Finite-time stability of set switched systems with non-instantaneous impulses

ABSTRACT

In this paper, we discuss the finite-time stability of set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems through introducing a revised mode-dependent average dwell time method. By designing time-dependent switching law and using the multiple Lyapunov-like functions method, the finite-time stability criteria of set switched systems with non-instantaneous impulses are given. Sufficient conditions which guarantee the finite-time boundedness of the switched systems with time-varying exogenous disturbances are also given. In our switching design strategy, slow switching and fast switching are, respectively, used among stable subsystems and unstable subsystems. The criteria obtained for the switched linear systems are all provided in terms of a set of linear matrix inequalities. Numerical examples are employed to verify the efficiency of the proposed method.

KEYWORDS:

• Switched system
• set-valued
• non-instantaneous impulses
• finite-time stability
• finite-time boundedness

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34A37
• 93C30
• 93D30

1. Introduction

Switched systems have been widely studied due to their extensive applications in power electronics, traffic, robot control systems and other areas [1–4]. Switched systems consist of a series of subsystems and switching laws that coordinate the switching. The earlier performance analysis of switched systems is the stability study, and most of the results involve the stability of the systems in an infinite time [5,6]. However, in practical engineering, there are systems which work in a short time, such as missile system, communication network system, robot operating and so on. Hence, it is important to investigate finite-time stability, which refers to that for the given time interval and initial state of the system, and the state trajectory is always kept within a given range [7]. Amato et al. [8] and Ariola et al. [9] considered the existence of certain external disturbances, then extended the finite-time stability to the finite-time boundedness. The concepts of finite-time stability and finite-time boundedness are introduced into switched systems for the first time and sufficient conditions are given in Ref. [10]. In addition, more practical situations will lead to the instability of the subsystem of the switching system. Zhai et al. [11] discussed the stability of the switched linear system with average dwell time switching and unstable subsystems. The stability criterion of switched systems by meshing together the quasi-alternating switching signal and the multiple Lyapunov functions method was proposed [12], which is effective for discussing the stability of ones with unstable subsystems. Li et al. [13] first discussed the sufficient conditions of finite-time stability of switched nonlinear systems with subsystems that are not finite-time stable by Lyapunov-like functions. Recently, in Refs. [14–17], finite-time stability and finite-time boundedness of linear switched systems with or without time-delays and finite-time stabilization of switched nonlinear singular systems with asynchronous switching are discussed.

Set differential equations (SDEs) are effective tools for describing uncertain systems, and ones generated by multivalued differential inclusions have been introduced in a semi-linear metric space, consisting of all nonempty, compact, convex subsets of an initial finite or infinite dimensional space [18]. It has been widely used in the fields of control science, biology, computer and information processing and so on. In 1969, Brandao Lopes Pinto et al. proposed the existence and uniqueness of solutions for SDEs [19]. For systematic work on SDEs, see Martynyuk's monograph [20]. Recently, in Refs. [21,22], the results of asymptotic stability of neutral set-valued functional differential equation and stability of perturbed SDEs involving causal operators are obtained. It is noted that most works are investigating the Lyapunov asymptotic stability of SDEs in an infinite time interval, and there are few results about finite-time stability of SDEs with switching.

Since Millman and Myshkis [23,24] first proposed impulsive differential equation in the 1960s, the theory of impulsive differential equation has become an important research field of differential equations. Impulses can be divided into instantaneous impulses and non-instantaneous ones according to the time of action of the impulses. The non-instantaneous impulse means that the interference process depends on the state and lasts for a period of time. The phenomenon of non-instantaneous impulse is ubiquitous, and it has been widely used in pharmacokinetics, population ecological dynamics, infectious disease dynamics and so on. In 2013, Hernández and O'Regan [25] first proposed the theory of non-instantaneous impulsive differential equations and studied the existence of weak solutions and classical solutions. The research on the theory and application of non-instantaneous ones has just started and has attracted the attention of scholars rapidly. The basic results can be found in Agarwal's literature [26]. Recently, in Refs. [27–31], several authors have discussed Lipschitz stability and finite-time stability of delay differential equations with non-instantaneous impulses and non-instantaneous impulsive fractional differential equations. Generally speaking, there is usually non-instantaneous impulsive effect during system switching. However, few existing results have noticed the switched systems with non-instantaneous impulses. Therefore, establishing the mathematical model of the set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems has important practical significance. That is to say, the problem of non-instantaneous impulsive systems has not been completely solved. The above analysis prompts us to consider the finite-time stability and finite-time boundedness of set switched systems with non-instantaneous impulses.

Compared to the existing literature, the main contribution of this work is to generalize switched systems to SDEs with non-instantaneous impulses. In this paper, we discuss the finite-time stability of set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems through introducing a revised mode-dependent average dwell time (MDADT) method and using the multiple Lyapunov-like functions. By designing time-dependent switching law and linear matrix inequality conditions, the finite-time stability and finite-time boundedness criteria of set switched systems with non-instantaneous impulses are given.

2. Preliminaries

Let $K_c\left({\mathbb{R}}^n\right)$ denote the collection of all convex, compact and nonempty subsets of ${\mathbb{R}}^n$. Define the Hausdorff metric $D[A,B]=max\left\{\underset{x\in B}{sup}\mathrm{d}\right(x,A),\underset{y\in A}{sup}\mathrm{d}(y,B\left)\right\},$ where $A,B\in K_c\left({\mathbb{R}}^n\right)$ and $\mathrm{d}(x,A)=inf\left\{\mathrm{d}\right(x,y):y\in A\}$. In particular, $D[A,\theta ]=\underset{y\in A}{sup}\mathrm{d}(y,\theta ),$ where θ is the zero element of $K_c\left({\mathbb{R}}^n\right)$.

Given any two sets $A,B\in K_c\left({\mathbb{R}}^n\right)$. We called the set C is the Hukuhara difference of the sets A and B, when A = B + C, where $C\in K_c\left({\mathbb{R}}^n\right)$, and it is denoted by A−B.

Definition 2.1

[18]

The mapping $F:I(=[0,T\left]\right)\to K_c\left({\mathbb{R}}^n\right)$ has a Hukuhara derivative $D_{H}F\left(t_0\right)$ at a point $t_0\in I$, if $D_{H}F\left(t_0\right)=\underset{h\to 0^{+}}{lim}\frac{F(t_0+h)-F\left(t_0\right)}{h}=\underset{h\to 0^{+}}{lim}\frac{F\left(t_0\right)-F(t_0-h)}{h},$

where h>0.

