname{T\,}}P_{22}^{(0)})))e^{-A_{22}\theta}.\end{gathered}$
(50)

The following proposition is a direct consequence of Theorem 4.1.

Proposition 6.1. Let $\Pi_{0}=(\pi_{ij}^{(0)})_{i,j=1,2}$ be positive definite
and
$Q:=\frac{\Theta_{0}(\theta)}{\lambda_{\min}(\Pi_{0})}+\ln\lambda_{\max}(P_{1}^{-1}B^{\operatorname{T\,}}P_{0}B)<0$,
then (1) is asymptotically stable.

Example.

$\begin{gathered}\dot{x}_{1}(t)=0.01x_{1}(t)-0.2\sin^{2}\frac{2\pi
t}{\theta}x_{2}(t),\\\ \dot{x}_{2}(t)=-0.1x_{2}(t)+0.2\cos^{2}\frac{2\pi
t}{\theta}x_{1}(t)\end{gathered}$ (51)

Choose $\theta=0.09$, $P_{11}^{(0)}=18$, $P_{12}^{(0)}=P_{21}^{(0)}=-7$,
$P_{22}^{(0)}=12$, then $\mu_{1}=0.01$, $\mu_{2}=-0.1$, $\delta_{1}=-0.01$,
$\delta_{2}=0.1$, $\gamma_{12}^{(0)}=\gamma_{21}^{(0)}=0.2$. By the direct
calculations we obtain that

$\begin{gathered}\Pi_{0}=\begin{pmatrix}17.208&-7\\\
-7&11.208\end{pmatrix},\quad P_{1}=\begin{pmatrix}18.0934&-7.0025\\\
-7.0024&12.0897\end{pmatrix}\end{gathered}$

and $\eta_{11}^{(0)}=26.3132$, $\eta_{22}^{(0)}=20.8924$,
$\eta_{12}^{(0)}=7.1037$, $\eta_{21}^{(0)}=7.1036$,
$\alpha_{11}^{(0)}=0.5263$, $\alpha_{12}^{(0)}=0.0526$,
$\alpha_{21}^{(0)}=0.0418$, $\alpha_{22}^{(0)}=0.4178$, $Q=-0.00004279<0$.
Therefore, the system is asymptotically stable. At the same time, the first
subsystem is unstable, which makes it impossible to apply a small-gain
theorem.

###  6.1 Comparison with the small-gain conditions

Consider (1) with $\int_{0}^{\theta}A_{ij}(t)\,dt=0$ for $i\neq j$ and
constant dwell-time $T_{k}=\theta$. Since the Lyapunov vector functions or
small-gain results are only applicable when the independent subsystems are
asymptotically stable, we assume that $r_{\sigma}(e^{\theta A_{ii}}B_{ii})<1$
for $i=1,2$. Now (47) reduces to two inequalities

$\begin{gathered}e^{\theta
A_{ii}^{\operatorname{T\,}}}B_{ii}^{\operatorname{T\,}}P_{ii}B_{ii}e^{\theta
A_{ii}}\prec P_{ii},\quad i=1,2.\end{gathered}$ (52)

To apply Theorem 4.1, we assume $P_{ii}^{(0)}=P_{ii}$, $i=1,2$
$P_{12}^{(0)}=0$, then from (48)–(50) we obtain

$\begin{gathered}P_{11}^{(1)}=e^{-\theta
A_{11}^{\operatorname{T\,}}}P_{11}e^{-\theta A_{11}},\quad
P_{22}^{(1)}=e^{-\theta A_{22}^{\operatorname{T\,}}}P_{22}e^{-\theta
A_{22}},\quad P_{12}^{(1)}=0.\end{gathered}$

