Exponential Stability in Mean Square of Singular Markovian Jump System with Saturating Actuators and Time-Varying Delay

Abstract

This paper investigates exponential stability in mean square of singular Markovian jump systems with saturating actuators and time-varying delay. The statistical property of the Markov process is fully used to derive the differential of the function. By using a delay decomposition method, a mode-dependent Lyapunov-Krasovskii function is established. A sufficient condition is proposed for exponential stability in mean square of the system designing the memoryless state feedback. A numerical example shows that the approach proposed is effective.

1. Introduction

Markovian jump systems are a special class of hybrid systems with both modes and state variables, which is described by a set of time-delayed linear systems with the transitions among models determined by a Markov chain in a finite mode set. In recent years, Markovian jump systems have received much attention, and many problems have been solved, including stability analysis, state feedback, and output feedback controller design [1, 2]. Both time-delay and saturating controls are commonly encountered in various engineering systems, which often lead to poor performance and instability of a control system. Therefore, much attention has been paid to the study on the stability problem of systems with saturating actuator. The problem of robust stabilization of uncertain time-delay systems with a saturating actuator was addressed by Niculescu et al. [3]. Liu has obtained the condition of stability for a class of time-varying delay systems with saturating actuators according to the usage of the linear matrix inequality and Leibniz-Newton formula [4]. For the systems with time-varying delays, the reported results are generally based on the assumption that the derivative of time-varying delays is less than one [3, 4]. Such restriction is very conservative. Furthermore, singular system model is a natural presentation of dynamic systems and can describe a larger class of systems than regular ones, such as large-scale systems, power systems, and constrained control systems, electrical circuits, power systems, and economics [5]. For a singular time-delay system, it is important to develop conditions which guarantee that the given singular system is not only stable but also regular and impulse free. Wu et al. discussed the stability problem on uncertain singular Markovian jump time-delay systems [6, 7]. Ma et al. investigated the robust stochastic stability problem for discrete-time uncertain singular Markov jump systems with actuator saturation [8]. Boukas et al. established an LMI condition for the singular time-delay systems with Markovian jump to be regular, impulse free, and stochastically stable. However, the result of [9] is delay independent. Generally speaking, delay-independent conditions are more conservative than delay-dependent ones, especially for small time delays. To the best of our knowledge, the robust stability for singular Markov jump systems with actuator saturation and time-varying delay has not been investigated in the literature; this problem is important in both theory and practice.

In this paper, we are concerned with stability of singular Markovian jump system with saturating actuators and time-varying delay. A new Lyapunov function can be constructed by using a delay decomposition method and considering the statistical property of Markovian process. In terms of LMI approach, stability conditions are proposed to guarantee the considered systems to be regular, impulse free, and exponentially stable in mean square. Free weighting matrix and reduction method are not introduced in the derivation of the stability criterion. Thus, the results reduce conservation.

2. Problem Formulation and Assumption

Consider the singular markovian jump system with saturating actuators and time-varying delay as follows: 𝐸𝑥^•=𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡𝑥((𝑡𝑟𝑡−𝜏))+𝐶_𝑡((𝑡𝑥(𝑡)(𝑡)[],Satu𝑡−𝜏)),𝑡≥0,=𝜙,𝑡∈−𝜏,0 (1) where𝑥(𝑡)∈𝑅^𝑁 and𝑢(𝑡)∈𝑅^𝑀 are the system states and inputs, respectively, 𝑢_𝑗(𝑡),𝑗=1,…,𝑀 is the𝑗th component of𝑢(𝑡), and 𝜙(𝑡)∈𝑅^𝑁 is a continuous vector-valued initial function defined on the interval[−𝜏,0]. The matrix𝐸∈𝑅^𝑁×𝑁 may be singular, and rank𝐸=𝑟<𝑛. (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) is the probability space. 𝐴(𝑟_𝑡), 𝐵(𝑟_𝑡), and 𝐶(𝑟_𝑡)are known real constant matrices with appropriate dimension. As 𝑟_𝑡=𝑖, for the sake of simplicity, 𝐴_𝑖, 𝐵_𝑖, and 𝐶_𝑖 denote the matrixes, respectively.𝜏(𝑡)is the time-varying delay satisfying 0≤𝜏(𝑡)≤𝜏,𝜏^•(𝑡)≤𝜇, (2) and ℎ, 𝜇 are real constant scalars.

The saturating function is defined as follows: Sat(𝑢(𝑡))=Sat𝑢₁(𝑡)Sat𝑢₂(𝑡)⋯Sat𝑢_𝑚(𝑡)^𝑇,Sat𝑢_𝑗(𝑡)=𝑈_𝑗,if𝑢_𝑗(𝑡)≤−𝑈_𝑗,𝑢_𝑗(𝑡)≤−𝑈_𝑗,𝑢_𝑗(𝑡),if−𝑈_𝑗<𝑢_𝑗(𝑡)<𝑈_𝑗,𝑈_𝑗,if𝑢_𝑗(𝑡)≥𝑈_𝑗,𝑢_𝑗(𝑡)≥𝑈_𝑗, (3)(4) where𝑈_𝑗>0(𝑗=1,…,𝑚)are known real constants.