Definition 2.2

[18]

The Hukuhara integral of F is given by ${\int }_{I}F\left(s\right)\mathrm{d}s=\left\{{\int }_{I}f\right(s)\mathrm{d}s:f\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{f}F\}.$ For the properties of Hukuhara derivative and Hukuhara integral of set-valued functions, please refer to the literature [18].

Let two increasing sequences of points $\{t_k{\}}_{k=1}^{\mathrm{\infty }}$ and $\{s_k{\}}_{k=0}^{\mathrm{\infty }}$ be given s.t. $0<s_0<t_k\le s_k<t_{k+1}$, $k=1,2,\dots$. Without loss of generality, we assume that $t_0\in [0,s_0)$.

For $X\left(t\right)\in K_c\left({\mathbb{R}}^n\right)$, let $\mathrm{\Gamma }X\left(t\right)=\left\{\mathrm{\Gamma }x\right(t\left)|x\right(t)\in X(t\left)\right\}$, where Γ is a constant real matrix.

Consider the following set switched system: (1) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(1) where $X\left(t\right)\in K_c\left({\mathbb{R}}^n\right)$, $F_{\sigma \left(t\right)}:{\cup }_{k=0}^{\mathrm{\infty }}[t_k,s_k]\times K_c\left({\mathbb{R}}^n\right)\to K_c\left({\mathbb{R}}^n\right)$, ${\varphi }_k:(s_k,t_{k+1}]\times K_c\left({\mathbb{R}}^n\right)\to K_c\left({\mathbb{R}}^n\right)$, $k=0,1,2,\dots$. The switching signal $\sigma \left(t\right):{\mathbb{R}}^{+}\to M=\{1,2,\dots ,m\}$, where $m\in {\mathbb{N}}^{+}$ is the number of subsystems. We assume that $M=\phi \cup \psi$, where $\phi =\{1,2,\dots ,u\}$ and $\psi =\{u+1,\dots ,m\}$ denote stable and unstable subsystems, respectively.

If each subsystem in $(t_k,s_k]$ is linear time-invariant system, the system (1(1) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(1) ) converts to (2) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(2) where $L_{\sigma \left(t\right)}$ is a constant real matrix, $k=0,1,2,\dots$.

When the system (1(1) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(1) ) exists the external perturbation $W\left(t\right)\in K_c\left({\mathbb{R}}^n\right)$, it converts to (3) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right)+H_{\sigma \left(t\right)}(t,W(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(3) where $H_{\sigma \left(t\right)}:{\cup }_{k=0}^{\mathrm{\infty }}[t_k,s_k]\times K_c\left({\mathbb{R}}^n\right)\to K_c\left({\mathbb{R}}^n\right)$, $k=0,1,2,\dots$.

Similarly, when the system (2(2) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(2) ) exists the exogenous disturbance, it converts to (4) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right)+G_{\sigma \left(t\right)}W\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(4) where $L_{\sigma \left(t\right)}$, $G_{\sigma \left(t\right)}$ are constant real matrices, $k=0,1,2,\dots$.

In order to prevent systems to exhibit Zeno behaviour, assume that for any given interval $[0,T]$, there are only a finite number of switching times. The switching law $\sigma \left(t\right)$ satisfies the following definition.

Definition 2.3

[12]

The switching law $\sigma \left(t\right)$ is called the quasi-alternative switching signal, if

1. $\sigma \left(t_k\right)\in \phi$, then $\sigma \left(t_{k+1}\right)\in M$, $\sigma \left(t_k\right)\ne \sigma \left(t_{k+1}\right)$;

2. $\sigma \left(t_k\right)\in \psi$, then $\sigma \left(t_{k+1}\right)\in \phi$.

Moreover, the definitions of the switching laws based on the MDADT property are given below.

Definition 2.4

[12]

For $N_{\sigma p}(T,t)$ which is expressed as the switching times of the pth subsystem and the overall running time $T_p(T,t)$ of ones in any $t\in [t_0,T]$, $p\in \phi$, we can find two constants $N_{0p}$ and ${\tau }_{ap}$ satisfying (5) $N_{\sigma p}(T,t)\le N_{0p}+\frac{T_p(T,t)}{{\tau }_{ap}}\mathrm{\forall }t\in [t_0,T],$(5) where ${\tau }_{ap}$ is called the slow MDADT of the switching signal $\sigma \left(t\right)$.

Definition 2.5

[12]

For $N_{\sigma q}(T,t)$ which is expressed as the switching times of the qth subsystem and the overall running time $T_q(T,t)$ of ones in any time interval $t\in [t_0,T]$, $q\in \psi$, we can find two constants $N_{0q}$ and ${\tau }_{aq}$ satisfying (6) $N_{\sigma q}(T,t)\ge N_{0q}+\frac{T_q(T,t)}{{\tau }_{aq}}\mathrm{\forall }t\in [t_0,T],$(6) where ${\tau }_{aq}$ is called the fast (reverse)MDADT of the switching signal $\sigma \left(t\right)$.

Note that in this paper we choose $N_{0p}=N_{0q}=0$, as commonly used in Ref. [32].

3. Main results

We now define the finite-time stable of set switched systems with non-instantaneous impulses and the finite-time bounded when ones are subject to the exogenous disturbance.

For convenience, we first give the following set required in this paper: $\mathcal{K}=\{a\in C[{\mathbb{R}}^{+},{\mathbb{R}}^{+}]:a\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{n}\mathrm{d}a(0)=0\}.$

Definition 3.1

Given three positive constants $c_2$, $c_1$, T, with $c_1<c_2$, and the given switching signal $\sigma \left(t\right)$, the switched systems (1(1) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(1) ) and (2(2) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(2) ) are said to be finite-time stable with respect to $(c_1,c_2,T,\sigma )$, if for any $t\in [t_0,T]$, (7) $D[X_0,\theta ]\le c_1\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}D\left[X\right(t),\theta ]<c_2.$(7)

Definition 3.2

Given three positive constants $c_2$, $c_1$, T, with $c_1<c_2$, and the given switching signal $\sigma \left(t\right)$, for any $W\left(t\right)\in \mathrm{\Omega }=\left\{{\int }_{t_0}^T{W}^T\right(t\left)W\right(t)dt\triangleq {\int }_{t_0}^T\underset{w\in W}{sup}{w}^T(t\left)w\right(t)dt\le d\}$, where $d\ge 0$, the switched systems (3(3) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right)+H_{\sigma \left(t\right)}(t,W(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(3) ) and (4(4) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right)+G_{\sigma \left(t\right)}W\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(4) ) are said to be finite-time bounded with respect to $(c_1,c_2,T,\sigma ,d)$, if for any $t\in [t_0,T]$, (8) $D[X_0,\theta ]\le c_1\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}D\left[X\right(t),\theta ]<c_2.$(8)

From the above definitions of $c_1$ and $c_2$, we have the following theorems.