Let us define

$\begin{gathered}\Phi=\begin{pmatrix}e^{\theta A_{11}}P_{11}^{-1}e^{\theta
A_{11}^{\operatorname{T\,}}}&0\\\ 0&e^{\theta A_{22}}P_{22}^{-1}e^{\theta
A_{22}^{\operatorname{T\,}}}\end{pmatrix}\begin{pmatrix}B_{11}^{\operatorname{T\,}}&B_{21}^{\operatorname{T\,}}\\\
B_{12}^{\operatorname{T\,}}&B_{22}^{\operatorname{T\,}}\end{pmatrix}\begin{pmatrix}P_{11}&0\\\
0&P_{22}\end{pmatrix}\begin{pmatrix}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{pmatrix}\end{gathered}$

A consequence of Theorem 4.1 is the following

Proposition 6.2. If $\int_{0}^{\theta}A_{ij}(t)\,dt=0$ for $i\neq j$,
$r_{\sigma}(\Phi)<1$, the conditions of Assumption 4.1 and the following
inequalities hold

$\begin{gathered}\theta<\min\Big{\\{}\frac{\lambda_{\min}(P_{11})}{\varrho},\frac{\lambda_{\min}(P_{22})}{\varrho}\Big{\\}},\\\
\frac{\Theta_{0}(\theta)}{\min\\{\lambda_{\min}(P_{11})-\theta\varrho,\lambda_{\min}(P_{22})-\theta\varrho\\}}<-\ln\lambda_{\max}(\Phi),\end{gathered}$
(53)

where $\varrho=\gamma_{12}\|P_{11}\|+\gamma_{21}\|P_{22}\|$. Then (1) is
asymptotically stable.

To compare the obtained results with the results known in the literature,
obtained on the basis of the ISS approach or Lyapunov vector function (small-
gain conditions), we consider (1) without impulsive action, i.e. $B_{ii}=I$,
$B_{ij}=0$ for $i,j=1,2$, $i\neq j$ and such that
$\int_{0}^{\theta}A_{ij}(t)\,dt=0$ for $i\neq j$.

Since the Lyapunov vector function or small-gain results are only applicable
when the independent subsystems are asymptotically stable, so we assume, that
$\max\\{\operatorname{Re\,}\lambda\,|\,\lambda\in\sigma(A_{ii})\\}<0$ for
$i=1,2$. For given symmetric and positive-definite matrices $Q_{i}$, $i=1,2$
consider the linear algebraic Lyapunov equations

$\begin{gathered}A_{ii}^{\operatorname{T\,}}P_{ii}+P_{ii}A_{ii}=-Q_{i}.\end{gathered}$
(54)

It is known that under our assumptions for $A_{ii}$, $i=1,2$ these equations
have unique solutions in the form of symmetric positive-definite matrices
$P_{ii}$.

Remark 6.1. Solutions of matrix algebraic equations (54) satisfy linear matrix
inequalities (52), up to $O(\theta^{2})$.

In this case $\Phi=\operatorname{diag\,}\\{e^{\theta
A_{11}}P_{11}^{-1}e^{\theta A_{11}^{\operatorname{T\,}}}P_{11},e^{\theta
A_{22}}P_{22}^{-1}e^{\theta A_{22}^{\operatorname{T\,}}}P_{22}\\}$. Since, we
assume that $\theta$ is a sufficiently small positive number, we note that

$\begin{gathered}\Phi=I-\theta\operatorname{diag\,}\\{P_{11}^{-1}Q_{1},P_{22}^{-1}Q_{2}\\}+O(\theta^{2}),\end{gathered}$

therefore

$\begin{gathered}-\ln\lambda_{\max}(\Phi)=\theta\min\\{\lambda_{\min}(P_{11}^{-1}Q_{1}),\lambda_{\min}(P_{22}^{-1}Q_{2})\\}+O(\theta^{2})\end{gathered}$

is positive-defined for sufficiently small $\theta>0$. On the other hand, it
is easy to show that
$\frac{\Theta_{0}(\theta)}{\min\\{\lambda_{\min}(P_{11})-\theta\varrho,\lambda_{\min}(P_{22})-\theta\varrho\\}}=O(\theta^{2})$,
hence it follows that there exists $\theta^{*}>0$, such that for all
$\theta\in(0,\theta^{*})$ the conditions of Proposition 6.1 are satisfied.
Thus, we come to an important Corollary.