Assumption 1. The markovian jumping parameter 𝑟_𝑡is right-continuous markovian process and takes values in finite set 𝑆={1,2,…,𝑁}, and is defined by 𝑃𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝜌=𝑖^𝑖𝑗Δ+𝑜(Δ),𝑖≠𝑗,1+𝜌_𝑖𝑗Δ+𝑜(Δ),𝑖=𝑗, (5) where Δ>0, lim_Δ→∞𝑜(Δ)/Δ=0,𝜌_𝑖𝑗 is the transition rate from 𝑖 to 𝑗 and 𝜌_𝑖𝑖=−^𝑆_{𝑗=1,𝑗≠𝑖}𝜌_𝑖𝑗,∀𝑖,𝑗∈𝑆𝜌_𝑖𝑗≥0,𝑗≠𝑖. (6)

Definition 2. The infinitesimal generator of the solution to system (1) is defined as 𝐿𝑉𝑥_𝑡,𝑟_𝑡=lim_Δ→0+𝐸𝑉𝑥_𝑡+Δ,𝑟_𝑡+Δ𝑥_𝑡,𝑟_𝑡−𝑉𝑥_𝑡,𝑟_𝑡Δ. (7)

Definition 3. (1) The singular Markovian jump system with saturating actuators and time-varying delay 𝐸𝑥^•(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴_𝑑𝑟_𝑡𝑥((𝑡,𝑥(𝑡)(𝑡)[]𝑡−𝜏))=𝜙,𝑡∈−𝜏,0 (8) is said to be regular and impulse free, if the pairs (𝐸,𝐴(𝑟_𝑡)) are singular and impulse free for every 𝑟_𝑡∈𝑆.
(2) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be exponential stable in mean square if there exist scalars𝛼>0, 𝛽>0, such that 𝐸‖𝑥(𝑡)‖²≤𝛼𝑒^−𝛽𝑡𝜙(𝑡)²_𝜏,𝑡>0. (9)
(3) The singular Markovian jump system with saturating actuators and time-varying delay (8) is said to be mean-square exponentially admissible, if it is regular, impulse free, and exponentially stable in mean square.

Lemma 4 (see [7]). Suppose that a positive continuous function 𝑓(𝑡) satisfies 𝐸𝑓(𝑡)≤𝜍₁𝐸sup_{𝑡−𝜏≤𝑠≤𝑡}𝑓(𝑠)+𝜍₂𝑒^−𝜀𝑡; (10) then 𝐸𝑓(𝑡)≤𝑒^−𝜀𝑡𝑓(𝑠)_𝜏+𝜍₂𝑒^−𝜀𝑡1−𝜍₁𝑒^𝜀𝜏, (11) where𝜀>0, 0<𝜍₁<1, 0<𝜍₁𝑒^𝜀𝜏<1, 𝜍₂>0 and 𝑑>0.

Lemma 5 (see [10]). 𝐷 is an 𝑀×𝑀 diagonal matrix whose entries are either 1 or 0. Note that there are2^𝑀 elements in𝐷. Suppose that each element in 𝐷 is labeled as 𝐷_𝑠, 𝑠∈[1,2^𝑀], and let𝐷⁻_𝑠=𝐼−𝐷_𝑠; it is obvious that𝐷⁻_𝑠∈𝐷, if 𝐷_𝑠∈𝐷. Given matrix𝐹,𝐻∈𝑅^𝑀×𝑁, one has 𝑆𝑎𝑡(𝐹𝑥)∈𝑐𝑜𝐷_𝑠𝐹𝑥+𝐷⁻_𝑠𝐻𝑥∶𝑠∈1,2^𝑀, (12) where𝑐𝑜 denotes convex hull.

Lemma 6 (see [11]). Given 𝑣>0, for a vector𝜔(𝑠) and a positive-definite matrixΖ with appropriate dimensions, and the integrations concerned are well defined, then 𝜐^𝜐₀𝜔(𝑠)^𝑇Ζ𝜔(𝑠)𝑑𝑠≥^𝜐₀𝜔(𝑠)𝑑𝑠^𝑇Ζ^𝜐₀𝜔(𝑠)𝑑𝑠. (13)

3. Main Results

Considering the system (1), for each region 𝑟_𝑡=𝑖∈𝑆, the system (1) becomes 𝐸𝑥^•(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏(𝑡))+𝐶_𝑖Satu(𝑡−𝜏(𝑡)). (14)

Taking the memoryless state feedback control for the systems (14), one has 𝑢(𝑡−𝜏(𝑡))=𝐾_𝑖𝑥(𝑡−𝜏(𝑡)). (15)

Substituting (15) into (14), then the corresponding closed-loop systems become 𝐸𝑥^•(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖𝑥(𝑡−𝜏(𝑡))+𝐶_𝑖Sat𝐾_𝑖𝑥(𝑡−𝜏(𝑡)). (16)

