Finite-time $H_{\mathrm{\infty }}$ control for a class of Markovian jump systems with mode-dependent time-varying delay

Abstract

This paper is concerned with the problem of finite-time $H_{\mathrm{\infty }}$ control for a class of Markovian jump system with mode-dependent time-varying delay. By using the new augmented multiple Lyapunov function with more general decomposition approach, a novel sufficient condition for finite-time bounded with an $H_{\mathrm{\infty }}$ performance index is derived. Based on the derived condition, the reliable $H_{\mathrm{\infty }}$ control problem is solved, and an explicit expression of the desired controller is also given, the system trajectory stays within a prescribed bound during a specified time interval. Finally, numerical examples are given to demonstrate that the proposed approach is more effective than some existing ones.

1 Introduction

Markovian jump systems were introduced by Krasovskii and Lidskii [1], which can be described by a set of systems with the transitions in a finite mode set. In the last few decades, there has been increasing interest in Markovian jump systems because this class of systems is appropriate to model many physical systems mainly those random failures, repairs and sudden environment disturbance [2–5]. Such class of systems is a special class of stochastic hybrid systems with finite operation modes, which may switch from one to another at different time. As a crucial factor, it is shown that such jumping can be determined by a Markovian chain [6]. For linear Markovian jumping systems, many important issues have been studied extensively such as stability, stabilization, control synthesis and filter design [6–12]. In finite operation modes, Markovian jump systems is a special class of stochastic systems that can switch from one to another at different time. It is worth pointing out that time delay is one of the instability sources for dynamical systems and is a common phenomenon in many industrial and engineering systems [13–18]. Hence, it is not surprising that much effort has been made to investigate of Markovian jump systems with time delay during the last two decades [19–23]. The exponential stabilization of Markovian jump systems with time delay was first studied in [19] where the decay rate was estimated by solving linear matrix inequalities [20]. However, in the aforementioned works, the network-induced delays have been commonly assumed to be deterministic, which is fairly unrealistic since delays resulting from network transmissions are typically time varying.

Generally speaking, the delay-dependent criterions are less conservative than delay-independent ones, especially when the time delay is small enough in Markovian jump systems. Thus, recent efforts were devoted to the delay-dependent Markovian jump systems stability analysis by employing Lyapunov-Krasovskii functionals [24–32]. However, most efforts have been given on how to construct an appropriate Lyapunov functional by dividing the delay interval $[-{\mu }_2,-{\mu }_1]$ into N equal length subintervals [22]. It should be pointed out that the delay decomposition method is not effective when the lower bound of time-varying delay is zero. Furthermore, although the decay rate can be computed, it is a fixed value that one cannot adjust to deduce if a larger decay rate is possible. Therefore, how to obtain the improved results without increasing the computational burden has greatly improved the current study.

Over the years, many research efforts have been devoted to the study of finite-time stability of systems. In finite-time interval, finite-time stability is investigated to address these transient performances of control systems. Recently, the concept of finite-time stability has been revisited in the light of linear matrix inequalities (LMIs) and Lyapunov function theory, some results are obtained to ensure that system is finite-time stable or finite-time bounded [33–49]. It is noted that there are still some related issues to be solved, to the best of our knowledge, the finite-time $H_{\mathrm{\infty }}$ control for a class of Markovian jump systems with time-varying delay has not been fully developed. The analysis method in the existing references seems still conservative to study Markovian jump system. There is room for further investigation.

The main contribution of this paper is as follows: Firstly, we present a new augmented Lyapunov functional by employing the more general decomposition of a delay interval for a class of Markovian jump systems with mode-dependent time-varying delay. Secondly, in order to reduce the possible conservativeness and computational burden, some slack matrices are introduced [18]. Several sufficient conditions are derived to guarantee the finite-time stability and boundedness of the resulting closed-loop system. Last but not the least, it is shown that less conservative and more general results can be derived since the time-varying delays are divided into a more general decomposition. We find that finite-time stability is a concept independent from Lyapunov stability and can always be affected by switching behavior significantly, thus it deserves our investigation. The finite-time boundedness criteria can be tackled in the form of LMIs. Finally, numerical examples illustrate the effectiveness of the developed techniques.

Notations: Throughout this paper, we let $P>0$ ($P\ge 0$, $P<0$, $P\le 0$) denote a symmetric positive definite matrix P (positive semi-definite, negative definite and negative semi-definite). For any symmetric matrix P, ${\lambda }_{max}(P)$ and ${\lambda }_{min}(P)$ denote the maximum and minimum eigenvalues of matrix P, respectively. ${\mathcal{R}}^n$ denotes the n-dimensional Euclidean space and ${\mathcal{R}}^{n\times m}$ refers to the set of all $n\times m$ real matrices and $\mathcal{N}=\{1,2,\dots ,N\}$. The identity matrix of order n is denoted as $I_n$. ∗ represents the elements below the main diagonal of a symmetric matrix. The superscripts ⊺ and −1 stand for matrix transposition and matrix inverse, respectively.

2 Preliminaries

Given a probability space $(\mathrm{\Omega },F,P)$ where Ω, F and P respectively represent the sample space, the algebra of events and the probability measure defined on F. In this paper, we consider the following Markov jump system over the space $(\mathrm{\Omega },F,P)$ described by

$$\{\begin{array}{l}\dot{x}(t)=A_{r_t}x(t)+A_{\tau r_t}x(t-{\tau }_{r_t}(t))+B_{r_t}u(t)+D_{r_t}\omega (t),\\ z(t)=C_{r_t}x(t)+C_{\tau r_t}x(t-{\tau }_{r_t}(t))+E_{r_t}u(t)+F_{r_t}\omega (t),\\ x(t)=\phi (t),t=[-{\mu }_2,0],\end{array}$$

(1)

where $x(t)\in {\mathcal{R}}^n$ is the state vector of the system, $z(t)\in {\mathcal{R}}^q$ is the controlled output, $u(t)\in {\mathcal{R}}^m$ is the control input and $\phi (t)$, $t=[-{\mu }_2,0]$ and $r_0\in \mathcal{N}$ are initial conditions of continuous state and the mode. $\omega (k)\in {\mathcal{R}}^q$ is the disturbance input satisfying

$${\int }_0^{\mathrm{\infty }}{\omega }^{⊺}(t)\omega (t)dt\le d.$$

(2)

Let the random form process $\{r_t,t\ge 0\}$ be the Markov stochastic process taking values on a finite set $\mathcal{N}=\{1,2,\dots ,N\}$ with the transition rate matrix $\mathrm{\Omega }=\{{\pi }_{ij}\}$, $i,j\in \mathcal{N}$, and the transition probabilities described as

$$Pr(r_{t+\mathrm{\Delta }}=j\mid r_t=i)=\{\begin{array}{ll}{\pi }_{ij}\mathrm{\Delta }+o(\mathrm{\Delta })& \text{if }i\ne j,\\ 1+{\pi }_{ii}\mathrm{\Delta }+o(\mathrm{\Delta })& \text{if }i=j,\end{array}$$

where $\mathrm{\Delta }>0$, ${\pi }_{ij}\ge 0$, for $i\ne j$, is the transition rate from mode i at time t to mode j at time $t+\mathrm{\Delta }$ and

$$-{\pi }_{ii}=\sum _{j=1,j\ne i}^N{\pi }_{ij}$$

for each mode $i\in \mathcal{N}$, ${lim}_{\mathrm{\Delta }\to 0_{+}}\frac{o(\mathrm{\Delta })}{\mathrm{\Delta }}=0$. ${\tau }_i(t)$ denotes the mode dependent time-varying state delay in the system and satisfies the following condition:

$$0<{\mu }_{1i}\le {\tau }_i(t)\le {\mu }_{2i}<\mathrm{\infty },$$

(3)

$$h_{1i}\le {\dot{\tau }}_i(t)\le h_{2i},$$

(4)

where ${\mu }_1=min\{{\mu }_{1i},i\in \mathcal{N}\}$ and ${\mu }_2=max\{{\mu }_{2i},i\in \mathcal{N}\}$ are prescribed integers representing the lower and upper bounds of time-varying delay ${\tau }_i(t)$. Similarly, $h_1=min\{h_{1i},i\in \mathcal{N}\}$ and $h_2=max\{h_{2i},i\in \mathcal{N}\}$ are prescribed integers representing the lower and upper bounds of time-varying delay ${\dot{\tau }}_i(t)$. $A_{r_t}$, $A_{\tau r_t}$, $B_{r_t}$, $D_{r_t}$, $C_{r_t}$, $C_{\tau r_t}$, $E_{r_t}$ and $F_{r_t}$ are known mode-dependent matrices with appropriate dimension functions of the random jumping process $\{r_t\}$ and represent the nominal systems for each $r_t\in \mathcal{N}$. For notation simplicity, when the system operates in the i th mode ($r_t=i$), $A_{r_t}$, $A_{\tau r_t}$, $B_{r_t}$, $D_{r_t}$, $C_{r_t}$, $C_{\tau r_t}$, $E_{r_t}$ and $F_{r_t}$ are denoted as $A_i$, $A_{\tau i}$, $B_i$, $D_i$, $C_i$, $C_{\tau i}$, $E_i$ and $F_i$, respectively.

Remark 1 In this paper, the lower bound of ${\dot{\tau }}_i(t)$ is required in order to implement the proposed delay decomposition method. If $h_{1i}=h_{2i}=0$, then ${\tau }_i(t)$ corresponds to the constant delay.

Moreover, the transient process of a system can be clearly characterized if its decay rate is available. The objective of this study is to develop a new approach to designing a state feedback controller

$$u(t)=K_{i}x(t)$$

(5)

via a novel Lyapunov functional such that the resulting closed-loop system is finite-time stable, where $K_i$ is the controller gains to be designed.

In this paper, we split the delay interval $[-{\mu }_2,-{\mu }_1]$ into two segments: $[-{\tau }_i(t),-{\mu }_1]\cup [-{\mu }_2,-{\tau }_i(t)]$. Moreover, we further subdivide each interval into l, m equal length subsegments $[-{\tau }_k,-{\tau }_{k-1}]$ and $[-{\tau }_{l+s},-{\tau }_{l+s-1}]$, respectively, where

$${\tau }_k={\mu }_1+\frac{k}{l}({\tau }_i(t)-{\mu }_1),{\tau }_{l+s}={\tau }_i(t)+\frac{s}{m}({\mu }_2-{\tau }_i(t)),k=0,1,\dots ,l,s=0,1,\dots ,m,$$

and l, m are given positive integers.