Corollary 6.1. System (1) without impulsive action and
$\int_{0}^{\theta}A_{ij}(t)\,dt=0$ is asymptotically stable for
$\theta\in(0,\theta^{*})$ if $\theta^{*}>0$ is small enough.

Note that the $\theta^{*}$ is determined from the conditions (53).

We apply the same Lyapunov functions
$V_{1}(x_{1})=x_{1}^{\operatorname{T\,}}P_{11}x_{1}$,
$V_{2}(x_{2})=x_{2}^{\operatorname{T\,}}P_{22}x_{2}$ to study the stability of
the coupled system (1) without impulsive action, using the small-gain theorem
in [27]. Consider estimates of derivatives $\dot{V}_{i}$ along solutions of
(1). Taking into account Assumption 4.1 and using the Cauchy–Bunyakovsky
inequality, we define

$\begin{gathered}\dot{V}_{i}(x_{i})=-x_{i}^{\operatorname{T\,}}Q_{i}x_{i}+2x_{i}^{\operatorname{T\,}}P_{ii}A_{ij}(t)x_{j}\\\
=-(P_{i}^{1/2}x_{i})^{\operatorname{T\,}}P_{ii}^{-1/2}Q_{i}P_{ii}^{-1/2}P_{ii}^{1/2}x_{i}+2(P_{ii}^{1/2}x_{i})^{\operatorname{T\,}}P_{ii}^{1/2}A_{ij}(t)P_{jj}^{-1/2}P_{ii}^{1/2}x_{j}\\\
\leq-\lambda_{\min}(P_{ii}^{-1/2}Q_{i}P_{ii}^{-1/2})V_{i}(x_{i})+2\|P_{ii}\|^{1/2}\|P_{jj}\|^{-1/2}\gamma_{ij}V_{ii}^{1/2}(x_{i})V_{jj}^{1/2}(x_{j})\\\
=-\lambda_{\min}(P_{ii}^{-1}Q_{i})V_{i}(x_{i})+2\|P_{ii}\|^{1/2}\|P_{jj}\|^{-1/2}\gamma_{ij}V_{i}^{1/2}(x_{i})V_{j}^{1/2}(x_{j}).\end{gathered}$
(55)

Here $i\neq j$, $i,j=1,2$. To check the conditions of the small-gain theorem
from [27] (Theorem 4), we choose

$\begin{gathered}\chi_{i}(r)=\Big{(}\frac{2\gamma_{ij}\|P_{ii}\|^{1/2}\|P_{jj}\|^{-1/2}}{\lambda_{\min}(P_{ii}^{-1}Q_{i})}+\varepsilon\Big{)}^{2}r.\end{gathered}$

Then Theorem 4 from [27] lead us to the following sufficient conditions for
the asymptotic stability of (1) (small-gain condition)

$\begin{gathered}\gamma_{12}\gamma_{21}<\frac{1}{4}\lambda_{\min}(P_{11}^{-1}Q_{1})\lambda_{\min}(P_{22}^{-1}Q_{2})\end{gathered}$
(56)

Remark 6.2. The method of Lyapunov vector functions lead us to the same
stability condition. Indeed, (55) in the new variables
$y_{i}(t)=V_{i}^{1/2}(x_{i}(t))$ leads to a linear system of differential
inequalities

$\begin{gathered}\dot{y}_{i}(t)\leq-\frac{1}{2}\lambda_{\min}(P_{ii}^{-1}Q_{i})y_{i}(t)+\|P_{ii}\|^{1/2}\|P_{jj}\|^{-1/2}\gamma_{ij}y_{j}(t).\end{gathered}$

Application of the comparison principle again leads to (56). From (56) follows
that the small-gain conditions are not depend on $\theta$. Therefore, it is
possible to choose the parameters of the system (1), such that (56) is not
satisfied, however, based on Corollary 6.1, this system is asymptotically
stable for sufficiently small $\theta$. Thus, we conclude that our approach
leads to less conservative stability conditions than the known small-gain
conditions.