Theorem 3.1

For the given constants ${\alpha }_r>0$, ${\mu }_p\ge 1$, $0<{\mu }_q<1$, $p\in \phi$, $q\in \psi$, assume that there exist multiple Lyapunov functions $V_r\left(X\right(t\left)\right):K_c\left({\mathbb{R}}^n\right)\to {\mathbb{R}}^{+}$, $r\in M$, and (9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) (10) $\begin{array}{rl}{\dot{V}}_r\left(X\right(t\left)\right)& \le {\alpha }_r{V}_r\left(X\right(t\left)\right)\mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(10) (11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) (12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) (13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) Then, the system (1(1) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(1) ) is finite-time stable with respect to $(c_1,c_2,T,\sigma )$ satisfying the following conditions: (14) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)-{\alpha }_{p}T},\end{array}$(14) (15) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(15)

Proof.

Let $t_0\in [0,s_0)$. From condition (10(10) $\begin{array}{rl}{\dot{V}}_r\left(X\right(t\left)\right)& \le {\alpha }_r{V}_r\left(X\right(t\left)\right)\mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(10) ), for $t\in [t_0,s_0]$, we get (16) $\frac{\mathrm{d}}{\mathrm{d}t}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t}\le 0,$(16) then integrating (16(16) $\frac{\mathrm{d}}{\mathrm{d}t}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t}\le 0,$(16) ) on $(t_0,t)$ gives (17) $V_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-t_0)}.$(17) Let $t\in (s_0,t_1]$. From condition (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and (17(17) $V_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-t_0)}.$(17) ), we get (18) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)& \le V_{\sigma \left(t_0\right)}\left(X\right(s_0-0\left)\right)=V_{\sigma \left(t_0\right)}\left(X\right(s_0\left)\right)\\ & \le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0)}.\end{array}$(18) Let $t\in (t_1,s_1]$. Similarly, we have (19) $V_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}.$(19) Then, from conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ), we get (20) $V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right).$(20) By (18(18) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)& \le V_{\sigma \left(t_0\right)}\left(X\right(s_0-0\left)\right)=V_{\sigma \left(t_0\right)}\left(X\right(s_0\left)\right)\\ & \le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0)}.\end{array}$(18) ), (19(19) $V_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}.$(19) ) and (20(20) $V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right).$(20) ), we obtain (21) $V_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(t-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}.$(21) Let $t\in (s_1,t_2]$. From conditions (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and (21(21) $V_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(t-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}.$(21) ), we get (22) $\begin{array}{rl}{V}_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)& \le V_{\sigma \left(t_1\right)}\left(X\right(s_1-0\left)\right)=V_{\sigma \left(t_1\right)}\left(X\right(s_1\left)\right)\\ & \le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(s_1-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}.\end{array}$(22) Let $t\in (t_2,s_2]$. Similarly, from conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ), we get $\begin{array}{rl}{V}_{\sigma \left(t_2\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_2\right)}{e}^{{\alpha }_{\sigma \left(t_2\right)}(t-t_2)}{V}_{\sigma \left(t_1\right)}\left(X\right(t_2^{-}\left)\right)\\ & \le {\mu }_{\sigma \left(t_1\right)}{\mu }_{\sigma \left(t_2\right)}{e}^{\left[{\alpha }_{\sigma \left(t_2\right)}\right(t-t_2)+{\alpha }_{\sigma \left(t_1\right)}(s_1-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right).\end{array}$ Continuing the above process, applying (5(5) $N_{\sigma p}(T,t)\le N_{0p}+\frac{T_p(T,t)}{{\tau }_{ap}}\mathrm{\forall }t\in [t_0,T],$(5) ) and (6(6) $N_{\sigma q}(T,t)\ge N_{0q}+\frac{T_q(T,t)}{{\tau }_{aq}}\mathrm{\forall }t\in [t_0,T],$(6) ), for any $t\in [0,T]$, let $t_u=max\{t_k\in [0,T]:t_k<t\}$, one derives (23) $\begin{array}{rl}V\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_1\right)}\cdots {\mu }_{\sigma \left(t_u\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\exp \left\{{\alpha }_{\sigma \left(t_u\right)}\right(t-t_u)+\sum _{k=0}^{u-1}{\alpha }_{\sigma \left(t_k\right)}(s_k-t_k\left)\right\}\\ & \le \prod _p{\mu }_p^{N_{\sigma p}(T,0)}\prod _q{\mu }_q^{N_{\sigma q}(T,0)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\\ & \times \exp \left\{\sum _p{\alpha }_p{T}_p\right(T,0)+\sum _q{\alpha }_q{T}_q(T,0\left)\right\}\\ & \le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\exp \left\{\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q\right(T,0)\\ & +\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)\},\end{array}$(23) then, from condition (9(9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) ), we get (24) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)& \le {\gamma }_2\left(D\right[X_0,\theta \left]\right),\end{array}$(24) (25) $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left(V_{\sigma \left(t_0\right)}\right(X\left(t_0\right))\exp \left\{\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q\right(T,0)\\ & +\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)\}).\end{array}$(25) Putting together (24(24) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)& \le {\gamma }_2\left(D\right[X_0,\theta \left]\right),\end{array}$(24) ) and (25(25) $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left(V_{\sigma \left(t_0\right)}\right(X\left(t_0\right))\exp \left\{\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q\right(T,0)\\ & +\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)\}).\end{array}$(25) ), we get $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left({\gamma }_2\right(D[X_0,\theta ])\exp \left\{\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q\right(T,0)\\ & +\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)\}).\end{array}$ Thus from $D[X_0,\theta ]\le c_1$ and switching laws (14(14) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)-{\alpha }_{p}T},\end{array}$(14) ) and (15(15) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(15) ), we get $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left({\gamma }_2\right(c_1)\exp \left\{\sum _p({\alpha }_p+\frac{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)}{T}-{\alpha }_p)T_p\right(T,0)\\ & +\sum _q({\alpha }_q-{\alpha }_q)T_q(T,0)\})\\ & \le {\gamma }_1^{-1}\left({\gamma }_2\right(c_1)\cdot {\left[\frac{{\gamma }_1\left(c_2\right)}{{\gamma }_2\left(c_1\right)}\right]}^{T_p/T})<c_2.\end{array}$ Therefore, Theorem 3.1 is proved.