Remark 2 The delay intervals are divided subsegments dependent on t, thus the proposed delay decomposition method is more general than those in [13–17, 19–24]. The conservatism will be reduced with the partitioning number l and m increase.

In order to more precisely describe the main objective, we introduce the following definitions and lemmas for the underlying system.

Definition 2.1 System (1) is said to be finite-time bounded with respect to $(c_1,c_2,T,{\overline{R}}_{r_t},d)$ if condition (2) and the following inequality hold:

$$\begin{array}{r}\underset{-{\mu }_2\le \upsilon \le 0}{sup}\mathbb{E}\{x^{⊺}(\upsilon ){\overline{R}}_{r_t}x(\upsilon ),{\dot{x}}^{⊺}(\upsilon ){\overline{R}}_{r_t}\dot{x}(\upsilon )\}\le c_1\\ \Rightarrow \mathbb{E}\{x^{⊺}(t){\overline{R}}_{r_t}x(t)\}<c_2,\mathrm{\forall }t\in [0,T],\end{array}$$

(6)

where $c_2>c_1\ge 0$ and ${\overline{R}}_{r_t}>0$.

Definition 2.2 [49]

Consider $V(x_t,r_t)$ as the stochastic Lyapunov function of the resulting system (1), its weak infinitesimal operator is defined as

$$\begin{array}{rl}\text{£}V(x_t,r_t,t)& =\underset{\mathrm{\Delta }t\to 0}{lim}\frac{1}{\mathrm{\Delta }t}[\mathbb{E}\{V(x_{t+\mathrm{\Delta }t},r_{t+\mathrm{\Delta }t},t+\mathrm{\Delta }t)\}-V(x_t,i,t)]\\ =\frac{\partial }{\partial t}V(x_t,i,t)+\frac{\partial }{\partial x}V(x_t,i,t)\dot{x}(t,i)+\sum _{j=1}^N{\pi }_{ij}V(x_t,j,t).\end{array}$$

Definition 2.3 Given a constant $T>0$, for all admissible $\omega (t)$ subject to condition (2), under zero initial conditions, if the closed-loop Markovian jump system (1) is finite-time bounded and the control outputs satisfy condition (5) with attenuation $\gamma >0$,

$$\mathbb{E}\{{\int }_0^T{z}^{⊺}(t)z(t)dt\}\le {\gamma }^2{e}^{\eta T}\mathbb{E}\{{\int }_0^T{\omega }^{⊺}(t)\omega (t)dt\},$$

then the controller system (1) is called the finite-time bounded with disturbance attenuation γ.

Remark 3 It should be pointed that the assumption of zero initial condition in system (1) is only for the purpose of technical simplification in the derivation, and it does not lose generality. In fact, if this assumption is lost, the same control result can still be got along the same lines, except adding extra manipulations in the derivation and extra terms in the control presentation. However, in real world applications, the initial condition of the underlying system is generally not zero.

Lemma 2.1 [18]

Let $f_i:{\mathcal{R}}^m\to \mathcal{R}$ ($i=1,2,\dots ,N$) have positive values in an open subset of ${\mathcal{R}}^m$. Then the reciprocally convex combination of $f_i$ over satisfies

$$\underset{\{{\beta }_i\mid {\beta }_i>0,{\sum }_i{\beta }_i=1\}}{min}\sum _i\frac{1}{{\beta }_i}{f}_i(t)=\sum _i{f}_i(t)+\underset{g_{i,j}(t)}{max}\sum _{i\ne j}{g}_{i,j}(t)$$

subject to

$$\{g_{i,j}:{\mathcal{R}}^m\to \mathcal{R},g_{j,i}(t)=g_{i,j}(t),\left[\begin{array}{cc}{f}_i(t)& g_{i,j}(t)\\ g_{i,j}(t)& f_j(t)\end{array}\right]\ge 0\}.$$

Lemma 2.2 For a given function ${\mu }_{1i}\le {\tau }_i(t)\le {\mu }_{2i}$, $h_{1i}\le {\dot{\tau }}_i(t)\le h_{2i}$ ($i\in \mathcal{N}$), there exist four functions ${\alpha }_1(t)\ge 0$, ${\alpha }_2(t)\ge 0$, ${\beta }_1(t)\ge 0$ and ${\beta }_2(t)\ge 0$ satisfying ${\alpha }_1(t)+{\alpha }_2(t)=1$ and ${\beta }_1(t)+{\beta }_2(t)=1$, respectively, such that $\mathrm{\forall }i\in \mathcal{N}$, the following equation holds:

$${\tau }_i(t)={\alpha }_1(t){\mu }_{1i}+{\alpha }_2(t){\mu }_{2i},{\dot{\tau }}_i(t)={\beta }_1(t)h_{1i}+{\beta }_2(t)h_{2i}.$$

Lemma 2.3 [50]

For matrices A, $Q=Q^{⊺}>0$ and $P>0$, the following matrix inequality

$$A^{⊺}PA-Q<0$$

holds if and only if there exists a matrix G of appropriate dimension such that

$$\left[\begin{array}{cc}-Q& A^{⊺}G\\ \ast & P-G-G^{⊺}\end{array}\right]<0.$$

Lemma 2.4 ([14] Schur complement)

Given constant matrices X, Y, Z, where $X=X^{⊺}$ and $0<Y=Y^{⊺}$, then $X+Z^{⊺}{Y}^{-1}Z<0$ if and only if

$$\left[\begin{array}{cc}X& Z^{⊺}\\ \ast & -Y\end{array}\right]<0\mathit{\text{or}}\left[\begin{array}{cc}-Y& Z\\ \ast & X\end{array}\right]<0.$$

(7)

3 Finite-time $H_{\mathrm{\infty }}$ performance analysis

We first consider the problem of stability analysis for system (1) with $u(t)=0$. The following results actually present the finite-time stability for the Markov jump system with time-varying delay.

Theorem 3.1 System (1) is finite-time bounded with respect to $(c_1,c_2,d,{\overline{R}}_i,T)$ if there exist matrices $P_i>0$, $Q_i^{(r)}>0$, $Q^{(r)}>0$ ($r=1,2,\dots ,(m+l)$), $R_i$, $R>0$, $W>0$, $S_i$, $\mathrm{\forall }i,j\in \mathcal{N}$, scalars $c_1<c_2$, $T>0$, ${\kappa }_1>0$, ${\kappa }_2>0$, ${\kappa }_3>0$, ${\lambda }_s>0$ ($s=1,2,\dots ,7$), $\lambda >0$, $\eta >0$ and $\mathrm{\Lambda }>0$, such that for all $i,j\in \mathcal{N}$, $k=1,2,\dots ,l$, $s=1,2,\dots ,m$, the following inequalities hold:

$${\mathrm{\Omega }}_i({\mu }_{qi},h_{qi})=\left[\begin{array}{ccc}{\mathrm{\Omega }}_{1i}({\mu }_{pi},h_{qi})& {\mathrm{\Upsilon }}_{1i}& {\mathrm{\Upsilon }}_{2i}\\ \ast & {\mathrm{\Omega }}_{2i}({\mu }_{pi},h_{qi})& {\mathrm{\Upsilon }}_{3i}\\ \ast & \ast & {\mathrm{\Upsilon }}_{4i}\end{array}\right]<0,p=1,2,q=1,2,$$

(8)

$$e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(k)}\le Q^{(k)},e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(l+s)}\le Q^{(l+s)},p=1,2,$$

(9)

$$e^{\lambda {\mu }_2}\sum _{j=1}^N{\pi }_{ij}{R}_j\le R,$$

(10)

$$c_1\mathrm{\Lambda }+d\lambda {\lambda }_7\frac{1}{\eta }(1-e^{-\eta T})<{\lambda }_1{c}_2{e}^{-\eta T},$$

(11)