Theorem 7. For given constants 𝜏, 𝜇 and two positive integers 𝑘, 𝑙, The system (16) is exponentially stable in mean square. If there exist symmetric positive matrices𝑃_𝑖=𝑃^𝑇_𝑖>0,  𝑁_𝑖=𝑁^𝑇_𝑖>0,  𝑅_𝑚𝑖=𝑅^𝑇_𝑚𝑖>0(𝑚=1,…,𝑙),  𝑀_𝑛𝑖=𝑀^𝑇_𝑛𝑖>0(𝑛=1,…,𝑘),  𝑇₁=𝑇^𝑇₁>0,  𝑇₂=𝑇^𝑇₂>0,  𝑄_𝑖=𝑄^𝑇_𝑖>0, where 𝑄_𝑖=𝑄^𝑇_𝑖=𝑄_11𝑖⋯𝑄_1𝑘𝑖𝑄⋮⋯⋮_{𝑇1𝑘𝑖}⋯𝑄_𝑘𝑘𝑖, (17)
for every𝑖∈𝑆, such that𝐸^𝑇𝑃_𝑖=𝑃_𝑖𝐸, (18a)ΣΣ=¹¹𝑄_12𝑖+𝑀_1𝑖−𝑀_2𝑖𝑄_13𝑖+𝑀_2𝑖−𝑀_3𝑖⋯𝑄_{1𝑘−1𝑖}+𝑀_𝑘−2𝑖−𝑀_𝑘−1𝑖𝑄_1𝑘𝑖+𝑀_𝑘−1𝑖−𝑀_𝑘𝑖𝑀_𝑘𝑖∗𝑄_22𝑖−𝑄_11𝑖𝑄_23𝑖−𝑄_12𝑖⋯𝑄_{2𝑘−1𝑖}−𝑄_{1𝑘−2𝑖}𝑄_2𝑘𝑖−𝑄_{1𝑘−1𝑖}−𝑄_1𝑘𝑖∗∗𝑄_33𝑖−𝑄_22𝑖⋯𝑄_{3𝑘−1𝑖}−𝑄_{2𝑘−2𝑖}𝑄_3𝑘𝑖−𝑄_{2𝑘−1𝑖}−𝑄_2𝑘𝑖⋮⋮⋮⋮⋮⋮⋮∗∗∗⋯𝑄_{𝑘−1𝑘−1𝑖}−𝑄_{𝑘−2𝑘−2𝑖}𝑄_{𝑘−1𝑘𝑖}−𝑄_{𝑘−2𝑘−1𝑖}−𝑄_{𝑘−2𝑘−1𝑖}∗∗∗⋯∗𝑄_𝑘𝑘𝑖−𝑄_{𝑘−1𝑘−1𝑖}−𝑄_{𝑘−1𝑘𝑖}∗∗∗⋯∗∗−𝑄_𝑘𝑘𝑖−1𝑙𝑇₂∗00⋯000000⋯000000⋯000⋮⋮⋮⋯⋮⋮⋮000⋯000000⋯000∗00⋯00∗2𝐸^𝑇𝑇₂𝐸00⋯00Σ_1𝑘+𝑙+11000⋯000000⋯000⋮⋮⋮⋮⋮⋮⋮000⋯000000⋯000000⋯00−𝑙𝐸^𝑇𝑇₂𝐸−2𝐸^𝑇𝑇₂𝜇𝐸−1−𝑙𝑅_𝑖1𝐸^𝑇𝑇₂𝐸0⋯000∗−2𝐸^𝑇𝑇₂𝐸−1−2𝜇𝑙𝑅_𝑖2𝐸^𝑇𝑇₂𝐸⋯0000∗−2𝐸^𝑇𝑇₂𝐸−1−3𝜇𝑙𝑅_𝑖3⋯000⋮⋮⋮⋮⋮⋮⋮000⋯−2𝐸^𝑇𝑇₂𝐸−1−𝑙−2𝑙𝑅_𝑖𝑙−2𝐸^𝑇𝑇₂𝐸0000⋯∗−2𝐸^𝑇𝑇₂𝐸−1−𝑙−1𝑙𝑅_𝑖𝑙−1𝑇₂000⋯0∗Σ_{𝑘+𝑙+1𝑘+𝑙+1}<0, (18b)where Σ₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑃_𝑖+^𝑙_𝑚=1𝑅_𝑚𝑖+𝜏²𝑘𝐴^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑙𝐴^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑘𝑇₁+𝜏²𝑙𝑇₂−𝑀_1𝑖−𝐸^𝑇𝑇₂𝐸+𝑄_11𝑖,Σ_1𝑘+𝑙+1=𝐵^𝑇_𝑖𝑃_𝑖+𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑃_𝑖+𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑃_𝑖+𝜏²𝑘𝐵^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑘𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑘𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑙𝐵^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑙𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑙𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑇₂𝐴_𝑖,Σ_{𝑘+𝑙𝑘+𝑙}=𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖^𝑇𝜏²𝑘𝑇₁+𝜏²𝑙𝑇₂×𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖−1𝑙𝐸^𝑇𝑇₂𝐸−1−𝜇𝑅_𝑖𝑙,−1ℎ𝑀_1𝑖0𝐴^𝑇_𝑖𝑀_𝑛𝑖+∑_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑀_𝑛𝑗0−𝐸^𝑇𝑇₂𝐸𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖^𝑇𝑀_𝑛𝑖∗∗−𝜏𝑇₁≤0,ℎΓ₁0𝑁_𝑖0ℎΓ₂−𝑁_𝑖∗∗∑_𝑗∈𝑆𝜌_𝑖𝑗𝑁_𝑗+𝑄_𝑗≤0,_𝑗∈𝑆𝜌_𝑖𝑗𝑅_𝑚𝑗−𝜏𝑇₂≤0. (18c)