Theorem 3.2

For the given constants ${\alpha }_r>0$, $r\in M$, ${\mu }_p\ge 1$, $0<{\mu }_q<1$, $p\in \phi$, $q\in \psi$, assume that there exist multiple Lyapunov functions $V_r\left(X\right(t\left)\right):K_c\left({\mathbb{R}}^n\right)\to {\mathbb{R}}^{+}$ satisfying (9(9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) ) and (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and a group of symmetric real matrices $P_r>0$, such that (26) $\begin{array}{rl}{P}_r{L}_r^T+L_r{P}_r-{\alpha }_r{P}_r& \le \mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(26) (27) $\begin{array}{rl}{P}_p^{-1}& \le {\mu }_p{P}_r^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(27) (28) $\begin{array}{rl}{P}_q^{-1}& \le {\mu }_q{P}_p^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(28) Then the system (2(2) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(2) ) is finite-time stable with respect to $(c_1,c_2,T,\sigma )$ satisfying the following conditions: (29) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)-{\alpha }_{p}T},\end{array}$(29) (30) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(30)

Proof.

Select the following multiple Lyapunov-like functions: $V_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)=X^T\left(t\right)P_{\sigma \left(t_k\right)}^{-1}X\left(t\right)\triangleq \underset{x\in X}{sup}{x}^T\left(t\right)P_{\sigma \left(t_k\right)}^{-1}x\left(t\right).$ From condition (26(26) $\begin{array}{rl}{P}_r{L}_r^T+L_r{P}_r-{\alpha }_r{P}_r& \le \mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(26) ), we get ${\dot{V}}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)\le \underset{x\in X}{sup}\left\{x^T\right(t\left){\alpha }_{\sigma \left(t_k\right)}{P}_{\sigma \left(t_k\right)}^{-1}x\right(t\left)\right\}={\alpha }_{\sigma \left(t_k\right)}{V}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right).$ In the same way, by (27(27) $\begin{array}{rl}{P}_p^{-1}& \le {\mu }_p{P}_r^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(27) ) and (28(28) $\begin{array}{rl}{P}_q^{-1}& \le {\mu }_q{P}_p^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(28) ), we can derive the conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ) in Theorem 3.1 and are satisfied. Therefore, the proof process is similar to Theorem 3.1 and is omitted here.

Theorem 3.3

For the given constant ${\alpha }_r>0$, $p\in \phi$, $q\in \psi$, $r\in M$, assume that there exist multiple Lyapunov functions $V_r\left(X\right(t\left)\right):K_c\left({\mathbb{R}}^n\right)\to {\mathbb{R}}^{+}$ satisfying (9(9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) ) and (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) )–(13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ), s.t. (31) $\begin{array}{rl}& {\dot{V}}_r\left(X\right(t\left)\right)\le {\alpha }_r{V}_r\left(X\right(t\left)\right)+{\rho }_r{W}^T\left(t\right)W\left(t\right)\mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(31) (32) $\begin{array}{rl}& {\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T}<{\gamma }_1\left(c_2\right)e^{-{\alpha }_{p}T},\end{array}$(32) where $\rho =\underset{r}{max}\left({\rho }_r\right)$ and $\alpha =\underset{r}{max}\left({\alpha }_r\right)$. Then the system (3(3) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right)+H_{\sigma \left(t\right)}(t,W(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(3) ) is finite-time bounded with respect to $(c_1,c_2,T,\sigma ,d)$ when the switching signals satisfy (33) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln {\gamma }_1\left(c_2\right)-\ln \left({\gamma }_2\right(c_1)+\rho d{e}^{\alpha T})-{\alpha }_{p}T},\end{array}$(33) (34) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(34)

Proof.