where

$$\begin{array}{c}{\mathrm{\Omega }}_{1i}({\mu }_{pi},h_{qi})\hfill \\ =\left[\begin{array}{cccccc}{\tilde{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})& 0& 0& 0& \cdots & 0\\ \ast & {\tilde{\mathrm{\Omega }}}_{2i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& 0& \cdots & 0\\ \ast & \ast & {\tilde{\mathrm{\Omega }}}_{3i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& \cdots & 0\\ \ast & \ast & \ast & \ddots & \ddots & ⋮\\ \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Omega }}}_{li}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i\\ \ast & \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Omega }}}_{(l+1)i}({\mu }_{pi},h_{qi})\end{array}\right],\hfill \\ {\mathrm{\Omega }}_{2i}({\mu }_{pi},h_{qi})\hfill \\ =\left[\begin{array}{ccccc}{\overline{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i& 0& \cdots & 0\\ \ast & {\overline{\mathrm{\Omega }}}_{2i}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i& \cdots & 0\\ \ast & \ast & \ddots & \ddots & 0\\ \ast & \ast & \ast & {\overline{\mathrm{\Omega }}}_{mi}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i\\ \ast & \ast & \ast & \ast & {\overline{\mathrm{\Omega }}}_{(m+1)i}({\mu }_{pi},h_{qi})\end{array}\right],\hfill \\ {\mathrm{\Upsilon }}_{1i}=\left[\begin{array}{ccccc}{P}_i{A}_{\tau i}& 0& \cdots & 0& 0\\ S_i& 2S_i& \cdots & 2S_i& S_i\\ S_i& 2S_i& \cdots & 2S_i& S_i\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ -\frac{l}{{\mu }_2-{\mu }_1}{R}_i+S_i& 2S_i& \cdots & 2S_i& S_i\end{array}\right],\hfill \\ {\mathrm{\Upsilon }}_{2i}=\left[\begin{array}{ccc}{P}_i{D}_i& A_i^{⊺}{R}_i& A_i^{⊺}R\\ 0& 0& 0\\ ⋮& ⋮& ⋮\\ 0& 0& 0\end{array}\right],{\mathrm{\Upsilon }}_{3i}=\left[\begin{array}{ccc}0& A_{\tau i}^{⊺}{R}_i& A_{\tau i}^{⊺}R\\ 0& 0& 0\\ ⋮& ⋮& ⋮\\ 0& 0& 0\end{array}\right],\hfill \\ {\mathrm{\Upsilon }}_{4i}=\left[\begin{array}{ccc}-\lambda W& D_i^{⊺}{R}_i& D_i^{⊺}R\\ \ast & -\frac{1}{{\kappa }_1}{R}_i& 0\\ \ast & \ast & -\frac{1}{{\kappa }_2}R\end{array}\right],\hfill \\ {\tilde{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})=-\lambda P_i+P_i{A}_i+A_i^{⊺}{P}_i+\sum _{j=1}^N{\pi }_{ij}{P}_j\hfill \\ \phantom{{\tilde{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})=}+\sum _{k=1}^l\frac{e^{\lambda [{\mu }_1+\frac{k({\mu }_{pi}-{\mu }_1)}{l}]}-e^{\lambda [{\mu }_1+\frac{(k-1)({\mu }_{pi}-{\mu }_1)}{l}]}}{\lambda }{Q}^{(k)}\hfill \\ \phantom{{\tilde{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})=}+\sum _{s=1}^m\frac{e^{\lambda [{\mu }_{pi}+\frac{(l+s)({\mu }_2-{\mu }_{pi})}{m}]}-e^{\lambda [{\mu }_{pi}+\frac{(l+s-1)({\mu }_2-{\mu }_{pi})}{m}]}}{\lambda }{Q}^{(l+s)},\hfill \\ {\tilde{\mathrm{\Omega }}}_{2i}({\mu }_{pi},h_{qi})=e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{Q}_i^{(1)}+\frac{l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Omega }}}_{3i}({\mu }_{pi},h_{qi})=-(1-\frac{1}{l}{h}_{qi})Q_i^{(1)}+(1-\frac{1}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{Q}_i^{(2)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Omega }}}_{li}({\mu }_{pi},h_{qi})=-(1-\frac{l-2}{l}{h}_{qi})Q_i^{(l-2)}+(1-\frac{l-2}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{Q}_i^{(l-1)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Omega }}}_{(l+1)i}({\mu }_{pi},h_{qi})=-(1-\frac{l-1}{l}{h}_{qi})Q_i^{(l-1)}+(1-\frac{l-1}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{Q}_i^{(l)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Omega }}}_{1i}({\mu }_{pi},h_{qi})=-(1-h_{qi})Q_i^{(l)}+(1-h_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{Q}_i^{(l+1)}+\frac{l+m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Omega }}}_{2i}({\mu }_{pi},h_{qi})=-(1-\frac{m-1}{m}{h}_{qi})Q_i^{(l+1)}+(1-\frac{m-1}{m}{h}_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{Q}_i^{(l+2)}+\frac{2m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Omega }}}_{mi}({\mu }_{pi},h_{qi})=-(1-\frac{1}{m}{h}_{qi})Q_i^{(l+m-1)}+(1-\frac{1}{m}{h}_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{Q}_i^{(l+m)}+\frac{2m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Omega }}}_{(m+1)i}({\mu }_{pi},h_{qi})=-Q_i^{(l+m)}+\frac{m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\kappa }_1=\frac{e^{\lambda {\mu }_2}-e^{\lambda {\mu }_1}}{\lambda },{\kappa }_2=\frac{e^{\lambda {\mu }_2}-e^{\lambda {\mu }_1}-\lambda ({\mu }_2-{\mu }_1)}{{\lambda }^2},\hfill \\ {\kappa }_3=\frac{2e^{\lambda {\mu }_2}-2e^{\lambda {\mu }_1}-{\lambda }^2({\mu }_2^2-{\mu }_1^2)-2\lambda ({\mu }_2-{\mu }_1)}{2{\lambda }^3},\hfill \\ \mathrm{\Lambda }={\lambda }_2+{\kappa }_1{\lambda }_3+{\kappa }_2({\lambda }_4+{\lambda }_5)+{\kappa }_3{\lambda }_6,\hfill \\ {\lambda }_1=\underset{i\in \mathcal{N}}{max}{\lambda }_{max}(P_i),{\lambda }_2=\underset{i\in \mathcal{N}}{max}{\lambda }_{max}({\tilde{P}}_i),{\lambda }_3=\underset{i\in \mathcal{N},1\le r\le (m+l)}{max}{\lambda }_{max}\left({\tilde{Q}}_i^{(r)}\right),\hfill \\ {\lambda }_4={\lambda }_{max}(\tilde{Q}),{\lambda }_5=\underset{i\in \mathcal{N}}{max}{\lambda }_{max}({\tilde{R}}_i),{\lambda }_6={\lambda }_{max}(\tilde{R}),{\lambda }_7={\lambda }_{max}(W),\hfill \\ {\tilde{P}}_i={\overline{R}}_i^{-\frac{1}{2}}{P}_i{\overline{R}}_i^{-\frac{1}{2}},{\tilde{Q}}_i^{(r)}={\overline{R}}_i^{-\frac{1}{2}}{Q}_i^{(r)}{\overline{R}}_i^{-\frac{1}{2}},\tilde{Q}={\overline{R}}_i^{-\frac{1}{2}}Q{\overline{R}}_i^{-\frac{1}{2}},\hfill \\ {\tilde{R}}_i={\overline{R}}_i^{-\frac{1}{2}}{R}_i{\overline{R}}_i^{-\frac{1}{2}},\tilde{R}={\overline{R}}_i^{-\frac{1}{2}}R{\overline{R}}_i^{-\frac{1}{2}}.\hfill \end{array}$$

Proof First, in order to cast our model into the framework of the Markov processes, we define a new process $\{(x_t,r_t),t\ge 0\}$ by

$$x_t(s)=x(t+s),s\in [-{\mu }_2,-{\mu }_1].$$

Now, we consider the following Lyapunov-Krasovskii functional:

$$V(x_t,r_t,t)=\sum _{l=1}^5{V}_l(x_t,r_t,t),$$

(12)

where

$$\begin{array}{c}{V}_1(x_t,r_t,t)=x{(t)}^{⊺}{e}^{\lambda t}{P}_{r_t}x(t),\hfill \\ \begin{array}{rl}{V}_2(x_t,r_t,t)=& \sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{e}^{\lambda (\upsilon +{\tau }_k)}{x}^{⊺}(\upsilon )Q_{r_t}^{(k)}x(\upsilon )d\upsilon \\ +\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{e}^{\lambda (\upsilon +{\tau }_s)}{x}^{⊺}(\upsilon )Q_{r_t}^{(l+s)}x(\upsilon )d\upsilon ,\end{array}\hfill \\ \begin{array}{rl}{V}_3(x_t,r_t,t)=& \sum _{k=1}^l{\int }_{-{\tau }_k}^{-{\tau }_{k-1}}{\int }_{t+\theta }^t{e}^{\lambda (\upsilon -\theta )}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon d\theta \\ +\sum _{s=1}^m{\int }_{-{\tau }_{l+s}}^{-{\tau }_{l+s-1}}{\int }_{t+\theta }^t{e}^{\lambda (\upsilon -\theta )}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon d\theta ,\end{array}\hfill \\ V_4(x_t,r_t,t)={\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{t+\theta }^t{e}^{\lambda (\upsilon -\theta )}{\dot{x}}^{⊺}(\upsilon )R_{r_t}\dot{x}(\upsilon )d\upsilon d\theta ,\hfill \\ V_5(x_t,r_t,t)={\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{\theta }^0{\int }_{t+\kappa }^t{e}^{\lambda (\upsilon -\kappa )}{\dot{x}}^{⊺}(\upsilon )R\dot{x}(\upsilon )d\upsilon d\kappa d\theta .\hfill \end{array}$$

Then, let the mode at time t be i, i.e., $r_t=i\in \mathcal{N}$, we have

$$\begin{array}{c}\begin{array}{rl}\text{£}{V}_1(x_t,i,t)=& \lambda e^{\lambda t}{x}^{⊺}(t)P_{i}x(t)+2e^{\lambda t}{x}^{⊺}(t)P_i(A_{i}x(t)+A_{\tau i}x(t-{\tau }_i(t))+D_i\omega (t))\\ +e^{\lambda t}{x}^{⊺}(t)\left(\sum _{j=1}^N{\pi }_{ij}{P}_j\right)x(t),\end{array}\hfill \\ \text{£}{V}_2(x_t,i,t)=\sum _{k=1}^l[(1-\frac{k-1}{l}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{k-1})e^{\lambda (t+{\tau }_k-{\tau }_{k-1})}{Q}_i^{(k)}x(t-{\tau }_{k-1})\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}-(1-\frac{k}{l}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_k)e^{\lambda t}{Q}_i^{(k)}x(t-{\tau }_k)]\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{e}^{\lambda (\upsilon +{\tau }_k)}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(k)}\right)x(\upsilon )d\upsilon \hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+\sum _{s=1}^m[(1-\frac{m-s+1}{m}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{l+s-1})e^{\lambda (t+{\tau }_{l+s}-{\tau }_{l+s-1})}{Q}_i^{(l+s)}x(t-{\tau }_{l+s-1})\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}-(1-\frac{m-s}{m}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{l+s})e^{\lambda t}{Q}_i^{(l+s)}x(t-{\tau }_{l+s})]\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{e}^{\lambda (\upsilon +{\tau }_{l+s})}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(l+s)}\right)x(\upsilon )d\upsilon \hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)}=e^{\lambda t}\sum _{k=1}^l[(1-\frac{k-1}{l}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{k-1})e^{\frac{\lambda [{\tau }_i(t)-{\mu }_1]}{l}}{Q}_i^{(k)}x(t-{\tau }_{k-1})\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}-(1-\frac{k}{l}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_k)Q_i^{(k)}x(t-{\tau }_k)]\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+e^{\lambda t}\sum _{s=1}^m[(1-\frac{m-s+1}{m}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{l+s-1})e^{\frac{\lambda }{m}[{\mu }_2-{\tau }_i(t)]}{Q}_i^{(l+s)}x(t-{\tau }_{l+s-1})\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}-(1-\frac{m-s}{m}{\dot{\tau }}_i(t))x^{⊺}(t-{\tau }_{l+s})Q_i^{(l+s)}x(t-{\tau }_{l+s})]\hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{e}^{\lambda (\upsilon +{\tau }_k)}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(k)}\right)x(\upsilon )d\upsilon \hfill \\ \phantom{\text{£}{V}_2(x_t,i,t)=}+\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{e}^{\lambda (\upsilon +{\tau }_{l+s})}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(l+s)}\right)x(\upsilon )d\upsilon ,\hfill \\ \begin{array}{rl}\text{£}{V}_3(x_t,i,t)=& e^{\lambda t}\sum _{k=1}^l\frac{e^{\lambda {\tau }_k}-e^{\lambda {\tau }_{k-1}}}{\lambda }{x}^{⊺}(t)Q^{(k)}x(t)-e^{\lambda t}\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon \\ +\sum _{k=1}^l{\dot{\tau }}_k{\int }_{t-{\tau }_k}^t{e}^{\lambda (\upsilon +{\tau }_k)}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon \\ -\sum _{k=1}^l{\dot{\tau }}_{k-1}{\int }_{t-{\tau }_{k-1}}^t{e}^{\lambda (\upsilon +{\tau }_{k-1})}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon \\ +e^{\lambda t}\sum _{s=1}^m\frac{e^{\lambda {\tau }_{l+s}}-e^{\lambda {\tau }_{l+s-1}}}{\lambda }{x}^{⊺}(t)Q^{(l+s)}x(t)\\ -e^{\lambda t}\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon \\ +\sum _{s=1}^m{\dot{\tau }}_{l+s}{\int }_{t-{\tau }_{l+s}}^t{e}^{\lambda (\upsilon +{\tau }_{l+s})}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon \\ -\sum _{s=1}^m{\dot{\tau }}_{l+s-1}{\int }_{t-{\tau }_{l+s-1}}^t{e}^{\lambda (\upsilon +{\tau }_{l+s-1})}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon \\ \le & e^{\lambda t}{x}^{⊺}(t)\{\sum _{k=1}^l\frac{e^{\lambda [{\mu }_1+\frac{k({\tau }_i(t)-{\mu }_1)}{l}]}-e^{\lambda [{\mu }_1+\frac{(k-1)({\tau }_i(t)-{\mu }_1)}{l}]}}{\lambda }{Q}^{(k)}\\ +\sum _{s=1}^m\frac{e^{\lambda [{\tau }_i(t)+\frac{(l+s)({\mu }_2-{\tau }_i(t))}{m}]}-e^{\lambda [{\tau }_i(t)+\frac{(l+s-1)({\mu }_2-{\tau }_i(t))}{m}]}}{\lambda }{Q}^{(l+s)}\}x(t)\\ -e^{\lambda t}\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon \\ -e^{\lambda t}\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon ,\end{array}\hfill \\ \begin{array}{rl}\text{£}{V}_4(x_t,i,t)=& {\dot{x}}^{⊺}(t)e^{\lambda t}\times \frac{e^{\lambda {\mu }_2}-e^{\lambda {\mu }_1}}{\lambda }{R}_i\dot{x}(t)-e^{\lambda t}{\int }_{t-{\mu }_2}^{t-{\mu }_1}{\dot{x}}^{⊺}(\upsilon )R_i\dot{x}(\upsilon )d\upsilon \\ +{\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{t+\theta }^t{e}^{\lambda (\upsilon -\theta )}{\dot{x}}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{R}_j\right)\dot{x}(\upsilon )d\upsilon \end{array}\hfill \\ \begin{array}{rl}\text{£}{V}_5(x_t,i,t)=& \frac{e^{\lambda {\mu }_2}-e^{\lambda {\mu }_1}-\lambda ({\mu }_2-{\mu }_1)}{{\lambda }^2}{e}^{\lambda t}{\dot{x}}^{⊺}(t)R\dot{x}(t)\\ -e^{\lambda t}{\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{t+\theta }^t{\dot{x}}^{⊺}(s)R\dot{x}(s)dsd\theta .\end{array}\hfill \end{array}$$