Proof. According to the definition of the saturating function and Lemma 5, we have Sat𝐾_𝑖𝑥(𝑡−𝜏(𝑡))=𝐷_𝑞𝐾_𝑖𝑥(𝑡−𝜏(𝑡))+𝐷⁻_𝑞𝐻_𝑖𝑥(𝑡−𝜏(𝑡))𝑞∈1,2^𝑀. (19)
Substituting (19) into (16) gives 𝐸𝑥^•(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖𝑥(𝑡−𝜏(𝑡)). (20)
Since rank𝐸=𝑟<𝑛, there exist nonsingular matrices 𝐺 and Υ, such that 𝐼𝐺𝐸Υ=^𝑟000. (21)
Let 𝐺𝐴_𝑖𝐴Υ=^𝑖1𝐴_𝑖2𝐴_𝑖3𝐴_𝑖4,𝐺^−𝑇𝑃_𝑖Υ=𝑃_𝑖1𝑃_𝑖2𝑃_𝑖3𝑃_𝑖4. (22)
According to (18a), it is easy to obtain that 𝑃_𝑖2=0 for every 𝑖∈𝑆. By pre- and postmultiplying Σ_11𝑖<0 byΥ^𝑇 and Υ, respectively, we have𝐴^𝑇_𝑖4𝑃_𝑖4+𝑃^𝑇_𝑖4𝐴_𝑖4<0, which implies that 𝐴_𝑖4 are nonsingular for every 𝑖∈𝑆. Thus, the pairs (𝐸,𝐴_𝑖)are regular and impulse free for every𝑖∈𝑆. Thus, by Definition 2, the system (8) is regular and impulse free for any time-varying delay𝜏(𝑡) satisfying (2).
Choose a Lyapunov-Krasovskii function candidate as 𝑉𝑥_𝑡,𝑟_𝑡,𝑡=𝑉₁𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₂𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₃𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₄𝑥₄,𝑟_𝑡,𝑡+𝑉₅𝑥_𝑡,𝑟_𝑡,𝑡+𝑉₆𝑥_𝑡,𝑟_𝑡,𝑡, (23) where 𝑉₁𝑥_𝑡,𝑟_𝑡,𝑡=𝑥(𝑡)^𝑇𝐸^𝑇𝑃𝑟_𝑡(𝑡),𝑉𝐸𝑥₂𝑥_𝑡,𝑟_𝑡,𝑡=^𝑡_𝑡−ℎ𝜉(𝑠)^𝑇𝑄𝑟_𝑡𝜉(𝑠)𝑉𝑑𝑠,₃𝑥_𝑡,𝑟_𝑡,𝑡=2𝑥(𝑡)^𝑇𝑘_𝑛=1𝐸^𝑇𝑀_𝑛𝑟_𝑡^{𝑡−(𝑛−1)ℎ}_{𝑡−𝑛ℎ}𝑥(𝑠)𝑉𝑑𝑠,₄𝑥_𝑡,𝑟_𝑡,𝑡=^𝑡_𝑡−ℎ𝑑𝑠^𝑡_𝑡−ℎ𝜉(𝑠)^𝑇𝑁𝑟_𝑡𝜉(𝜃)𝑉𝑑𝜃,₅𝑥_𝑡,𝑟_𝑡,𝑡=^𝑙_𝑚=1^𝑡_{(𝑡)𝑡−𝑚𝜎}𝑥(𝑠)^𝑇𝑅_𝑚𝑟_𝑡𝑥(𝑠)𝑉𝑑𝑠,₆𝑥_𝑡,𝑟_𝑡,𝑡=ℎ⁰_−𝜏^𝑡•_𝑡+𝜃𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)+𝑥(𝑠)^𝑇𝑇₁𝑥(𝑠)𝑑𝑠𝑑𝜃+𝜎⁰_−𝜏^𝑡•_𝑡+𝜃𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)+𝑥(𝑠)^𝑇𝑙𝑇₂𝑥(𝑠)𝜉𝑑𝑠𝑑𝜃,_𝑇(𝑡)=[𝑥(𝑡),𝑥(𝑡−ℎ),…,𝑥(𝑡−(𝑘−1)ℎ)]^𝑇,𝜎(𝑡)=𝜏(𝑡)𝑙𝜏,ℎ=𝑘𝜏,𝜎=𝑙. (24)