Let $t_0\in [0,s_0)$. From condition (31(31) $\begin{array}{rl}& {\dot{V}}_r\left(X\right(t\left)\right)\le {\alpha }_r{V}_r\left(X\right(t\left)\right)+{\rho }_r{W}^T\left(t\right)W\left(t\right)\mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(31) ), for $t\in [t_0,s_0]$, we get (35) $\frac{\mathrm{d}}{\mathrm{d}t}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t}\le {\rho }_{\sigma \left(t_0\right)}{W}^T\left(t\right)W\left(t\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t},$(35) then integrating (35(35) $\frac{\mathrm{d}}{\mathrm{d}t}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t}\le {\rho }_{\sigma \left(t_0\right)}{W}^T\left(t\right)W\left(t\right)e^{-{\alpha }_{\sigma \left(t_0\right)}t},$(35) ) on $(t_0,t)$ gives (36) $V_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-t_0)}+{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-s)}\mathrm{d}s.$(36) Let $t\in (s_0,t_1]$. From conditions (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and (36(36) $V_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-t_0)}+{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(t-s)}\mathrm{d}s.$(36) ), we get (37) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)& \le V_{\sigma \left(t_0\right)}\left(X\right(s_0-0\left)\right)=V_{\sigma \left(t_0\right)}\left(X\right(s_0\left)\right)\\ & \le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0)}+{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s.\end{array}$(37) Let $t\in (t_1,s_1]$. Similarly, we have (38) $V_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)\le V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}+{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-s)}\mathrm{d}s.$(38) Then, from conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ), we get (39) $V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right).$(39) By (37(37) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t\left)\right)& \le V_{\sigma \left(t_0\right)}\left(X\right(s_0-0\left)\right)=V_{\sigma \left(t_0\right)}\left(X\right(s_0\left)\right)\\ & \le V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0)}+{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s.\end{array}$(37) ) –(39(39) $V_{\sigma \left(t_1\right)}\left(X\right(t_1\left)\right)\le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right).$(39) ), we obtain (40) $\begin{array}{rl}{V}_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}+{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-s)}\mathrm{d}s\\ & \le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(t-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}\\ & +{\mu }_{\sigma \left(t_1\right)}{\rho }_{\sigma \left(t_0\right)}{e}^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s\\ & +{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-s)}\mathrm{d}s.\end{array}$(40) Let $t\in (s_1,t_2]$. From conditions (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and (40(40) $\begin{array}{rl}{V}_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_1^{-}\left)\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}+{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-s)}\mathrm{d}s\\ & \le {\mu }_{\sigma \left(t_1\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(t-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}\\ & +{\mu }_{\sigma \left(t_1\right)}{\rho }_{\sigma \left(t_0\right)}{e}^{{\alpha }_{\sigma \left(t_1\right)}(t-t_1)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s\\ & +{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(t-s)}\mathrm{d}s.\end{array}$(40) ), we get (41) $\begin{array}{rl}{V}_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_1\right)}{e}^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(s_1-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\\ & +{\mu }_{\sigma \left(t_1\right)}{e}^{{\alpha }_{\sigma \left(t_1\right)}(s_1-t_1)}{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s\\ & +{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^{s_1}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(s_1-s)}\mathrm{d}s.\end{array}$(41) Let $t\in (t_2,s_2]$. Similarly, (42) $V_{\sigma \left(t_2\right)}\left(X\right(t\left)\right)\le e^{{\alpha }_{\sigma \left(t_2\right)}(t-t_2)}{V}_{\sigma \left(t_2\right)}\left(X\right(t_2\left)\right)+{\rho }_{\sigma \left(t_2\right)}{\int }_{t_2}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_2\right)}(t-s)}\mathrm{d}s.$(42) Then, from conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ), one derives (43) $V_{\sigma \left(t_2\right)}\left(X\right(t_2\left)\right)\le {\mu }_{\sigma \left(t_2\right)}{V}_{\sigma \left(t_1\right)}\left(X\right(t_2^{-}\left)\right).$(43) By (41(41) $\begin{array}{rl}{V}_{\sigma \left(t_1\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_1\right)}{e}^{\left[{\alpha }_{\sigma \left(t_1\right)}\right(s_1-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\\ & +{\mu }_{\sigma \left(t_1\right)}{e}^{{\alpha }_{\sigma \left(t_1\right)}(s_1-t_1)}{\rho }_{\sigma \left(t_0\right)}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s\\ & +{\rho }_{\sigma \left(t_1\right)}{\int }_{t_1}^{s_1}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(s_1-s)}\mathrm{d}s.\end{array}$(41) )–(43(43) $V_{\sigma \left(t_2\right)}\left(X\right(t_2\left)\right)\le {\mu }_{\sigma \left(t_2\right)}{V}_{\sigma \left(t_1\right)}\left(X\right(t_2^{-}\left)\right).$(43) ), we obtain $\begin{array}{rl}{V}_{\sigma \left(t_2\right)}\left(X\right(t\left)\right)& \le {\mu }_{\sigma \left(t_2\right)}{V}_{\sigma \left(t_1\right)}\left(X\right(t_2^{-}\left)\right)e^{{\alpha }_{\sigma \left(t_2\right)}(t-t_2)}+{\rho }_{\sigma \left(t_2\right)}{\int }_{t_2}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_2\right)}(t-s)}\mathrm{d}s\\ & \le {\mu }_{\sigma \left(t_1\right)}{\mu }_{\sigma \left(t_2\right)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)e^{\left[{\alpha }_{\sigma \left(t_2\right)}\right(t-t_2)+{\alpha }_{\sigma \left(t_1\right)}(s_1-t_1)+{\alpha }_{\sigma \left(t_0\right)}(s_0-t_0\left)\right]}\\ & +{\mu }_{\sigma \left(t_1\right)}{\mu }_{\sigma \left(t_2\right)}{\rho }_{\sigma \left(t_0\right)}{e}^{\left[{\alpha }_{\sigma \left(t_2\right)}\right(t-t_2)+{\alpha }_{\sigma \left(t_1\right)}(s_1-t_1\left)\right]}{\int }_{t_0}^{s_0}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_0\right)}(s_0-s)}\mathrm{d}s\\ & +{\mu }_{\sigma \left(t_2\right)}{\rho }_{\sigma \left(t_1\right)}{e}^{{\alpha }_{\sigma \left(t_2\right)}(t-t_2)}{\int }_{t_1}^{s_1}{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_1\right)}(s_1-s)}\mathrm{d}s\\ & +{\rho }_{\sigma \left(t_2\right)}{\int }_{t_2}^t{W}^T\left(s\right)W\left(s\right)e^{{\alpha }_{\sigma \left(t_2\right)}(t-s)}\mathrm{d}s.\end{array}$ Continue this process and from induction argument, applying (5(5) $N_{\sigma p}(T,t)\le N_{0p}+\frac{T_p(T,t)}{{\tau }_{ap}}\mathrm{\forall }t\in [t_0,T],$(5) ) and (6(6) $N_{\sigma q}(T,t)\ge N_{0q}+\frac{T_q(T,t)}{{\tau }_{aq}}\mathrm{\forall }t\in [t_0,T],$(6) ), $\rho =\underset{r}{max}\left({\rho }_r\right)$, $0<{\mu }_q<1$, $W\left(t\right)\in \mathrm{\Omega }$, for any $t\in [0,T]$, let $t_u=max\{t_k\in [0,T]:t_k<t\}$, we obtain (44) $\begin{array}{rl}V\left(X\right(t\left)\right)& \le \prod _p{\mu }_p^{N_{\sigma p}(T,0)}\prod _q{\mu }_q^{N_{\sigma q}(T,0)}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)\\ & \times \exp \left\{\sum _p{\alpha }_p{T}_p\right(T,0)+\sum _q{\alpha }_q{T}_q(T,0\left)\right\}\\ & +\rho {\int }_{t_0}^T\prod _p{\mu }_p^{N_{\sigma p}(t,s)}\prod _q{\mu }_q^{N_{\sigma q}(t,s)}{W}^T\left(s\right)W\left(s\right)\\ & \times \exp \left\{\sum _{k=1}^u{\alpha }_{\sigma \left(t_k\right)}\right(s_k-t_k)+{\alpha }_{\sigma \left(t_0\right)}(s_0-s\left)\right\}\mathrm{d}s\\ & \le \prod _p{\mu }_p^{N_{\sigma p}(T,0)}\left[V_{\sigma \left(t_0\right)}\right(X\left(t_0\right))\exp \left\{\sum _p{\alpha }_p{T}_p\right(T,0)\\ & +\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q(T,0)\}\\ & +\rho \mathrm{d}\exp \left\{\sum _p{\alpha }_p{T}_p\right(T,0)+\sum _q{\alpha }_q{T}_q(T,0\left)\right\}].\end{array}$(44) According to the switching law (34(34) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(34) ), we have $\sum _q({\alpha }_q+\frac{\ln {\mu }_q}{{\tau }_{aq}})T_q(T,0)<0.$ Putting together the above two formulas and $\alpha =\underset{r}{max}\left({\alpha }_r\right)$, one derives $V\left(X\right(t\left)\right)\le \left(V_{\sigma \left(t_0\right)}\right(X\left(t_0\right))+\rho d{e}^{\alpha T})\exp \left\{\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p\right(T,0\left)\right\},$ then, from condition (9(9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) ), we get (45) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)& \le {\gamma }_2\left(D\right[X_0,\theta \left]\right),\end{array}$(45) (46) $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left(\right(V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)+\rho d{e}^{\alpha T})\exp \left\{\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p\right(T,0\left)\right\}).\end{array}$(46) Putting together (45(45) $\begin{array}{rl}{V}_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)& \le {\gamma }_2\left(D\right[X_0,\theta \left]\right),\end{array}$(45) ), (46(46) $\begin{array}{rl}D\left[X\right(t),\theta ]& \le {\gamma }_1^{-1}\left(\right(V_{\sigma \left(t_0\right)}\left(X\right(t_0\left)\right)+\rho d{e}^{\alpha T})\exp \left\{\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p\right(T,0\left)\right\}).\end{array}$(46) ) and $D[X_0,\theta ]\le c_1$, we get (47) $D\left[X\right(t),\theta ]\le {\gamma }_1^{-1}\left(\right({\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T})\exp \left\{\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p\right(T,0\left)\right\}).$(47) From condition (32(32) $\begin{array}{rl}& {\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T}<{\gamma }_1\left(c_2\right)e^{-{\alpha }_{p}T},\end{array}$(32) ), we get $\ln {\gamma }_1\left(c_2\right)-\ln \left({\gamma }_2\right(c_1)+\rho d{e}^{\alpha T})-{\alpha }_{p}T>0,$ then combining switching law (33(33) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln {\gamma }_1\left(c_2\right)-\ln \left({\gamma }_2\right(c_1)+\rho d{e}^{\alpha T})-{\alpha }_{p}T},\end{array}$(33) ), we obtain (48) $\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)<\ln \left({\gamma }_1\right(c_2)\cdot \frac{1}{{\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T}}).$(48) Obviously, combining (47(47) $D\left[X\right(t),\theta ]\le {\gamma }_1^{-1}\left(\right({\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T})\exp \left\{\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p\right(T,0\left)\right\}).$(47) ) and (48(48) $\sum _p({\alpha }_p+\frac{\ln {\mu }_p}{{\tau }_{ap}})T_p(T,0)<\ln \left({\gamma }_1\right(c_2)\cdot \frac{1}{{\gamma }_2\left(c_1\right)+\rho d{e}^{\alpha T}}).$(48) ), we have $D\left[X\right(t),\theta ]<c_2$. Therefore, Theorem 3.3 is proved.