Moreover, denote

$$\eta (k)={\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}\dot{x}(\upsilon )d\upsilon ,\eta (l+s)={\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}\dot{x}(\upsilon )d\upsilon .$$

By using Lemma 2.1, it yields that

$$\begin{array}{r}-{\int }_{t-{\mu }_2}^{t-{\mu }_1}{\dot{x}}^{⊺}(\upsilon )R_i\dot{x}(\upsilon )\\ =-\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{\dot{x}}^{⊺}(\upsilon )R_i\dot{x}(\upsilon )-\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{\dot{x}}^{⊺}(\upsilon )R_i\dot{x}(\upsilon )\\ \le -\frac{{\mu }_2-{\mu }_1}{{\tau }_i(t)-{\mu }_1}\sum _{k=1}^l{\eta }^{⊺}(k)\frac{l}{{\mu }_2-{\mu }_1}{R}_i\eta (k)-\frac{{\mu }_2-{\mu }_1}{{\mu }_2-{\tau }_i(t)}\sum _{s=1}^m{\eta }^{⊺}(l+s)\frac{m}{{\mu }_2-{\mu }_1}{R}_i\eta (l+s)\\ =-\sum _{k=1}^l{\eta }^{⊺}(k)\frac{l}{{\mu }_2-{\mu }_1}{R}_i\eta (k)-\frac{{\mu }_2-{\tau }_i(t)}{{\tau }_i(t)-{\mu }_1}\sum _{k=1}^l{\eta }^{⊺}(k)\frac{l}{{\mu }_2-{\mu }_1}{R}_i\eta (k)\\ -\sum _{k=1}^l{\eta }^{⊺}(l+s)\frac{m}{{\mu }_2-{\mu }_1}{R}_i\eta (l+s)-\frac{{\tau }_i(t)-{\mu }_1}{{\mu }_2-{\tau }_i(t)}\sum _{s=1}^m{\eta }^{⊺}(l+s)\frac{m}{{\mu }_2-{\mu }_1}{R}_i\eta (l+s)\\ \le -{\left[\begin{array}{c}{\sum }_{k=1}^l\eta (k)\\ {\sum }_{s=1}^m\eta (l+s)\end{array}\right]}^{⊺}\left[\begin{array}{cc}\frac{l}{{\mu }_2-{\mu }_1}{R}_i& S_i\\ \ast & \frac{m}{{\mu }_2-{\mu }_1}{R}_i\end{array}\right]\left[\begin{array}{c}{\sum }_{k=1}^l\eta (k)\\ {\sum }_{s=1}^m\eta (l+s)\end{array}\right].\end{array}$$

(13)

It follows from (9) and $Q^{(r)}>0$ ($r=1,2,\dots ,(m+l)$) that

$$\begin{array}{r}\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{e}^{\lambda (\upsilon +{\tau }_k)}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(k)}\right)x(\upsilon )d\upsilon \\ \le e^{\lambda t}\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{x}^{⊺}(\upsilon )e^{\lambda ({\tau }_k-{\tau }_{k-1})}\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(k)}\right)x(\upsilon )d\upsilon \\ \le e^{\lambda t}\sum _{k=1}^l{\int }_{t-{\tau }_k}^{t-{\tau }_{k-1}}{x}^{⊺}(\upsilon )Q^{(k)}x(\upsilon )d\upsilon ,\end{array}$$

(14)

$$\begin{array}{r}\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{e}^{\lambda (\upsilon +{\tau }_{l+s})}{x}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(l+s)}\right)x(\upsilon )d\upsilon \\ \le e^{\lambda t}\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{x}^{⊺}(\upsilon )e^{\lambda ({\tau }_{l+s}-{\tau }_{l+s-1})}\left(\sum _{j=1}^N{\pi }_{ij}{Q}_j^{(l+s)}\right)x(\upsilon )d\upsilon \\ \le e^{\lambda t}\sum _{s=1}^m{\int }_{t-{\tau }_{l+s}}^{t-{\tau }_{l+s-1}}{x}^{⊺}(\upsilon )Q^{(l+s)}x(\upsilon )d\upsilon .\end{array}$$

(15)

Similarly, (10) implies

$${\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{t+\theta }^t{e}^{\lambda (\upsilon -\theta )}{\dot{x}}^{⊺}(\upsilon )\left(\sum _{j=1}^N{\pi }_{ij}{R}_j\right)\dot{x}(\upsilon )d\upsilon \le e^{\lambda t}{\int }_{-{\mu }_2}^{-{\mu }_1}{\int }_{t+\theta }^t{\dot{x}}^{⊺}(s)R\dot{x}(s)dsd\theta .$$

(16)

From (12)-(16), we can eventually obtain

$$\begin{array}{r}\text{£}V(x_t,r_t,t)-\lambda {\omega }^{⊺}(t)W\omega (t)\le e^{\lambda t}{\xi }^{⊺}(t){\mathrm{\Xi }}_i({\tau }_i(t),{\dot{\tau }}_i(t))\xi (t),\end{array}$$

(17)

where

$${\xi }^{⊺}(t)=[x^{⊺}(t),x^{⊺}(t-{\mu }_1),x^{⊺}(t-{\tau }_1),\dots ,x^{⊺}(t-{\tau }_l),x^{⊺}(t-{\tau }_{l+1}),\dots ,x^{⊺}(t-{\tau }_{l+m}),{\omega }^{⊺}(t)],$$