Moreover, the action of infinitesimal generator (7) on each term function (23) could be expressed as 𝐿𝑉₁𝑥_𝑡𝑖,𝑡=2𝑥(𝑡)^𝑇𝐸^𝑇𝑃𝑥_𝑖^•(𝑡)+𝑥(𝑡)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑃_𝑗𝑥(𝑡),𝐿𝑉₂𝑥_𝑡,𝑖,𝑡=𝜉(𝑡)^𝑇𝑄_𝑖𝜉(𝑡)−𝜉(𝑡−ℎ)^𝑇𝑄_𝑖𝜉(𝑡−ℎ)+^𝑡_𝑡−ℎ𝜉(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑄_𝑗𝜉(𝑠)𝑑𝑠,𝐿𝑉₃𝑥_𝑡,𝑖,𝑡=2𝑥^•(𝑡)^𝑇𝑘_𝑛=1𝐸^𝑇𝑀_𝑛𝑖^{𝑡−(𝑛−1)ℎ}_{𝑡−𝑛ℎ}𝑥(𝑠)(𝑡)𝑑𝑠−2𝑥^𝑇𝑘_𝑛=1𝑀_𝑛𝑖[𝑥(()ℎ)((𝑡)𝑡−𝑛−1−𝑥𝑡−𝑛ℎ)]+2𝑥^𝑇𝑘_{𝑛=1𝑗∈𝑆}𝜌_𝑖𝑗𝐸^𝑇𝑀_𝑛𝑗^{𝑡−(𝑛−1)ℎ}_{𝑡−𝑛ℎ}𝑥(𝑠)𝑑𝑠,𝐿𝑉₄𝑥_𝑡,𝑖,𝑡=2𝜉(𝑡)−𝜉(𝑡−ℎ)^𝑇𝑁_𝑖^𝑡_𝑡−ℎ𝜉(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ𝑡𝑡−ℎ}𝜉(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑁_𝑗𝜉(𝜃)𝑑𝑠𝑑𝜃,𝐿𝑉₅𝑥_𝑡,𝑖,𝑡=𝑥(𝑡)^𝑇𝑙_𝑚=1𝑅_𝑚𝑖𝑥(𝑡)−^𝑙_𝑚=11−𝑚𝜎^•(𝑡)×𝑥(𝑡−𝑚𝜎(𝑡))^𝑇𝑅_𝑚𝑖𝑥((𝑡+𝑡−𝑚𝜎))_{𝑙𝑚=1𝑡(𝑡)𝑡−𝑚𝜎}𝑥(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑅_𝑚𝑗𝑥(𝑠)𝑑𝑠,𝐿𝑉₆𝑥_𝑡,𝑖,𝑡=ℎ𝜏𝑥^•(𝑡)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑡)+ℎ𝜏𝑥(𝑡)^𝑇𝑇₁𝑥(𝑡)−ℎ^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁(𝑠)(𝑠)𝐸𝑥+𝑥^𝑇𝑇₁𝑥(𝑠)𝑑𝑠+𝜎𝜏𝑥^•(𝑡)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑡)+𝜎𝜏𝑥(𝑡)^𝑇𝑙𝑇₂𝑥(𝑡)−𝜎^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)+𝑥(𝑠)^𝑇𝑙𝑇₂𝑥(𝑠)𝑑𝑠. (25) According to (20), (23), and (25), we have 𝐿𝑉𝑥_𝑡,𝑖,𝑡≤𝑥(𝑡)^𝑇𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑃_𝑗+^𝑙_𝑚=1𝑅_𝑖𝑚+ℎ𝜏𝐴^𝑇_𝑖𝑇₁𝐴_𝑖+𝜎𝜏𝐴^𝑇_𝑖𝑇₂𝐴_𝑖+ℎ𝜏𝑇₁+𝜎𝜏𝑇₂×𝑥(𝑡)+2𝑥(𝑡)^𝑇𝑃_𝑖𝐵_𝑖+𝑃_𝑖𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝑃_𝑖𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖+ℎ𝜏𝐴^𝑇_𝑖𝑇₁𝐵_𝑖+ℎ𝜏𝐴^𝑇_𝑖𝑇₁𝐶_𝑖𝐷_𝑞𝐾_𝑖+ℎ𝜏𝐴^𝑇_𝑖𝑇₁𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖+𝜎𝜏𝐴^𝑇_𝑖𝑇₂𝐵_𝑖+𝜎𝜏𝐴^𝑇_𝑖𝑇₂𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝜎𝜏𝐴^𝑇_𝑖𝑇₂𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖((𝑡((𝑡×𝑥𝑡−𝜏))+𝑥𝑡−𝜏))^𝑇𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖×ℎ𝜏𝑇₁+𝜎𝜏𝑇₂𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖×𝑥(𝑡−𝜏(𝑡))+𝜉(𝑡)^𝑇𝑄_𝑖𝜉(𝑡)−𝜉(𝑡−ℎ)^𝑇𝑄_𝑖𝜉()+𝑡−ℎ_{𝑡𝑡−ℎ}𝜉(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑄_𝑗𝜉(𝑠)+𝑑𝑠_{𝑘𝑛=1𝑡−(𝑛−1)ℎ𝑡−𝑛ℎ}𝑥(𝑡)𝑥(𝑡−𝜏(𝑡))𝑥(𝑠)^𝑇×−1ℎ𝑀_1𝑖0𝐴^𝑇_𝑖𝑀_𝑛𝑖+∑_