Theorem 3.4

For the given constants ${\alpha }_r>0$, $\gamma >0$, ${\mu }_p\ge 1$, $p\in \phi$, $0<{\mu }_q<1$, $q\in \psi$ and any $W\in \mathrm{\Omega }$, assume that there exist multiple Lyapunov functions $V_r\left(X\right(t\left)\right):K_c\left({\mathbb{R}}^n\right)\to {\mathbb{R}}^{+}$ satisfying (9(9) $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)\le V_r\left(X\right(t\left)\right)& \le {\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right)\mathrm{\forall }r\in M,{\gamma }_1,{\gamma }_2\in \mathcal{K},t\in (t_0,T],\end{array}$(9) ) and (11(11) $\begin{array}{rl}{V}_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& \le V_r\left(X\right(s_k-0\left)\right)\mathrm{\forall }r\in M,t\in (s_k,t_{k+1}],\end{array}$(11) ) and a group of symmetric real matrices ${\overline{P}}_r>0$, $Q_r>0$, $r\in M$, s.t. (49) $\left[\begin{array}{cc}{L}_r{\overline{P}}_r+{\overline{P}}_r{L}_r^T-{\alpha }_r{\overline{P}}_r& G_r{Q}_r\\ \ast & -\gamma Q_r\end{array}\right]\le 0\mathrm{\forall }r\in M,$(49) where $\ast$ does not affect the study in the formula, so it is not necessary to know. (50) $\begin{array}{rl}& {\overline{P}}_p^{-1}\le {\mu }_p{\overline{P}}_r^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(50) (51) $\begin{array}{rl}& {\overline{P}}_q^{-1}\le {\mu }_q{\overline{P}}_p^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi ,\end{array}$(51) (52) $\begin{array}{rl}& {\gamma }_2\left(c_1\right)+\frac{1}{\lambda }{e}^{\alpha T}\gamma d<{\gamma }_1\left(c_2\right)e^{-{\alpha }_{p}T},\end{array}$(52) where $\lambda =\underset{r}{min}\left({\lambda }_{min}{Q}_r\right)\mathrm{a}\mathrm{n}\mathrm{d}\alpha =\underset{r}{max}\left({\lambda }_{max}{\alpha }_r\right)$.

Then the system (4(4) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right)+G_{\sigma \left(t\right)}W\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\varphi }_k(t,X(s_k-0\left)\right),t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(4) ) is finite-time bounded with respect to $(c_1,c_2,T,\sigma ,d)$ when the switching signals satisfy $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln {\gamma }_1\left(c_2\right)-\ln \left({\gamma }_2\right(c_1)+(1/\lambda \left)e^{\alpha T}\gamma d\right)-{\alpha }_{p}T},\\ {\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$

Proof.