and

$$\begin{array}{c}{\mathrm{\Xi }}_i({\tau }_i(t),{\dot{\tau }}_i(t))=\left[\begin{array}{ccc}{\mathrm{\Xi }}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))& {\mathrm{\Sigma }}_{1i}& {\mathrm{\Sigma }}_{2i}\\ \ast & {\mathrm{\Xi }}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))& {\mathrm{\Sigma }}_{3i}\\ \ast & \ast & -\lambda W+D_i^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)D_i\end{array}\right],\hfill \\ {\mathrm{\Xi }}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))\hfill \\ =\left[\begin{array}{cccccc}{\tilde{\mathrm{\Xi }}}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))& 0& 0& 0& \cdots & 0\\ \ast & {\tilde{\mathrm{\Xi }}}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& 0& \cdots & 0\\ \ast & \ast & {\tilde{\mathrm{\Xi }}}_{3i}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& \cdots & 0\\ \ast & \ast & \ast & \ddots & \ddots & ⋮\\ \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Xi }}}_{(l)i}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i\\ \ast & \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Xi }}}_{(l+1)i}({\tau }_i(t),{\dot{\tau }}_i(t))\end{array}\right],\hfill \\ {\mathrm{\Xi }}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))\hfill \\ =\left[\begin{array}{ccccc}{\overline{\mathrm{\Xi }}}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i& 0& \cdots & 0\\ \ast & {\overline{\mathrm{\Xi }}}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i& \cdots & 0\\ \ast & \ast & \ddots & \ddots & 0\\ \ast & \ast & \ast & {\overline{\mathrm{\Xi }}}_{mi}({\tau }_i(t),{\dot{\tau }}_i(t))& -\frac{m}{{\mu }_2-{\mu }_1}{R}_i\\ \ast & \ast & \ast & \ast & {\overline{\mathrm{\Xi }}}_{(m+1)i}({\tau }_i(t),{\dot{\tau }}_i(t))\end{array}\right],\hfill \\ {\mathrm{\Sigma }}_{1i}=\left[\begin{array}{ccccc}{P}_i{A}_{\tau i}+A_i^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)A_{\tau i}& 0& \cdots & 0& 0\\ S_i& 2S_i& \cdots & 2S_i& S_i\\ S_i& 2S_i& \cdots & 2S_i& S_i\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ -\frac{l}{{\mu }_2-{\mu }_1}{R}_i+S_i& 2S_i& \cdots & 2S_i& S_i\end{array}\right],\hfill \\ {\mathrm{\Sigma }}_{2i}=\left[\begin{array}{c}{P}_i{D}_i+A_i^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)D_i\\ 0\\ ⋮\\ 0\end{array}\right],{\mathrm{\Sigma }}_{3i}=\left[\begin{array}{c}{A}_{\tau i}^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)D_i\\ 0\\ ⋮\\ 0\end{array}\right],\hfill \\ \begin{array}{rl}{\tilde{\mathrm{\Xi }}}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))=& \lambda P_i+P_i{A}_i+A_i^{⊺}{P}_i+\sum _{j=1}^N{\pi }_{ij}{P}_j++A_i^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)A_i\\ +\sum _{k=1}^l\frac{e^{\lambda [{\mu }_1+\frac{k({\tau }_i(t)-{\mu }_1)}{l}]}-e^{\lambda [{\mu }_1+\frac{(k-1)({\tau }_i(t)-{\mu }_1)}{l}]}}{\lambda }{Q}^{(k)}\\ +\sum _{s=1}^m\frac{e^{\lambda [{\tau }_i(t)+\frac{(l+s)({\mu }_2-{\tau }_i(t))}{m}]}-e^{\lambda [{\tau }_i(t)+\frac{(l+s-1)({\mu }_2-{\tau }_i(t))}{m}]}}{\lambda }{Q}^{(l+s)},\end{array}\hfill \\ {\tilde{\mathrm{\Xi }}}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))=e^{\frac{\lambda }{l}({\tau }_i(t)-{\mu }_1)}{Q}_i^{(1)}+\frac{l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Xi }}}_{3i}({\tau }_i(t),{\dot{\tau }}_i(t))=-(1-\frac{1}{l}{\dot{\tau }}_i(t))Q_i^{(1)}+(1-\frac{1}{l}{\dot{\tau }}_i(t))e^{\frac{\lambda }{l}({\tau }_i(t)-{\mu }_1)}{Q}_i^{(2)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Xi }}}_{li}({\tau }_i(t),{\dot{\tau }}_i(t))=-(1-\frac{l-2}{l}{\dot{\tau }}_i(t))Q_i^{(l-2)}\hfill \\ \phantom{{\tilde{\mathrm{\Xi }}}_{li}({\tau }_i(t),{\dot{\tau }}_i(t))=}+(1-\frac{l-2}{l}{\dot{\tau }}_i(t))e^{\frac{\lambda }{l}({\tau }_i(t)-{\mu }_1)}{Q}_i^{(l-1)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\tilde{\mathrm{\Xi }}}_{(l+1)i}({\tau }_i(t),{\dot{\tau }}_i(t))=-(1-\frac{l-1}{l}{\dot{\tau }}_i(t))Q_i^{(l-1)}\hfill \\ \phantom{{\tilde{\mathrm{\Xi }}}_{(l+1)i}({\tau }_i(t),{\dot{\tau }}_i(t))=}+(1-\frac{l-1}{l}{\dot{\tau }}_i(t))e^{\frac{\lambda }{l}({\tau }_i(t)-{\mu }_1)}{Q}_i^{(l)}+\frac{2l}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ \begin{array}{rl}{\overline{\mathrm{\Xi }}}_{1i}({\tau }_i(t),{\dot{\tau }}_i(t))=& -(1-{\dot{\tau }}_i(t))Q_i^{(l)}+(1-{\dot{\tau }}_i(t))e^{\frac{\lambda }{m}({\mu }_2-{\tau }_i(t))}{Q}_i^{(l+1)}\\ +\frac{l+m}{{\mu }_2-{\mu }_1}{R}_i+A_{\tau i}^{⊺}({\kappa }_1{R}_i+{\kappa }_{2}R)A_{\tau i},\end{array}\hfill \\ {\overline{\mathrm{\Xi }}}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))=-(1-\frac{m-1}{m}{\dot{\tau }}_i(t))Q_i^{(l+1)}\hfill \\ \phantom{{\overline{\mathrm{\Xi }}}_{2i}({\tau }_i(t),{\dot{\tau }}_i(t))=}+(1-\frac{m-1}{m}{\dot{\tau }}_i(t))e^{\frac{\lambda }{m}({\mu }_2-{\tau }_i(t))}{Q}_i^{(l+2)}+\frac{2m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Xi }}}_{mi}({\tau }_i(t),{\dot{\tau }}_i(t))=-(1-\frac{1}{m}{\dot{\tau }}_i(t))Q_i^{(l+m-1)}\hfill \\ \phantom{{\overline{\mathrm{\Xi }}}_{mi}({\tau }_i(t),{\dot{\tau }}_i(t))=}+(1-\frac{1}{m}{\dot{\tau }}_i(t))e^{\frac{\lambda }{m}({\mu }_2-{\tau }_i(t))}{Q}_i^{(l+m)}+\frac{2m}{{\mu }_2-{\mu }_1}{R}_i,\hfill \\ {\overline{\mathrm{\Xi }}}_{(m+1)i}({\tau }_i(t),{\dot{\tau }}_i(t))=-Q_i^{(l+m)}+\frac{m}{{\mu }_2-{\mu }_1}{R}_i.\hfill \end{array}$$

By Lemma 2.2, there exist functions ${\alpha }_1(t)\ge 0$, ${\alpha }_2(t)\ge 0$, ${\beta }_1(t)\ge 0$ and ${\beta }_2(t)\ge 0$ satisfying ${\alpha }_1(t)+{\alpha }_2(t)=1$ and ${\beta }_1(t)+{\beta }_2(t)=1$, respectively. Using the Schur complement such that

$$\begin{array}{rl}{\mathrm{\Xi }}_i({\tau }_i(t),{\dot{\tau }}_i(t))=& {\alpha }_1(t){\beta }_1(t){\mathrm{\Omega }}_i({\mu }_{1i},h_{1i})+{\alpha }_2(t){\beta }_1(t){\mathrm{\Omega }}_i({\mu }_{2i},h_{1i})\\ +{\alpha }_1(t){\beta }_2(t){\mathrm{\Omega }}_i({\mu }_{1i},h_{2i})+{\alpha }_2(t){\beta }_2(t){\mathrm{\Omega }}_i({\mu }_{2i},h_{2i}),\end{array}$$

(18)

where ${\mathrm{\Omega }}_i({\mu }_{1i},h_{1i})$, ${\mathrm{\Omega }}_i({\mu }_{2i},h_{1i})$, ${\mathrm{\Omega }}_i({\mu }_{1i},h_{2i})$ and ${\mathrm{\Omega }}_i({\mu }_{2i},h_{2i})$ are defined in Theorem 3.1. Substituting (18) into (17), then (18) can be rewritten as

$$\text{£}V(x_t,r_t,t)-\lambda {\omega }^{⊺}(t)W\omega (t)\le \sum _{p=1}^2\sum _{q=1}^2{\mathrm{\Omega }}_i({\mu }_{pi},h_{qi}).$$

(19)

Therefore, the following relation holds by condition (8) and (19):

$$\mathbb{E}\{\text{£}V(x_t,r_t,t)\}\le \mathbb{E}[\eta V(x_t,r_t,t)]+\lambda {\omega }^{⊺}(t)W\omega (t).$$

Multiplying the aforementioned inequality by $e^{-\eta t}$, we can get

$$\mathbb{E}\left\{\text{£}[e^{-\eta t}V(x_t,r_t,t)]\right\}\le e^{-\eta t}\lambda {\omega }^{⊺}(t)W\omega (t).$$

By integrating the aforementioned inequality between 0 and t, it follows that

$$e^{-\eta t}\mathbb{E}[V(x_t,r_t,t)]-\mathbb{E}[V(x_0,r_0,0)]\le \lambda {\int }_0^t{e}^{-\eta s}{\omega }^{⊺}(s)W\omega (s)ds.$$

Denote ${\tilde{P}}_i={\overline{R}}_i^{-\frac{1}{2}}{P}_i{\overline{R}}_i^{-\frac{1}{2}}$, ${\tilde{Q}}_i^{(r)}={\overline{R}}_i^{-\frac{1}{2}}{Q}_i^{(r)}{\overline{R}}_i^{-\frac{1}{2}}$, $\tilde{Q}={\overline{R}}_i^{-\frac{1}{2}}Q{\overline{R}}_i^{-\frac{1}{2}}$, ${\tilde{R}}_i={\overline{R}}_i^{-\frac{1}{2}}{R}_i{\overline{R}}_i^{-\frac{1}{2}}$, $\tilde{R}={\overline{R}}_i^{-\frac{1}{2}}R{\overline{R}}_i^{-\frac{1}{2}}$, it yields that

$$\begin{array}{r}\mathbb{E}[V(x_0,r_0,0)]\\ \le \{\underset{i\in \mathcal{N}}{max}{\lambda }_{max}({\tilde{P}}_i)+{\kappa }_1\underset{i\in \mathcal{N},1\le r\le (m+l)}{max}{\lambda }_{max}\left({\tilde{Q}}_i^{(r)}\right)+{\kappa }_2{\lambda }_{max}(\tilde{Q})\\ +{\kappa }_2\underset{i\in \mathcal{N}}{max}{\lambda }_{max}({\tilde{R}}_i)+{\kappa }_3{\lambda }_{max}(\tilde{R})\}\times \underset{-{\mu }_2\le \upsilon \le 0}{sup}\{x^{⊺}(\upsilon ){\overline{R}}_{i}x(\upsilon ),{\dot{x}}^{⊺}(\upsilon ){\overline{R}}_i\dot{x}(\upsilon )\}\\ =({\lambda }_2+{\kappa }_1{\lambda }_3+{\kappa }_2({\lambda }_4+{\lambda }_5)+{\kappa }_3{\lambda }_6)\times \underset{-{\mu }_2\le \upsilon \le 0}{sup}\{x^{⊺}(\upsilon ){\overline{R}}_{i}x(\upsilon ),{\dot{x}}^{⊺}(\upsilon ){\overline{R}}_i\dot{x}(\upsilon )\}\\ =c_1\mathrm{\Lambda }.\end{array}$$

(20)