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑀_𝑛𝑗00𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖^𝑇𝑀_𝑛𝑖∗∗−ℎ𝑇₁×𝑥(𝑡)𝑥((𝑡𝑥(𝑠)(𝑡)𝑡−𝜏))−𝑥^𝑇𝑀_1𝑖𝑥(𝑡)+2𝑥(𝑡)^𝑇𝑀_𝑖1𝑥()(𝑡)𝑡−ℎ−2𝑥^𝑇𝑘_𝑛=2𝑀_𝑛𝑖[𝑥(()ℎ)(+𝑡−𝑛−1−𝑥𝑡−𝑛ℎ)]_{𝑡𝑡−ℎ}𝜉(𝑡)−𝜉(𝑡−ℎ)^𝑇𝑁_𝑖𝜉(𝑠)+𝜉(𝑠)𝑁_𝑖𝜉(𝑡)((𝑠)−𝜉𝑡−ℎ)+𝜉^𝑇_𝑗∈𝑆𝜌_𝑖𝑗ℎ𝑁_𝑗𝜉(𝑠)−𝑑𝑠_𝑙𝑚=11−𝑚𝜎^•(𝑡)𝑥(𝑡−𝑚𝜎(𝑡))^𝑇×𝑅_𝑚𝑖𝑥((𝑡+𝑡−𝑚𝜎))_{𝑙𝑚=1𝑡(𝑡)𝑡−𝑚𝜎}𝑥(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑅_𝑚𝑗𝑥(𝑠)−𝑥(𝑠)^𝑇𝜏𝑇₂𝑥(𝑠)𝑑𝑠−ℎ^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)𝑑𝑠×𝜎^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝑑𝑠, (26)𝜉(𝑡)^𝑇𝑄_𝑖𝜉(𝑡)−𝜉(𝑡−ℎ)^𝑇𝑄_𝑖𝜉(𝑡−ℎ)=𝜁(𝑡)^𝑇(𝑡),𝜁(𝑡)Ξ𝜁_𝑇=𝜉(𝑡)^𝑇,𝑥(𝑡−𝜏)^𝑇𝑇,𝑄Ξ=^11𝑖𝑄_12𝑖𝑄_13𝑖𝑄_14𝑖⋅⋅⋅𝑄_{1𝑘−1𝑖}𝑄_1𝑘𝑖0∗𝑄_22𝑖−𝑄_11𝑖𝑄_23𝑖−𝑄_12𝑖𝑄_24𝑖−𝑄_13𝑖⋅⋅⋅𝑄_{2𝑘−1𝑖}−𝑄_{1𝑘−2𝑖}𝑄_2𝑘𝑖−𝑄_{1𝑘−1𝑖}−𝑄_1𝑘𝑖∗∗𝑄_33𝑖−𝑄_22𝑖𝑄_34𝑖−𝑄_23𝑖⋅⋅⋅𝑄_{3𝑘−1𝑖}−𝑄_{2𝑘−2𝑖}𝑄_3𝑘𝑖−𝑄_{2𝑘−1𝑖}−𝑄_2𝑘𝑖∗∗∗𝑄_44𝑖−𝑄_33𝑖⋅⋅⋅𝑄_{4𝑘−1𝑖}−𝑄_{3𝑘−2𝑖}𝑄_4𝑘𝑖−𝑄_{3𝑘−1𝑖}−𝑄_3𝑘𝑖⋮⋮⋮⋮⋮⋮⋮⋮∗∗∗∗∗𝑄_{𝑘−1𝑘−1𝑖}−𝑄_{𝑘−2𝑘−2𝑖}𝑄_{𝑘−1𝑘𝑖}−𝑄_{𝑘−2𝑘−1𝑖}−𝑄_{𝑘−2𝑘𝑖}∗∗∗∗∗∗𝑄_𝑘𝑘𝑖−𝑄_{𝑘−1𝑘−1𝑖}−𝑄_{𝑘−1𝑘𝑖}∗∗∗∗∗∗∗−𝑄_𝑘𝑘𝑖, (27)−𝑙𝜎^𝑡_𝑡−𝜏𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠≤−𝑙𝜎^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠≤−𝜎^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠−𝜎^𝑡_{()𝜎(𝑡)𝑡−𝑙−1}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠−⋯−𝜎^𝑡_{(𝑡)𝑡−2𝜎}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠−𝜎^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠=−𝜎^𝑙_𝑚=1^𝑡_{(𝑡)𝑡−𝑚𝜎}𝑥(𝑠)^𝑇𝑇₂𝑥(𝑠)𝑑𝑠,−ℎ^𝑡_𝑡−𝜏𝑥(𝑠)^𝑇𝑇₁𝑥(𝑠)𝑑𝑠=−ℎ^𝑘_𝑛=1^{𝑡−(𝑛−1)ℎ}_{𝑡−𝑛ℎ}𝑥(𝑠)^𝑇𝑇₁𝑥(𝑠)𝑑𝑠,−ℎ^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)𝑑𝑠=−ℎ^𝑡•_𝑡−ℎ𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)𝑑𝑠−ℎ^{•𝑡−ℎ}_𝑡−2ℎ𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•𝑑𝑠−⋯−ℎ_()ℎ^{•𝑡−(𝑘−1)ℎ}_{𝑡−𝑘−2}𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)𝑑𝑠−ℎ^{•𝑡−(𝑘−1)ℎ}_