Selecting the following multiple Lyapunov-like functions: $V_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)=X^T\left(t\right){\overline{P}}_{\sigma \left(t_k\right)}^{-1}X\left(t\right)\triangleq \underset{x\in X}{sup}{x}^T\left(t\right){\overline{P}}_{\sigma \left(t_k\right)}^{-1}x\left(t\right),$ then, we can derive (53) $\begin{array}{rl}{\dot{V}}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)& =\underset{x\in X,w\in W}{sup}\left\{x^T\right(t\left)\right(L_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}\left)x\right(t)+2x^T(t\left){\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}w\right(t\left)\right\}\\ & =\underset{x\in X,w\in W}{sup}\left[\begin{array}{cc}{x}^T\left(t\right)& w^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & 0\end{array}\right]\\ & \times \left[\begin{array}{c}x\left(t\right)\\ w\left(t\right)\end{array}\right].\end{array}$(53) For condition (49(49) $\left[\begin{array}{cc}{L}_r{\overline{P}}_r+{\overline{P}}_r{L}_r^T-{\alpha }_r{\overline{P}}_r& G_r{Q}_r\\ \ast & -\gamma Q_r\end{array}\right]\le 0\mathrm{\forall }r\in M,$(49) ), let $r=\sigma \left(t_k\right)$, by pre- and post-multiplying $\left[\begin{array}{cc}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}& 0\\ 0& Q_{\sigma \left(t_k\right)}^{-1}\end{array}\right],$ we obtain $\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}-{\alpha }_{\sigma \left(t_k\right)}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & -\gamma Q_{\sigma \left(t_k\right)}^{-1}\end{array}\right]\le 0,$ then, $\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & 0\end{array}\right]+\left[\begin{array}{cc}-{\alpha }_{\sigma \left(t_k\right)}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}& 0\\ 0& -\gamma Q_{\sigma \left(t_k\right)}^{-1}\end{array}\right]\le 0.$ This leads to (54) $\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & 0\end{array}\right]\le \left[\begin{array}{cc}{\alpha }_{\sigma \left(t_k\right)}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}& 0\\ 0& \gamma Q_{\sigma \left(t_k\right)}^{-1}\end{array}\right].$(54) Combining (53(53) $\begin{array}{rl}{\dot{V}}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)& =\underset{x\in X,w\in W}{sup}\left\{x^T\right(t\left)\right(L_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}\left)x\right(t)+2x^T(t\left){\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}w\right(t\left)\right\}\\ & =\underset{x\in X,w\in W}{sup}\left[\begin{array}{cc}{x}^T\left(t\right)& w^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & 0\end{array}\right]\\ & \times \left[\begin{array}{c}x\left(t\right)\\ w\left(t\right)\end{array}\right].\end{array}$(53) ), (54(54) $\left[\begin{array}{cc}{L}_{\sigma \left(t_k\right)}^T{\overline{P}}_{\sigma \left(t_k\right)}^{-1}+{\overline{P}}_{\sigma \left(t_k\right)}^{-1}{L}_{\sigma \left(t_k\right)}& {\overline{P}}_{\sigma \left(t_k\right)}^{-1}{G}_{\sigma \left(t_k\right)}\\ \ast & 0\end{array}\right]\le \left[\begin{array}{cc}{\alpha }_{\sigma \left(t_k\right)}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}& 0\\ 0& \gamma Q_{\sigma \left(t_k\right)}^{-1}\end{array}\right].$(54) ) and $\lambda =\underset{r}{min}\left({\lambda }_{min}{Q}_r\right)$ leads to $\begin{array}{rl}{\dot{V}}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)& \le \underset{x\in X,w\in W}{sup}\left\{x^T\right(t\left){\alpha }_{\sigma \left(t_k\right)}{\overline{P}}_{\sigma \left(t_k\right)}^{-1}x\right(t)+\gamma w^T(t\left)Q_{\sigma \left(t_k\right)}^{-1}w\right(t\left)\right\}\\ & \le {\alpha }_{\sigma \left(t_k\right)}{V}_{\sigma \left(t_k\right)}\left(X\right(t\left)\right)+\frac{\gamma }{\lambda }{W}^T\left(t\right)W\left(t\right).\end{array}$ By (50(50) $\begin{array}{rl}& {\overline{P}}_p^{-1}\le {\mu }_p{\overline{P}}_r^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(50) ) and (51(51) $\begin{array}{rl}& {\overline{P}}_q^{-1}\le {\mu }_q{\overline{P}}_p^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi ,\end{array}$(51) ), we can derive conditions (12(12) $\begin{array}{rl}{V}_p\left(X\right(t_k\left)\right)& \le {\mu }_p{V}_r\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }r\in M,\end{array}$(12) ) and (13(13) $\begin{array}{rl}{V}_q\left(X\right(t_k\left)\right)& \le {\mu }_q{V}_p\left(X\right(t_k^{-}\left)\right)\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(13) ) and are satisfied. Therefore, the proof process is similar to Theorem 3.3 and is omitted here.

4. Examples

To verify the validity of the results in this paper, we give two examples.

Example 4.1

Consider the following set switched system: (55) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\displaystyle \frac{X(s_k-0)}{t+1}},t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(55) where $F_1(t,X(t\left)\right)=1+0.5{\sin }^2\left(X\right(t\left)\right)X\left(t\right)$, $F_2(t,X(t\left)\right)=2+{\cos }^2\left(X\right(t\left)\right)X\left(t\right)$ , $\sigma \left(t\right)=\{1,2\}$, $k=0,1,2,\dots$, let $X_0=0.9$, T = 1, $c_1=1$, and $c_2=9$. It is not difficult to verify that the first subsystem is finite-time stable and the second subsystem is not finite-time stable. Simulation results are presented as follows: Figure 1 shows the value of $D\left[X\right(t),\theta ]$ for the first subsystem in $[0,T]$ and Figure 2 shows the value of $D\left[X\right(t),\theta ]$ for the second subsystem in $[0,T]$.

Figure 1. $D\left[X\right(t),\theta ]$ of the first subsystem.

Figure 2. $D\left[X\right(t),\theta ]$ of the second subsystem.

Choosing multiple Lyapunov-like functions $V_1\left(X\right(t\left)\right)=X^2\left(t\right)$, $V_2\left(X\right(t\left)\right)=0.9X^2\left(t\right)$, then let ${\gamma }_1,{\gamma }_2\in \mathcal{K}$ satisfy the following inequality: ${\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)=0.8X^2\left(t\right)\le V_r\left(X\right(t\left)\right)\le 1.1X^2\left(t\right)={\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right),r=1,2.$ Obviously, when $t\in (t_k,s_k]$, taking the derivative of $V\left(X\right(t\left)\right)$ yields $\begin{array}{rl}{\dot{V}}_1\left(X\right(t\left)\right)& =2X\left(t\right)\dot{X}\left(t\right)=2(1+0.5{\sin }^2(X\left(t\right)\left)X^2\right(t)\le 3V_1(X\left(t\right)\left)\right),\\ {\dot{V}}_2\left(X\right(t\left)\right)& =1.8X\left(t\right)\dot{X}\left(t\right)=1.8(2+{\cos }^2(X\left(t\right)\left)X^2\right(t)\le 6V_2(X\left(t\right)\left)\right),\end{array}$ when $t\in (s_k,t_{k+1}]$, we get $\begin{array}{rl}{V}_1\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& =\frac{X^2(s_k-0)}{(t+1{)}^2}\le V_1\left(X\right(s_k-0\left)\right),\\ V_2\left({\varphi }_k\right(t,X\left(t\right)\left)\right)& =\frac{0.9X^2(s_k-0)}{(t+1{)}^2}\le V_2\left(X\right(s_k-0\left)\right),\end{array}$ where $k=0,1,2,\dots$.