Noting that $\eta >0$ and $0\le t\le T$, we have

$$\begin{array}{rl}\mathbb{E}[V(x_t,r_t,t)]\le & \mathbb{E}[e^{\eta t}V(x_0,r_0,0)]+e^{\eta t}\lambda {\int }_0^t{e}^{-\eta s}{\omega }^{⊺}(s)W\omega (s)ds\\ \le & e^{\eta T}{c}_1\mathrm{\Lambda }+d\lambda e^{\eta T}{\lambda }_{max}(W){\int }_0^T{e}^{-\eta s}ds\\ \le & e^{\eta T}\{c_1\mathrm{\Lambda }+d\lambda {\lambda }_7\frac{1}{\eta }(1-e^{-\eta T})\}.\end{array}$$

(21)

On the other hand, it follows from (12) that

$$\begin{array}{rl}\mathbb{E}[V(x_t,r_t,t)]& \ge \mathbb{E}[x^{⊺}(t)e^{\lambda t}{P}_{i}x(t)]\ge \underset{i\in \mathcal{N}}{max}{\lambda }_{min}(P_i)\mathbb{E}[x^{⊺}(t){\overline{R}}_{i}x(t)]\\ ={\lambda }_1\mathbb{E}[x^{⊺}(t){\overline{R}}_{i}x(t)].\end{array}$$

(22)

It can be derived from (21)-(22) that

$$\begin{array}{r}\mathbb{E}[x^{⊺}(t){\overline{R}}_{i}x(t)]\le \frac{e^{\eta T}}{{\lambda }_1}\{c_1\mathrm{\Lambda }+d\lambda {\lambda }_7\frac{1}{\eta }(1-e^{-\eta T})\}.\end{array}$$

(23)

From (11) and (23), we have

$$\mathbb{E}[x^{⊺}(t){\overline{R}}_{i}x(t)]<c_2.$$

(24)

Then the system is finite-time bounded with respect to $(c_1,c_2,d,{\overline{R}}_i,T)$. □

Remark 4 It should be mentioned that novel terms $V_2(x_t,i,t)$ and $V_3(x_t,i,t)$ are continuous at ${\tau }_i(t)={\tau }_l$ is included in the Lyapunov-Krasovskii functional (12), which plays an important role in reducing conservativeness of the derived result.

Remark 5 In this paper, ${\tau }_i(t)$ and ${\dot{\tau }}_i(t)$ may have different upper bounds in various delay intervals satisfying (3) and (4), respectively. While in previous work such as [16, 17], ${\tau }_i(t)$ and ${\dot{\tau }}_i(t)$ are enlarged to ${\tau }_i(t)\le {\mu }_2=max\{{\mu }_{2i},i\in \mathcal{N}\}$ and ${\dot{\tau }}_i(t)\le h_2=max\{h_{2i},i\in \mathcal{N}\}$, respectively, which may lead to conservativeness inevitably. However, the case above can be taken fully into account by employing the Lyapunov-Krasovskii functional (12).

Remark 6 When dealing with term $-{\int }_{t-{\mu }_2}^{t-{\mu }_1}{\dot{x}}^{⊺}(\upsilon )R_i\dot{x}(\upsilon )d\upsilon$, the convex combination is not employed, Lemma 2.1 is used in this paper, then the free-weighting matrices-dependent null add items are necessary to be introduced in our proof, which leads to the decrease in the number of LMIs and LMIs scalar decision variables.

Remark 7 The feature of this paper is the way to deal with the integral term. Many researchers have enlarged the derivative of the Lyapunov functional in order to deal with the integral term in mathematical operations. In this paper, we transform different integral intervals with the same integral length into an integral interval. It is worth pointing out that in the proof of the theorem no extra inequality is introduced. We propose a novel delay-dependent sufficient criterion, which ensures that the Markovian jump system with time-varying delays is finite-time stable.

Remark 8 One can clearly see from the proof of Theorem 3.1 that neither free-weighting matrices nor model transformation has been employed to deal with the sum terms, and none of useful items are ignored, resulting in better results with the less number of LMIs scalar decision variables, which deduces some conservatism in some sense.

By using the novel Lyapunov functionals with the more general decomposition of delay interval, a state feedback controller (3) can be designed such that the resulting closed-loop system is finite-time bounded with $H_{\mathrm{\infty }}$ performance. When $r_t=i$, the closed-loop system is expressed by

$$\{\begin{array}{l}\dot{x}(t)={\overline{A}}_{i}x(t)+A_{i}x(t-{\tau }_{r_t}(t))+D_i\omega (t),\\ z(t)={\overline{C}}_{i}x(t)+C_{\tau i}x(t-{\tau }_i(t))+F_i\omega (t),\end{array}$$

(25)

where

$${\overline{A}}_i=A_i+B_i{K}_i,{\overline{C}}_i=C_i+E_i{K}_i.$$

Theorem 3.2 System (25) is finite-time bounded with respect to $(c_1,c_2,d,{\overline{R}}_i,T)$ if there exist matrices $P_i>0$, $Q_i^{(r)}>0$, $Q^{(r)}>0$ ($r=1,2,\dots ,(m+l)$), $R_i$, $R>0$, $S_i$, $\mathrm{\forall }i,j\in \mathcal{N}$, scalars $\gamma >0$, $c_1<c_2$, $T>0$, ${\kappa }_1>0$, ${\kappa }_2>0$, ${\kappa }_3>0$, ${\lambda }_s>0$ ($s=1,2,\dots ,6$), $\lambda >0$, $\eta >0$ and $\mathrm{\Lambda }>0$, such that for all $i,j\in \mathcal{N}$, $k=1,2,\dots ,l$, $s=1,2,\dots ,m$, the following inequalities hold:

$${\mathrm{\Theta }}_i({\mu }_{qi},h_{qi})=\left[\begin{array}{ccc}{\mathrm{\Theta }}_{1i}({\mu }_{pi},h_{qi})& {\mathrm{\Upsilon }}_{1i}& {\tilde{\mathrm{\Upsilon }}}_{2i}\\ \ast & {\mathrm{\Omega }}_{2i}({\mu }_{pi},h_{qi})& {\tilde{\mathrm{\Upsilon }}}_{3i}\\ \ast & \ast & {\tilde{\mathrm{\Upsilon }}}_{4i}\end{array}\right]<0,p=1,2,q=1,2,$$

(26)

$$c_1\mathrm{\Lambda }+d\lambda {\gamma }^2\frac{1}{\eta }(1-e^{-\eta T})<{\lambda }_1{c}_2{e}^{-\eta T},$$

(27)

where

$$\begin{array}{c}{\mathrm{\Theta }}_{1i}({\mu }_{pi},h_{qi})\hfill \\ =\left[\begin{array}{cccccc}{\tilde{\mathrm{\Theta }}}_{1i}({\mu }_{pi},h_{qi})& 0& 0& 0& \cdots & 0\\ \ast & {\tilde{\mathrm{\Omega }}}_{2i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& 0& \cdots & 0\\ \ast & \ast & {\tilde{\mathrm{\Omega }}}_{3i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i& \cdots & 0\\ \ast & \ast & \ast & \ddots & \ddots & ⋮\\ \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Omega }}}_{li}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{R}_i\\ \ast & \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Omega }}}_{(l+1)i}({\mu }_{pi},h_{qi})\end{array}\right],\hfill \\ {\tilde{\mathrm{\Upsilon }}}_{2i}=\left[\begin{array}{cccc}{P}_i{D}_i& {\overline{C}}_i& {\overline{A}}_i^{⊺}{R}_i& {\overline{A}}_i^{⊺}R\\ 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0\end{array}\right],\hfill \\ {\tilde{\mathrm{\Upsilon }}}_{3i}=\left[\begin{array}{cccc}0& C_{\tau i}& A_{\tau i}^{⊺}{R}_i& A_{\tau i}^{⊺}R\\ 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0\end{array}\right],\hfill \\ {\tilde{\mathrm{\Upsilon }}}_{4i}=\left[\begin{array}{cccc}-{\gamma }^{2}I& F_i& D_i^{⊺}{R}_i& D_i^{⊺}R\\ \ast & -I& 0& 0\\ \ast & \ast & -\frac{1}{{\kappa }_1}{R}_i& 0\\ \ast & \ast & \ast & -\frac{1}{{\kappa }_2}R\end{array}\right],\hfill \\ \begin{array}{rl}{\tilde{\mathrm{\Theta }}}_{1i}({\mu }_{pi},h_{qi})=& -\lambda P_i+P_i{\overline{A}}_i+{\overline{A}}_i^{⊺}{P}_i+\sum _{j=1}^N{\pi }_{ij}{P}_j\\ +\sum _{k=1}^l\frac{e^{\lambda [{\mu }_1+\frac{k({\mu }_{pi}-{\mu }_1)}{l}]}-e^{\lambda [{\mu }_1+\frac{(k-1)({\mu }_{pi}-{\mu }_1)}{l}]}}{\lambda }{Q}^{(k)}\\ +\sum _{s=1}^m\frac{e^{\lambda [{\mu }_{pi}+\frac{(l+s)({\mu }_2-{\mu }_{pi})}{m}]}-e^{\lambda [{\mu }_{pi}+\frac{(l+s-1)({\mu }_2-{\mu }_{pi})}{m}]}}{\lambda }{Q}^{(l+s)}.\end{array}\hfill \end{array}$$

Proof We now consider the $H_{\mathrm{\infty }}$ performance of system (25). Select the same Lyapunov-Krasovskii functional as Theorem 3.1 and the Schur complement, it yields that

$$\text{£}V(x_t,i,t)+z^{⊺}(t)z(t)-{\gamma }^2{\omega }^{⊺}(t)\omega (t)\le {\xi }^{⊺}(t){\mathrm{\Theta }}_i({\mu }_{pi},h_{qi})\xi (t).$$

(28)

It follows from (26) that

$$\mathbb{E}\{\text{£}V(x_t,i,t)\}\le \mathbb{E}[\eta V(x_t,i,t)]+{\gamma }^2{\omega }^{⊺}(t)\omega (t)-\mathbb{E}[z^{⊺}(t)z(t)].$$

(29)

Multiplying the aforementioned inequality by $e^{-\eta t}$, one has

$$\mathbb{E}\left\{\text{£}\left[e^{-\eta tV(x_t,i,t)}\right]\right\}\le e^{-\eta t}[{\gamma }^2{\omega }^{⊺}(t)\omega (t)-z^{⊺}(t)z(t)].$$

(30)

Under zero initial condition and $\mathbb{E}[V(x_t,i,t)]>0$, by integrating the aforementioned inequality between 0 and T, we can get

$$\mathbb{E}[{\int }_0^T{e}^{-\eta \upsilon }{z}^{⊺}(\upsilon )z(\upsilon )d\upsilon ]\le {\gamma }^2\mathbb{E}[{\int }_0^T{e}^{-\eta \upsilon }{\omega }^{⊺}(\upsilon )\omega (\upsilon )d\upsilon ].$$

(31)

Then it yields

$$\mathbb{E}[{\int }_0^T{z}^{⊺}(\upsilon )z(\upsilon )d\upsilon ]\le {\gamma }^2{e}^{\eta T}\mathbb{E}[{\int }_0^T{\omega }^{⊺}(\upsilon )\omega (\upsilon )d\upsilon ].$$

(32)

Thus it is concluded by Definition 2.3 that system (25) is finite-time bounded with an $H_{\mathrm{\infty }}$ performance γ. The proof is completed. □

Remark 9 From the proof process of Theorem 3.1 and Theorem 3.2, it is easy to see that neither bounding technique for cross terms nor model transformation is involved. In other words, the obtained result is expected to be less conservative.