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₁𝐸𝑥^•(𝑠)𝑑𝑠≤−^𝑡•_𝑡−ℎ𝑥(𝑠)𝑑𝑠^𝑇𝐸^𝑇𝑇₁𝐸^𝑡•_𝑡−ℎ𝑥(𝑠)−𝑑𝑠_{•𝑡−ℎ𝑡−2ℎ}𝑥(𝑠)𝑑𝑠^𝑇𝐸^𝑇𝑇₁𝐸×^{•𝑡−ℎ}_𝑡−2ℎ𝑥(𝑠)𝑑𝑠−⋯−^{•𝑡−(𝑘−1)ℎ}_𝑡−𝜏𝑥(𝑠)𝑑𝑠^𝑇×𝐸^𝑇𝑇₁𝐸^{•𝑡−(𝑘−1)ℎ}_𝑡−𝜏𝑥(𝑠)[𝑥(𝑡)(𝑑𝑠=−−𝑥𝑡−ℎ)]^𝑇𝐸^𝑇𝑇₁𝐸[𝑥(𝑡)(−[𝑥()(−𝑥𝑡−ℎ)]𝑡−ℎ−𝑥𝑡−2ℎ)]^𝑇×𝐸^𝑇𝑇₁𝐸[𝑥()([𝑥(()ℎ)(𝑡−ℎ−𝑥𝑡−2ℎ)]−⋯−𝑡−𝑘−1−𝑥𝑡−𝜏)]^𝑇×𝐸^𝑇𝑇₁𝐸[𝑥(()ℎ)((𝑡)𝑡−𝑘−1−𝑥𝑡−𝜏)]=𝜉^𝑇Γ₁𝜉(𝑡)+𝜉(𝑡−ℎ)^𝑇Γ₂𝜉(),Γ𝑡−ℎ₁=−𝐸^𝑇𝑇₁𝐸𝐸^𝑇𝑇₁𝐸0⋯00∗−𝐸^𝑇𝑇₁𝐸𝐸^𝑇𝑇₁𝐸⋯000∗−𝐸^𝑇𝑇₁𝐸⋯00⋮⋮⋮⋮⋮⋮000⋯−𝐸^𝑇𝑇₁𝐸𝐸^𝑇𝑇₁𝐸000⋯𝐸^𝑇𝑇₁𝐸−𝐸^𝑇𝑇₁𝐸,Γ₂=−𝐸^𝑇𝑇₁𝐸00⋯000−𝐸^𝑇𝑇₁𝐸0⋯0000−𝐸^𝑇𝑇₁𝐸⋯00⋮⋮⋮⋮⋮⋮000⋯−𝐸^𝑇𝑇₁𝐸0000⋯0−𝐸^𝑇𝑇₁𝐸,−𝜎^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝑑𝑠=−𝜎^𝑡_(𝑡)^•_𝑡−𝜎𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝑑𝑠−_(𝑡)^{•𝑡−𝜎(𝑡)}_𝑡−2𝜎𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥(𝑠)𝑑𝑠−⋯−𝜎^{•𝑡−𝜏(𝑡)}_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂(𝑠)(𝑡)𝐸𝑥𝑑𝑠≤−𝜎^𝑡_(𝑡)^•_𝑡−𝜎𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)(𝑡)𝑑𝑠−𝜎_(𝑡)^{•𝑡−𝜎(𝑡)}_𝑡−2𝜎𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝜎𝑑𝑠−⋯−𝜏((𝑡×𝜏−𝜏))_{•𝑡−𝜏(𝑡)𝑡−𝜏}𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝑑𝑠. (28)(29)(30)(31)(32)(33) Similar to (33), by Lemma 6, it is easy to obtain−𝜎^𝑡•_𝑡−𝜏𝑥(𝑠)^𝑇𝐸^𝑇𝑇₂𝐸𝑥^•(𝑠)𝑑𝑠≤𝜁₂(𝑡)^𝑇−𝐸^𝑇𝑇₂𝐸0𝐸^𝑇𝑇₂1𝐸0⋯000−𝑙𝐸^𝑇𝑇₂1𝐸00⋯0𝑙𝐸^𝑇𝑇₂𝐸∗0−2𝐸^𝑇𝑇₂𝐸𝐸^𝑇𝑇₂𝐸⋯0000∗−2𝐸^𝑇𝑇₂𝐸⋯00⋮⋮⋮⋮⋯⋮⋮0000⋯−2𝐸^𝑇𝑇₂𝐸𝐸^𝑇𝑇₂𝐸10∗00⋯∗−1+𝑙𝐸^𝑇𝑇₂𝐸𝜁₂(𝑡), (34)where𝜁₂(𝑡)=𝑥(𝑡)^𝑇𝑥(𝑡−𝜏)^𝑇𝑥(𝑡−𝜎(𝑡))^𝑇𝑥(𝑡−2𝜎(𝑡))^𝑇⋅⋅⋅𝑥(𝑡−(𝑙−1)𝜎(𝑡))^𝑇𝑥(𝑡−𝜏(𝑡))^𝑇. (35)In conclusion, one has 𝐿𝑉𝑥_𝑡≤,𝑖,𝑡_{𝑘𝑛=1𝑡−(𝑛−1)ℎ𝑡−𝑛ℎ}𝑥(𝑡)^𝑇𝑥(𝑡−𝜏(𝑡))^𝑇𝑥(𝑠)^𝑇𝑇×−1ℎ𝑀_𝑖10𝐴^𝑇_𝑖𝑀_𝑛𝑖+∑_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑀_𝑛𝑗0−𝐸^𝑇𝑇₂𝐸𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖𝑀_𝑛𝑖∗∗−𝜏𝑇₁×𝑥(𝑡)𝑥((𝑡𝑥(𝑠)+𝑡−𝜏))𝑑𝑠_{𝑡𝑡−ℎ}𝜉(𝑡)^𝑇𝜉(𝑡−ℎ)^𝑇𝜉(𝑠)^𝑇×ℎΓ₁0𝑁_𝑖0ℎΓ₂−𝑁_𝑖∗∗∑_𝑗∈𝑆𝜌_𝑖𝑗ℎ𝑁_𝑗+𝑄_𝑗𝜉(𝑡)𝜉()𝜉(𝑠)+𝑡−ℎ𝑑𝑠_{𝑙𝑚=1𝑡(𝑡)𝑡−𝑚𝜎}𝑥(𝑠)^𝑇_𝑗∈𝑆𝜌_𝑖𝑗𝑅_𝑗𝑚−𝜏𝑇₂𝑥(𝑠)(𝑡)𝑑𝑠+𝜋^𝑇Σ𝜋(𝑡), (36) where 