Example. Consider a second-order linear system

$\begin{gathered}\dot{x}_{1}(t)=-0.2x_{1}(t)+0.15a_{12}(t)x_{2}(t),\\\
\dot{x}_{2}(t)=-0.1x_{2}(t)+0.2a_{21}(t)x_{1}(t)\end{gathered}$ (57)

where $a_{ij}\in C(\mathbb{R})$, $\|a_{ij}\|_{C[0,\theta]}\leq 1$,
$\int\limits_{0}^{\theta}a_{ij}(t)\,dt=0$, then $\mu_{1}=-0.2$,
$\mu_{2}=-0.1$, $\delta_{1}=0.2$, $\delta_{2}=0.1$ and the small-gain
condition $0.03\|a_{12}\|_{C[0,\theta]}\|a_{21}\|_{C[0,\theta]}<0.02$ is not
satisfied.

Choose $\theta=0.5$, $P_{11}^{(0)}=2.5$, $P_{22}^{(0)}=5$, $P_{12}^{(0)}=0$,
$M_{1}=M_{2}=N_{1}=N_{2}=1$. By a direct calculation $\gamma_{12}^{(0)}=0.15$,
$\gamma_{21}^{(0)}=0.2$,

$\begin{gathered}\Pi_{0}=\begin{pmatrix}1.8125&0\\\
0&4.3125\end{pmatrix},\quad P_{1}=\begin{pmatrix}3.0535&0\\\
0&5.5259\end{pmatrix}\end{gathered}$

Since $Q=-0.0050178428<0$, the considered system asymptotically stable.

Note that in our example (57), the functions $a_{ij}$ are actually unknown, we
know only restrictions on their norm and mean value. Thus, Theorem 4.1
provides conditions for a whole class of systems (57). In this sense, Theorem
4.1 is similar to the well-known small-gain theorems for coupled systems.

##  7 Numerical examples

Here we consider some examples of application of the main results from
Sections 4 and 5 to fourth order systems. Example 1 is an impulsive system,
with one unstable subsystem. In this case, small-gain conditions are not
applicable. Example 2 illustrates the possibility to apply Theorem 4.1 in the
case when the continuous and discrete dynamics are both unstable at constant
dwell-times, and example 3 is the general case of non-constant dwell-times.

Example 1. Consider a linear fourth-order impulsive system (1) with the
matrices

$\begin{gathered}A_{11}=\begin{pmatrix}0.1&0.05\\\ 0.05&0.1\end{pmatrix},\quad
A_{22}=\begin{pmatrix}-1&0.01\\\ 0.01&-1\end{pmatrix}\end{gathered}$
$\begin{gathered}A_{12}(t)=-2\sin^{2}(\omega t)I,\quad
A_{21}(t)=2\sin^{2}(\omega t)I;\end{gathered}$

$\begin{gathered}B_{11}=\begin{pmatrix}0.98&0\\\ 0&0.98\end{pmatrix},\quad
B_{22}=\begin{pmatrix}1.02&0\\\ -0.01&1.01\end{pmatrix}\end{gathered}$
$\begin{gathered}B_{12}=\begin{pmatrix}-0.01&0\\\ 0.02&0\end{pmatrix},\quad
B_{21}=\begin{pmatrix}0&0.01\\\ -0.05&0\end{pmatrix}.\end{gathered}$

Here, $\tau_{k+1}-\tau_{k}=\theta=0.2$, $\omega=\frac{2\pi}{\theta}$. To check
the asymptotic stability conditions of Theorem 6.1, we choose $N=50$,
$P_{11}^{(0)}=18I$, $P_{12}^{(0)}=-7I$, $P_{22}^{(0)}=12I$. Then
$\min_{k}\lambda_{\min}(\Pi_{k})=7.0322$, $Q=-0.0464<0$. Hence (1) is
asymptotically stable. Note, that the independent subsystem