Remark 10 Lyapunov asymptotic stability and finite-time stability of a class of systems are independent concepts. Lyapunov asymptotically stable system may not be finite-time stable. Moreover, finite-time stable system may also not be Lyapunov asymptotically stable. There exist some results on Lyapunov stability, while finite-time stability also needs our full investigation, which was neglected by most previous work.

4 Finite-time $H_{\mathrm{\infty }}$ control

Theorem 4.1 System (25) is finite-time bounded with respect to $(c_1,c_2,d,{\overline{R}}_i,T)$ if there exist matrices ${\hat{P}}_i>0$, ${\hat{Q}}_i^{(r)}>0$, ${\hat{Q}}^{(r)}>0$ ($r=1,2,\dots ,(m+l)$), ${\hat{R}}_i$, $\hat{R}>0$, ${\hat{S}}_i$, $\mathrm{\forall }i,j\in \mathcal{N}$, scalars $c_1<c_2$, $T>0$, ${\kappa }_1>0$, ${\kappa }_2>0$, ${\kappa }_3>0$, ${\sigma }_s>0$ ($s=1,2,\dots ,5$), $\lambda >0$, $\eta >0$ and $\overline{\mathrm{\Lambda }}>0$, such that for all $i,j\in \mathcal{N}$, $k=1,2,\dots ,l$, $s=1,2,\dots ,m$, the following inequalities hold:

$${\mathrm{\Phi }}_i({\mu }_{qi},h_{qi})=\left[\begin{array}{cccc}{\mathrm{\Phi }}_{1i}({\mu }_{pi},h_{qi})& {\tilde{\mathrm{\Psi }}}_{1i}& {\tilde{\mathrm{\Psi }}}_{2i}& {\mathrm{\Pi }}_{1i}\\ \ast & {\mathrm{\Phi }}_{2i}({\mu }_{pi},h_{qi})& {\tilde{\mathrm{\Psi }}}_{3i}& 0\\ \ast & \ast & {\tilde{\mathrm{\Psi }}}_{4i}& 0\\ \ast & \ast & \ast & {\mathrm{\Pi }}_{2i}\end{array}\right]<0,p=1,2,q=1,2,$$

(33)

$$\begin{array}{l}{e}^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}\sum _{j=1}^N{\pi }_{ij}{\hat{Q}}_j^{(k)}\le {\hat{Q}}^{(k)},\\ e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}\sum _{j=1}^N{\pi }_{ij}{\hat{Q}}_j^{(l+s)}\le {\hat{Q}}^{(l+s)},p=1,2,\end{array}$$

(34)

$$e^{\lambda {\mu }_2}\sum _{j=1}^N{\pi }_{ij}{\hat{R}}_{ij}\le {\hat{R}}_i,$$

(35)

$$\begin{array}{l}{\sigma }_1{\overline{R}}_i^{-1}<X_i<{\overline{R}}_i^{-1},0<{\hat{Q}}_{ii}<{\sigma }_2{\overline{R}}_i,0<{\hat{Q}}_i<{\sigma }_3{\overline{R}}_i,\\ 0<{\hat{R}}_{ii}<{\sigma }_4{\overline{R}}_i,0<{\hat{R}}_i<{\sigma }_5{\overline{R}}_i,\end{array}$$

(36)

$$\left[\begin{array}{cc}{c}_1\overline{\mathrm{\Lambda }}+d\lambda {\gamma }^2\frac{1}{\eta }(1-e^{-\eta T})-c_2{e}^{-\eta T}& \sqrt{c_1}\\ \sqrt{c_1}& {\sigma }_1\end{array}\right]<0,$$

(37)

where

$$\begin{array}{c}{\mathrm{\Phi }}_{1i}({\mu }_{pi},h_{qi})\hfill \\ =\left[\begin{array}{cccccc}{\tilde{\mathrm{\Phi }}}_{1i}({\mu }_{pi},h_{qi})& 0& 0& 0& \cdots & 0\\ \ast & {\tilde{\mathrm{\Phi }}}_{2i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}& 0& \cdots & 0\\ \ast & \ast & {\tilde{\mathrm{\Phi }}}_{3i}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}& \cdots & 0\\ \ast & \ast & \ast & \ddots & \ddots & ⋮\\ \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Phi }}}_{li}({\mu }_{pi},h_{qi})& -\frac{l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}\\ \ast & \ast & \ast & \ast & \ast & {\tilde{\mathrm{\Phi }}}_{(l+1)i}({\mu }_{pi},h_{qi})\end{array}\right],\hfill \\ {\mathrm{\Phi }}_{2i}({\mu }_{pi},h_{qi})\hfill \\ =\left[\begin{array}{ccccc}{\overline{\mathrm{\Phi }}}_{1i}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}& 0& \cdots & 0\\ \ast & {\overline{\mathrm{\Phi }}}_{2i}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}& \cdots & 0\\ \ast & \ast & \ddots & \ddots & 0\\ \ast & \ast & \ast & {\overline{\mathrm{\Phi }}}_{mi}({\mu }_{pi},h_{qi})& -\frac{m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}\\ \ast & \ast & \ast & \ast & {\overline{\mathrm{\Phi }}}_{(m+1)i}({\mu }_{pi},h_{qi})\end{array}\right],\hfill \\ {\tilde{\mathrm{\Psi }}}_{1i}=\left[\begin{array}{ccccc}{A}_{\tau i}{X}_i& 0& \cdots & 0& 0\\ {\hat{S}}_{ii}& 2{\hat{S}}_{ii}& \cdots & 2{\hat{S}}_{ii}& {\hat{S}}_{ii}\\ {\hat{S}}_{ii}& 2{\hat{S}}_{ii}& \cdots & 2{\hat{S}}_{ii}& {\hat{S}}_{ii}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ -\frac{l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii}+{\hat{S}}_{ii}& 2{\hat{S}}_{ii}& \cdots & 2{\hat{S}}_{ii}& {\hat{S}}_{ii}\end{array}\right],\hfill \\ {\tilde{\mathrm{\Psi }}}_{2i}=\left[\begin{array}{cccc}{D}_i{X}_i& X_i{C}_i^{⊺}+Y_i{E}_i^{⊺}& X_i{A}_i^{⊺}+Y_i{B}_i^{⊺}& X_i{A}_i^{⊺}+Y_i{B}_i^{⊺}\\ 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0\end{array}\right],\hfill \\ {\tilde{\mathrm{\Psi }}}_{3i}=\left[\begin{array}{cccc}0& X_i{C}_{\tau i}^{⊺}& X_i{A}_{\tau i}^{⊺}& X_i{A}_{\tau i}^{⊺}\\ 0& 0& 0& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& 0\end{array}\right],\hfill \\ {\tilde{\mathrm{\Psi }}}_{4i}=\left[\begin{array}{cccc}-{\gamma }^2(2X_i-I)& X_i{F}_i& X_i{D}_i^{⊺}& X_i{D}_i^{⊺}\\ \ast & -I& 0& 0\\ \ast & \ast & -\frac{1}{{\kappa }_1}(2X_i-{\hat{R}}_{ii})& 0\\ \ast & \ast & \ast & -\frac{1}{{\kappa }_2}(2X_i-{\hat{R}}_i)\end{array}\right],\hfill \\ {\mathrm{\Pi }}_{1i}=[\sqrt{{\pi }_{i1}}{X}_i,\dots ,\sqrt{{\pi }_{i(i-1)}}{X}_i,\sqrt{{\pi }_{i(i+1)}}{X}_i,\dots ,\sqrt{{\pi }_{iN}}{X}_i],\hfill \\ {\mathrm{\Pi }}_{2i}=diag\{X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_N\},\hfill \\ \begin{array}{rl}{\tilde{\mathrm{\Phi }}}_{1i}({\mu }_{pi},h_{qi})=& -\lambda X_i+{\overline{A}}_i{X}_i+X_i{\overline{A}}_i^{⊺}+{\pi }_{ii}{X}_i\\ +\sum _{k=1}^l\frac{e^{\lambda [{\mu }_1+\frac{k({\mu }_{pi}-{\mu }_1)}{l}]}-e^{\lambda [{\mu }_1+\frac{(k-1)({\mu }_{pi}-{\mu }_{1i})}{l}]}}{\lambda }{\hat{Q}}^{(k)}\\ +\sum _{s=1}^m\frac{e^{\lambda [{\mu }_{pi}+\frac{(l+s)({\mu }_2-{\mu }_{pi})}{m}]}-e^{\lambda [{\mu }_{pi}+\frac{(l+s-1)({\mu }_2-{\mu }_{pi})}{m}]}}{\lambda }{\hat{Q}}^{(l+s)},\end{array}\hfill \\ {\tilde{\mathrm{\Phi }}}_{2i}({\mu }_{pi},h_{qi})=e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{\hat{Q}}_{ii}^{(1)}+\frac{l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\tilde{\mathrm{\Phi }}}_{3i}({\mu }_{pi},h_{qi})=-(1-\frac{1}{l}{h}_{qi}){\hat{Q}}_{ii}^{(1)}+(1-\frac{1}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{\hat{Q}}_{ii}^{(2)}+\frac{2l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\tilde{\mathrm{\Phi }}}_{li}({\mu }_{pi},h_{qi})=-(1-\frac{l-2}{l}{h}_{qi}){\hat{Q}}_{ii}^{(l-2)}+(1-\frac{l-2}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{\hat{Q}}_{ii}^{(l-1)}+\frac{2l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\tilde{\mathrm{\Phi }}}_{(l+1)i}({\mu }_{pi},h_{qi})=-(1-\frac{l-1}{l}{h}_{qi}){\hat{Q}}_{ii}^{(l-1)}+(1-\frac{l-1}{l}{h}_{qi})e^{\frac{\lambda }{l}({\mu }_{pi}-{\mu }_1)}{\hat{Q}}_{ii}^{(l)}+\frac{2l}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\overline{\mathrm{\Phi }}}_{1i}({\mu }_{pi},h_{qi})=-(1-h_{qi}){\hat{Q}}_{ii}^{(l)}+(1-h_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{\hat{Q}}_{ii}^{(l+1)}+\frac{l+m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\overline{\mathrm{\Phi }}}_{2i}({\mu }_{pi},h_{qi})=-(1-\frac{m-1}{m}{h}_{qi}){\hat{Q}}_{ii}^{(l+1)}+(1-\frac{m-1}{m}{h}_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{\hat{Q}}_{ii}^{(l+2)}+\frac{2m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\overline{\mathrm{\Phi }}}_{mi}({\mu }_{pi},h_{qi})=-(1-\frac{1}{m}{h}_{qi}){\hat{Q}}_{ii}^{(l+m-1)}+(1-\frac{1}{m}{h}_{qi})e^{\frac{\lambda }{m}({\mu }_2-{\mu }_{pi})}{\hat{Q}}_{ii}^{(l+m)}+\frac{2m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ {\overline{\mathrm{\Phi }}}_{(m+1)i}({\mu }_{pi},h_{qi})=-{\hat{Q}}_{ii}^{(l+m)}+\frac{m}{{\mu }_2-{\mu }_1}{\hat{R}}_{ii},\hfill \\ \overline{\mathrm{\Lambda }}={\kappa }_1{\sigma }_2+{\kappa }_2({\sigma }_3+{\sigma }_4)+{\kappa }_3{\sigma }_5.\hfill \end{array}$$