Therefore, we select parameters ${\alpha }_1=3$, ${\alpha }_2=6$, ${\mu }_1=1.2$, ${\mu }_2=0.9$ according to (14(14) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)-{\alpha }_{p}T},\end{array}$(14) ) and (15(15) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(15) ), and let the average dwell time ${\tau }_{a1}\ge {\tau }_{a1}^{\ast }=0.1694$, ${\tau }_{a2}\le {\tau }_{a2}^{\ast }=0.0176$. By Theorem 3.1, the system (55(55) $\{\begin{array}{l}{D}_{H}X\left(t\right)=F_{\sigma \left(t\right)}(t,X(t\left)\right),t\in (t_k,s_k],\\ X\left(t\right)={\displaystyle \frac{X(s_k-0)}{t+1}},t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(55) ) is finite-time stable. The simulation results are presented as follows, see Figure 3.

Figure 3. Time response of $D\left[X\right(t),\theta ]$.

Example 4.2

Consider the following set switched system: (56) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\displaystyle \frac{X(s_k-0)}{t+1}},t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(56) where $L_1=\left[\begin{array}{cc}0.1& 0\\ -0.5& 0.5\end{array}\right],$ $L_2=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right],$ $\sigma \left(t\right)=\{1,2\}$, $k=0,1,2,\dots$, let $X_0=[0.7,0.7]$, T = 1, $c_1=1$, and $c_2=7$. It is not difficult to verify that the first subsystem is finite-time stable and the second subsystem is not finite-time stable. Simulation results are presented as follows: Figure 4 shows the value of $D\left[X\right(t),\theta ]$ for the first subsystem in $[0,T]$ and Figure 5 shows the value of $D\left[X\right(t),\theta ]$ for the second subsystem in $[0,T]$.

Figure 4. $D\left[X\right(t),\theta ]$ of the first subsystem.

Figure 5. $D\left[X\right(t),\theta ]$ of the second subsystem.

Let parameters ${\alpha }_1=1.25$, ${\alpha }_2=5$, ${\mu }_1=1.9$, ${\mu }_2=0.75$, then we choose $P_1=\left[\begin{array}{cc}99& 50\\ 50& 137\end{array}\right],$ $P_2=\left[\begin{array}{cc}159& 81\\ 81& 216\end{array}\right]$ satisfying (26(26) $\begin{array}{rl}{P}_r{L}_r^T+L_r{P}_r-{\alpha }_r{P}_r& \le \mathrm{\forall }r\in M,t\in (t_k,s_k],\end{array}$(26) )–(28(28) $\begin{array}{rl}{P}_q^{-1}& \le {\mu }_q{P}_p^{-1}\mathrm{\forall }p\in \phi \mathrm{\forall }q\in \psi .\end{array}$(28) ). Therefore, choosing multiple Lyapunov-like functions as follows: $\begin{array}{rl}{V}_1\left(X\right(t\left)\right)& =X^T\left(t\right)P_1^{-1}X\left(t\right)=X^T\left(t\right)\left[\begin{array}{cc}0.0124& -0.0045\\ -0.0045& 0.0089\end{array}\right]X\left(t\right),\\ V_2\left(X\right(t\left)\right)& =X^T\left(t\right)P_2^{-1}X\left(t\right)=X^T\left(t\right)\left[\begin{array}{cc}0.0078& -0.0029\\ -0.0029& 0.0057\end{array}\right]X\left(t\right),\end{array}$ then let ${\gamma }_1,{\gamma }_2\in \mathcal{K}$ satisfy the following inequality: $\begin{array}{rl}{\gamma }_1\left(D\right[X\left(t\right),\theta \left]\right)& =0.003X^T\left(t\right)X\left(t\right)\le V_r\left(X\right(t\left)\right)\\ & \le 0.016X^T\left(t\right)X\left(t\right)={\gamma }_2\left(D\right[X\left(t\right),\theta \left]\right),r=1,2.\end{array}$ Obviously, when $t\in (s_k,t_{k+1}]$, we get $V_r\left({\varphi }_k\right(t,X\left(t\right)\left)\right)=\frac{X^T(s_k-0)P_r^{-1}X(s_k-0)}{(t+1{)}^2}\le V_r\left(X\right(s_k-0\left)\right),$ where r = 1, 2, $k=0,1,2,\dots$.

Therefore, according to (29(29) $\begin{array}{rl}{\tau }_{ap}& \ge {\tau }_{ap}^{\ast }=\frac{T\ln {\mu }_p}{\ln \left({\gamma }_1\left(c_2\right)/{\gamma }_2\left(c_1\right)\right)-{\alpha }_{p}T},\end{array}$(29) ) and (30(30) $\begin{array}{rl}{\tau }_{aq}& \le {\tau }_{aq}^{\ast }=-\frac{\ln {\mu }_q}{{\alpha }_q}.\end{array}$(30) ), let the average dwell time ${\tau }_{a1}\ge {\tau }_{a1}^{\ast }=0.6632$ and ${\tau }_{a2}\le {\tau }_{a2}^{\ast }=0.0575$. By Theorem 3.2, the system (56(56) $\{\begin{array}{l}{D}_{H}X\left(t\right)=L_{\sigma \left(t\right)}X\left(t\right),t\in (t_k,s_k],\\ X\left(t\right)={\displaystyle \frac{X(s_k-0)}{t+1}},t\in (s_k,t_{k+1}],\\ X\left(t_0\right)=X_0,\end{array}$(56) ) is finite-time stable. The simulation results are presented as follows, see Figure 6.

Figure 6. Time response of $D\left[X\right(t),\theta ]$.

5. Conclusions

In this paper, we mainly investigate the finite-time stability and finite-time boundedness of set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems through introducing a revised MDADT method and using the multiple Lyapunov-like functions. The results in this paper will enrich the existing results on switched systems. However, only the case of quasi-alternative switched systems are considered, and how to expand the results to more general cases and further construct a realistic set switched systems with non-instantaneous impulses which consist of stable and unstable subsystems in the real world is worthy of our further research.

Author's contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Availability of data and materials

Data sharing not applicable to the paper as no datasets were generated or analysed during the current study.

Additional information

Funding

This paper was supported by the National Natural Science Foundation of China (12171135 and 11771115).