$\begin{gathered}\dot{x}_{2}(t)=A_{22}x_{2}(t),\quad t\neq\tau_{k},\\\
x_{2}(t^{+})=B_{22}x_{2}(t),\quad t=\tau_{k},\\\ \end{gathered}$ (58)

is not stable due to the fact that $r_{\sigma}(e^{A_{22}\theta}B_{22})>1$.

Example 2. Consider a linear impulsive system (1) with the matrices

$\begin{gathered}A_{11}=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix},\quad
A_{22}=\begin{pmatrix}0.1&0\\\ 0&0.1\end{pmatrix}\end{gathered}$
$\begin{gathered}A_{12}(t)=\begin{pmatrix}0.2\cos(\omega t)&-0.2\sin(\omega
t)\\\ 0.2\sin(\omega t)&0.2\cos(\omega t)\end{pmatrix},\\\
A_{21}(t)=\begin{pmatrix}0.1\cos(\omega t)&-0.1\sin(\omega t)\\\
0.1\sin(\omega t)&0.1\cos(\omega t)\end{pmatrix}\end{gathered}$
$\begin{gathered}B_{11}=\begin{pmatrix}1.2&0.1\\\ -0.1&1.5\end{pmatrix},\quad
B_{22}=\begin{pmatrix}0.5&0.05\\\ -0.05&-0.5\end{pmatrix}\end{gathered}$
$\begin{gathered}B_{12}=\begin{pmatrix}0.04&0.1\\\ 0.1&0.04\end{pmatrix},\quad
B_{21}=\begin{pmatrix}0.05&0.1\\\ 0.2&0.05\end{pmatrix}\end{gathered}$

Here, $\tau_{k+1}-\tau_{k}=\theta=0.5$, $\omega=\frac{2\pi}{\theta}$. To check
the asymptotic stability conditions, obtained in Theorem 6.1, we choose $N=3$,
$P_{11}^{(0)}=I$, $P_{12}^{(0)}=0$, $P_{22}^{(0)}=I$. Then,

$\begin{gathered}P_{3}=\begin{pmatrix}2.7172&0&-0.0008&-0.0207\\\
0&2.7172&0.0207&-0.0008\\\ -0.0008&0.0207&0.902&0\\\
-0.0207&-0.0008&0&0.902\end{pmatrix},\end{gathered}$

$\min_{k}\lambda_{\min}(\Pi_{k})=0.7793$, $Q=-0.0383<0$. Therefore, the linear
impulsive system (1) is asymptotically stable. Consider separately the
continuous dynamics of a linear impulsive system, which is described by a
linear time-variant ODEs

$\begin{gathered}\dot{x}_{1}(t)=A_{11}x_{1}(t)+A_{12}(t)x_{2}(t),\\\
\dot{x}_{2}(t)=A_{21}(t)x_{1}(t)+A_{22}x_{2}(t),\end{gathered}$ (59)

We denote

$U(\omega t)=\begin{pmatrix}\cos(\omega t)&-\sin(\omega t)\\\ \sin(\omega
t)&\cos(\omega t)\end{pmatrix}$

and rewrite (59) as

$\begin{gathered}\dot{x}_{1}(t)=-x_{1}(t)+0.2U(\omega t)x_{2}(t),\\\
\dot{x}_{2}(t)=0.1U(\omega t)x_{1}(t)+0.1x_{2}(t),\end{gathered}$ (60)

Consider the Lyapunov–Chetaev function
$v(x_{1},x_{2})=2\|x_{2}\|^{2}-\|x_{1}\|^{2}$, the total derivative of which
is

$\begin{gathered}\dot{v}(x_{1},x_{2})=2(0.2x_{2}\|^{2}+\|x_{1}\|^{2}+0.4\sin(\omega
t)x_{1}^{\operatorname{T\,}}Jx_{2})\\\ \geq
2(0.2\|x_{2}\|^{2}+\|x_{1}\|^{2}-0.4\|x_{1}\|\|x_{2}\|)>0,\quad\text{for all
}(x_{1},x_{2})\neq(0,0).\end{gathered}$

Here, $J=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ is the symplectic unit.
Hence, the ODE (60) is unstable. The discrete dynamics is unstable since
$r_{\sigma}(B)=1.4722$.