Moreover, the state feedback gain matrices can be designed as

$$K_i=Y_i{X}_i^{-1},\mathrm{\forall }i=1,2,\dots ,N.$$

Proof Consider Theorem 3.2 and the overall closed-loop Markov jump system (25). Pre- and post-multiplying inequality (26) by block-diagonal matrix $diag\{P_i^{-1},\dots ,P_i^{-1},I,R_i^{-1},R^{-1}\}$ and its transpose, respectively. Letting

$$\begin{array}{r}{X}_i=P_i^{-1},Y_i=K_i{X}_i,{\hat{Q}}_{ij}^{(r)}=X_i{Q}_j^{(r)}{X}_i,{\hat{Q}}_i^{(r)}=X_i{Q}^{(r)}{X}_i,\\ {\hat{R}}_{ij}=X_i{R}_j{X}_i,{\hat{R}}_i=X_{i}R{X}_i,{\hat{S}}_{ij}=X_i{S}_j{X}_i.\end{array}$$

(38)

It can be easily obtained that

$${\mathrm{\Gamma }}_i({\mu }_{qi},h_{qi})=\left[\begin{array}{cccc}{\mathrm{\Phi }}_{1i}({\mu }_{pi},h_{qi})& {\tilde{\mathrm{\Psi }}}_{1i}& {\tilde{\mathrm{\Psi }}}_{2i}& {\mathrm{\Pi }}_{1i}\\ \ast & {\mathrm{\Phi }}_{2i}({\mu }_{pi},h_{qi})& {\tilde{\mathrm{\Psi }}}_{3i}& 0\\ \ast & \ast & {\tilde{\mathrm{\Gamma }}}_i& 0\\ \ast & \ast & \ast & {\mathrm{\Pi }}_{2i}\end{array}\right]<0,p=1,2,q=1,2,$$

(39)

where

$${\tilde{\mathrm{\Gamma }}}_i=\left[\begin{array}{cccc}-{\gamma }^2{X}_i{X}_i& X_i{F}_i& X_i{D}_i^{⊺}& X_i{D}_i^{⊺}\\ \ast & -I& 0& 0\\ \ast & \ast & -\frac{1}{{\kappa }_1}{X}_i{\hat{R}}_{ii}^{-1}{X}_i& 0\\ \ast & \ast & \ast & -\frac{1}{{\kappa }_2}{X}_i{\hat{R}}_i^{-1}{X}_i\end{array}\right].$$

From Lemma 2.3, for any $X_i>0$, ${\hat{R}}_{ii}>0$ and ${\hat{R}}_i>0$, one can obtain $-X_i{X}_i\le 2X_i-I$, $-X_i{\hat{R}}_{ii}{X}_i\le 2X_i-{\hat{R}}_{ii}$, $-X_i{\hat{R}}_i{X}_i\le 2X_i-{\hat{R}}_i$. Then (33) is equivalent to (39). Therefore, if (33) holds, system (1) is finite-time bounded with a prescribed $H_{\mathrm{\infty }}$ performance index γ. The proof is completed. □

Remark 11 By solving the Markovian jumping system with finite-time observer-based controller, the mode-dependent positive-definite weighting matrices ${\overline{R}}_i$ in inequalities (33)-(37) should be known first. For convenience, we always choose the initial value for ${\overline{R}}_i=I$.

Remark 12 In many actual applications, the minimum value of ${\gamma }_{min}^2$ is of interest. In Theorem 3.2, with a fixed λ, ${\gamma }_{min}$ can be obtained through the following optimization procedure:

$$\begin{array}{r}min{\gamma }^2\\ \text{s.t. (26)-(27)}.\end{array}$$

In Theorem 4.1, as for finite-time stability and boundedness, once the state bound $c_2$ is not ascertained, the minimum value $c_{2min}$ is of interest. With a fixed λ, and define ${\lambda }_1=1$, ${\lambda }_2={\sigma }_1$, then the following optimization problem can be formulated to get the minimum value $c_{2min}$

$$\begin{array}{r}min\varsigma {\gamma }^2+(1-\varsigma )c_2\\ \text{s.t. (33)-(37)},\end{array}$$

where ς is weighted factor, and $\varsigma \in [0,1]$.

5 Illustrative example

Example 1 Consider the Markovian jump system (1) with two operation modes and the following data:

$$\begin{array}{c}{A}_1=\left[\begin{array}{cc}-0.9& 0.5\\ -0.32& -0.8\end{array}\right],A_{\tau 1}=\left[\begin{array}{cc}-0.5& -0.3\\ 0.3& -0.2\end{array}\right],\hfill \\ B_1=\left[\begin{array}{cc}-1.05& 0.8\\ -0.15& -1.3\end{array}\right],C_1=\left[\begin{array}{cc}0.6& -0.4\\ 0.35& -0.41\end{array}\right],\hfill \end{array}$$

and the transition probability matrix is

$$\mathrm{\Omega }=\left[\begin{array}{cc}-0.2& 0.2\\ 0.8& -0.8\end{array}\right].$$

Under different levels of the upper bound ${\mu }_2$ and λ, Table 1 and Table 2 list the results of the maximum allowable upper bound, the decay rate λ for different time delays and maximum values of ${\mu }_2$ derived from various methods including the one proposed in this paper, respectively. One can see from Table 1 and Table 2 that the same results are obtained in [21, 22]. It is clear from Table 1 and Table 2 that the performance achieved by our method is much better than those by [21, 22]. Therefore, our results not only are less conservative, but also require the less number of scalar decision variables.

Table 1 Comparison of the upper bounds of the decay rate for different delays

Table 2 Comparison of the allowable values of time delay ${\mathit{\mu }}_{\mathbf{2}}$ for different decay rates

Example 2 Consider a two-mode Markovian jump system (1) with

$$\begin{array}{c}{A}_1=\left[\begin{array}{cc}-0.8& 1.5\\ 2& 3\end{array}\right],A_{\tau 1}=\left[\begin{array}{cc}-0.45& 1\\ -0.5& 2\end{array}\right],\hfill \\ B_1=\left[\begin{array}{cc}-1& 0.2\\ 0.5& -0.1\end{array}\right],D_1=\left[\begin{array}{c}0.2\\ 0.1\end{array}\right],\hfill \\ C_1=\left[\begin{array}{cc}0.2& 0\\ 0& 0.1\end{array}\right],C_{\tau 1}=\left[\begin{array}{cc}0.03& 0\\ 0.01& 0.02\end{array}\right],\hfill \\ E_1=\left[\begin{array}{cc}0.02& 0\\ 0.01& 0.01\end{array}\right],D_1=\left[\begin{array}{c}0.01\\ 0.001\end{array}\right],\hfill \\ A_2=\left[\begin{array}{cc}-2& 1.2\\ 1& 4\end{array}\right],A_{\tau 2}=\left[\begin{array}{cc}-1& 1.2\\ 0& -0.5\end{array}\right],\hfill \\ B_2=\left[\begin{array}{cc}-1& 1\\ 0.5& -2\end{array}\right],D_2=\left[\begin{array}{c}0.2\\ 0.3\end{array}\right],\hfill \\ C_2=\left[\begin{array}{cc}0.1& 0.02\\ 0& 0.1\end{array}\right],C_{\tau 2}=\left[\begin{array}{cc}0.02& 0\\ 0.1& 0.02\end{array}\right],\hfill \\ E_2=\left[\begin{array}{cc}0.04& 0\\ 0.1& 0.01\end{array}\right],F_2=\left[\begin{array}{c}0.04\\ 0.01\end{array}\right].\hfill \end{array}$$

In addition, the transition rate matrix is given by

$$\mathrm{\Omega }=\left[\begin{array}{cc}-1.2& 1.2\\ 1& -1\end{array}\right].$$

Then we choose ${\overline{R}}_1={\overline{R}}_1=I$, $T=2$, $c_1=1$, $d=0.01$, through Theorem 4.1, it yields that $c_2=152.4231$. Moreover, we also can obtain the following controller gains:

$$K_1=\left[\begin{array}{cc}-1.1562& 2.5461\\ -0.2712& 3.2523\end{array}\right],K_2=\left[\begin{array}{cc}0.9613& -1.7163\\ -2.2864& 4.3842\end{array}\right].$$

It confirms the effectiveness of Theorem 4.1 for the state feedback controller design to finite-time Markovian jump systems with time-varying delay.

6 Conclusions

In this paper, we have examined the problems of finite-time $H_{\mathrm{\infty }}$ control for a class of Markovian jump systems with mode-dependent time-varying delay. Based on a novel approach, a sufficient condition is derived such that the closed-loop Markovian jump system is finite-time bounded and satisfies the prescribed level of $H_{\mathrm{\infty }}$ disturbance attenuation in a finite time interval. The controller and observer gains can be solved directly by using the existing LMIs optimization techniques. Finally, numerical examples are also given to illustrate the effectiveness of the proposed design approach.