Example 3. Consider an example, which illustrate the application of Theorem
5.1 to the case of a non-periodic impulsive system. Let

$\begin{gathered}A_{11}=\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix},\quad
A_{22}=\begin{pmatrix}0.1&0\\\ 0&0.1\end{pmatrix}\end{gathered}$
$\begin{gathered}A_{12}(t)=\begin{pmatrix}0.05\cos(\omega t)&-0.05\sin(\omega
t)\\\ 0.05\sin(\omega t)&0.05\cos(\omega t)\end{pmatrix},\\\
A_{21}(t)=\begin{pmatrix}0.05\cos(\omega t)&-0.05\sin(\omega t)\\\
0.05\sin(\omega t)&0.05\cos(\omega t)\end{pmatrix}\end{gathered}$
$\begin{gathered}B_{11}=\begin{pmatrix}1.2&0.1\\\ -0.1&1.5\end{pmatrix},\quad
B_{22}=\begin{pmatrix}0.25&0.05\\\ -0.05&-0.25\end{pmatrix}\end{gathered}$
$\begin{gathered}B_{12}=\begin{pmatrix}0.04&0.1\\\ 0.1&0.04\end{pmatrix},\quad
B_{21}=\begin{pmatrix}0.05&0.1\\\ 0.2&0.05\end{pmatrix}\end{gathered}$

For $\theta=1$, $\theta_{1}=0.8$, $\theta_{2}=1.2$, we choose $N=7$,
$P_{11}^{(0)}=I$, $P_{12}^{(0)}=0$, $P_{22}^{(0)}=I$. Then, $N_{3}=4$,
$N_{4}=8$, $\min_{m=\overline{0,11}}\lambda_{\min}(\Pi_{m}^{(l)})=0.7585>0$,
$\max_{l=\overline{0,9},m=\overline{6,12}}Q_{m}^{(l)}=-0.1399<0$. Thus, by
Theorem 5.1, the linear system (1) is asymptotically stable only if the dwell-
times $T_{k}=\tau_{k+1}-\tau_{k}$ belong to the interval
$[\theta_{1},\theta_{2}]$. We note that in this example both continuous and
discrete dynamics are not stable. The instability of continuous dynamics is
proved using the Lyapunov–Chetaev function
$v(x_{1},x_{2})=2\|x_{2}\|^{2}-\|x_{1}\|^{2}$ as done above. The instability
of discrete dynamics follows from the fact that $r_{\sigma}(B)=1.4746$.

##  8 Discussion

Theorems 4.1 and 5.1 are the main result of the paper and establish sufficient
conditions of the exponential stability of the linear time-variant impulsive
system (1). The proposed approach to the study of a coupled impulsive system
significantly expands the capabilities of the method of Lyapunov vector
functions and allows one to study the asymptotic stability of a linear
impulsive system with unstable subsystems. Given examples in Section 6 show
that our results significantly expand the known methods for studying impulsive
systems developed in [15, 14, 9, 5]. In addition, the obtained results are
applied in the case when the continuous and discrete dynamics of an impulsive
system are both unstable. In the case when dwell-times are non constant for
studying the stability of a linear impulsive system, the classical Floquet
theory turns out to be inapplicable. These results significantly expand the
possibilities of the direct Lyapunov method in the context of theorems from
[1].

In this case, Theorem 5.1 allows us to deduce the Lyapunov asymptotic
stability. It is of interest to generalize the proposed approaches for the
construction of Lyapunov functions for nonlinear coupled systems, as well as
when expanding the assumptions about the number of independent subsystems.

##  9 Appendix

Proof of Lemma 4.1. Using the expressions for $P_{ij}^{(m)}$ we obtain

$\begin{gathered}v(t,x_{1},x_{2})=z_{1m}^{\operatorname{T\,}}\Big{(}P_{11}^{(m)}-\int\limits_{mh}^{t}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,ds\Big{)}z_{1m}\\\
+2z_{1m}^{\operatorname{T\,}}\Big{(}P_{12}^{(m)}-\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,ds\Big{)}z_{2m}\\\
+z_{2m}^{\operatorname{T\,}}\Big{(}P_{22}^{(m)}-\int\limits_{mh}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,ds\Big{)}z_{2m}.\end{gathered}$

Applying the Cauchy-Bunyakovsky inequality, we obtain

$\begin{gathered}\Big{|}z_{1m}^{\operatorname{T\,}}\int\limits_{mh}^{t}(P_{12}^{(m)}A_{21}(s)+A_{21}^{\operatorname{T\,}}(s)P_{21}^{(m)})\,dsz_{1m}\Big{|}\leq
2\|P_{12}^{(m)}\|\gamma_{21}^{(m)}\|z_{1m}\|^{2},\end{gathered}$
$\begin{gathered}\Big{|}z_{1m}^{\operatorname{T\,}}\int\limits_{mh}^{t}(P_{11}^{(m)}A_{12}(s)+A_{21}^{\operatorname{T\,}}(s)P_{22}^{(m)})\,dsz_{2m}\Big{|}\leq(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})\|z_{1m}\|\|z_{2m}\|,\end{gathered}$
$\begin{gathered}\Big{|}z_{2m}^{\operatorname{T\,}}\int\limits_{mh}^{t}(A_{12}^{\operatorname{T\,}}(s)P_{12}^{(m)}+P_{21}^{(m)}A_{12}(s))\,dsz_{2m}\Big{|}\leq
2\|P_{12}^{(m)}\|\gamma_{12}^{(m)}\|z_{2m}\|^{2}\end{gathered}$

Thus,

$\begin{gathered}v(t,x_{1},x_{2})\geq
z_{1m}^{\operatorname{T\,}}P_{11}^{(m)}z_{1m}+2z_{1m}^{\operatorname{T\,}}P_{12}^{(m)}z_{2m}+z_{2m}^{\operatorname{T\,}}P_{22}^{(m)}z_{2m}-2\|P_{12}^{(m)}\|\gamma_{21}^{(m)}\|z_{1m}\|^{2}\\\
-2(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)})\|z_{1m}\|\|z_{2m}\|-2\|P_{12}^{(m)}\|\gamma_{12}^{(m)}\|z_{2m}\|^{2}.\end{gathered}$

Applying the Cauchy inequality, we get

$\begin{gathered}v(t,x_{1},x_{2})\geq
z_{1m}^{\operatorname{T\,}}P_{11}^{(m)}z_{1m}+2z_{1m}^{\operatorname{T\,}}P_{12}^{(m)}z_{2m}+z_{2m}^{\operatorname{T\,}}P_{22}^{(m)}z_{2m}\\\
-(2\|P_{12}^{(m)}\|\gamma_{21}^{(m)}+(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)}))\|z_{1m}\|^{2}\\\
-(2\|P_{12}^{(m)}\|\gamma_{12}^{(m)}+(\|P_{11}^{(m)}\|\gamma_{12}^{(m)}+\|P_{22}^{(m)}\|\gamma_{21}^{(m)}))\|z_{2m}\|^{2}\geq\lambda_{\min}(\Pi_{m})\|z_{m}\|^{2}.\end{gathered}$

The upper bound for the Lyapunov function $v(t,x_{1},x_{2})$ is proved
similarly.

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