𝜋(𝑡)=𝜉(𝑡)^𝑇,𝑥(𝑡−𝜏)^𝑇,𝑥(𝑡−𝜎(𝑡))^𝑇,𝑥(𝑡−2𝜎(𝑡))^𝑇,…,𝑥(𝑡−(𝑙−1)𝜎(𝑡))^𝑇,𝑥(𝑡−𝜏(𝑡))^𝑇𝑇,_{ΣΣ=11𝑄12𝑖+𝑀𝑖1−𝑀𝑖2𝑄13𝑖+𝑀𝑖2−𝑀𝑖3⋯𝑄1𝑘−1𝑖+𝑀𝑖𝑘−2−𝑀𝑖𝑘−1𝑄1𝑘𝑖+𝑀𝑖𝑘−1−𝑀𝑖𝑘𝑀𝑖𝑘𝐸𝑇𝑇2𝐸∗𝑄22𝑖−𝑄11𝑖𝑄23𝑖−𝑄12𝑖⋯𝑄2𝑘−1𝑖−𝑄1𝑘−2𝑖𝑄2𝑘𝑖−𝑄1𝑘−1𝑖−𝑄1𝑘𝑖0∗∗𝑄33𝑖−𝑄22𝑖⋯𝑄3𝑘−1𝑖−𝑄2𝑘−2𝑖𝑄3𝑘𝑖−𝑄2𝑘−1𝑖−𝑄2𝑘𝑖0⋮⋮⋮⋮⋮⋮⋮⋮∗∗∗⋯𝑄𝑘−1𝑘−1𝑖−𝑄𝑘−2𝑘−2𝑖𝑄𝑘−1𝑘𝑖−𝑄𝑘−2𝑘−1𝑖−𝑄𝑘−2𝑘−1𝑖0∗∗∗⋯∗𝑄𝑘𝑘𝑖−𝑄𝑘−1𝑘−1𝑖−𝑄𝑘−1𝑘𝑖0∗∗∗⋯∗∗−𝑄𝑘𝑘𝑖−1𝑙𝐸𝑇𝑇2𝐸0∗00⋯000−2𝐸𝑇𝑇2𝜇𝐸−1−𝑙𝑅𝑖1000⋯000∗000⋯0000⋮⋮⋮⋯⋮⋮⋮⋮000⋯0000000⋯0000∗00⋯00∗000⋯00Σ1𝑘+𝑙+1100⋯00000⋯000⋮⋮⋮⋮⋮⋮00⋯00000⋯00000⋯00−𝑙𝐸𝑇𝑇2𝐸𝐸𝑇𝑇2𝐸0⋯000−2𝐸𝑇𝑇2𝐸−1−2𝜇𝑙𝑅𝑖2𝐸𝑇𝑇2𝐸⋯000∗−2𝐸𝑇𝑇2𝐸−1−3𝜇𝑙𝑅𝑖3⋯000⋮⋮⋮⋮⋮⋮00⋯−2𝐸𝑇𝑇2()𝜇𝐸−1−𝑙−2𝑙𝑅𝑖𝑙−2𝐸𝑇𝑇2𝐸000⋯∗−2𝐸𝑇𝑇2()𝜇𝐸−1−𝑙−1𝑙𝑅𝑖𝑙−1𝐸𝑇𝑇2𝐸00⋯0∗Σ𝑘+𝑙+1𝑘+𝑙+1,}Σ₁₁=𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+_𝑗∈𝑆𝜌_𝑖𝑗𝐸^𝑇𝑃_𝑖+^𝑙_𝑚=1𝑅_𝑖𝑚+𝜏²𝑘𝐴^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑙𝐴^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑘𝑇₁+𝜏²𝑙𝑇₂−𝑀_𝑖1−𝐸^𝑇𝑇₂𝐸+𝑄_11𝑖,Σ_{1𝑘+𝑙−1}=𝐵^𝑇_𝑖𝑃_𝑖+𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑃_𝑖+𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑃_𝑖+𝜏²𝑘𝐵^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑘𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑘𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑇₁𝐴_𝑖+𝜏²𝑙𝐵^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑙𝐾^𝑇_𝑖𝐷^𝑇_𝑞𝐶^𝑇_𝑖𝑇₂𝐴_𝑖+𝜏²𝑙𝐻^𝑇_𝑖𝐷_𝑞^−𝑇𝐶^𝑇_𝑖𝑇₂𝐴_𝑖,Σ_{𝑘+𝑙𝑘+𝑙}=𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖^𝑇𝜏²𝑘𝑇₁+𝜏²𝑙𝑇₂𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖−1𝑙𝐸^𝑇𝑇₂𝐸−1−𝜇𝑅_𝑖𝑙; (37)
using (18b) and (18c), it is easy to obtain𝐿𝑉𝑥_𝑡,𝑖,𝑡≤0. (38)
Since 𝐴_𝑖4 are nonsingular for every 𝑖∈𝑆, we set 𝐺^∧=𝐼_𝑟−𝐴_𝑖2𝐴⁻¹_𝑖40𝐴⁻¹_𝑖4𝐺. (39)
It is easy to get 𝐺^∧𝐼𝐸Υ=^𝑟0,00^∧𝐺𝐴_𝑖Υ=^∧𝐴_𝑖10_∧𝐴_𝑖3𝐼, (40)
where 𝐴^∧_𝑖1=𝐴_𝑖1−𝐴_𝑖2𝐴⁻¹_𝑖4𝐴_𝑖3,𝐴^∧_𝑖3=𝐴⁻¹_𝑖4𝐴_𝑖3. (41)
Denote that 𝐺^∧𝐵_𝑖+𝐶_𝑖𝐷_𝑞𝐾_𝑖+𝐶_𝑖𝐷⁻_𝑞𝐻_𝑖𝐴Υ=^𝑖𝑑1𝐴_𝑖𝑑2𝐴_𝑖𝑑3𝐴_𝑖𝑑4∧,𝐺^−𝑇𝑃_𝑖𝑃Υ=^𝑖1𝑃_𝑖2𝑃_𝑖3𝑃_𝑖4,Υ^𝑇𝑅_𝑖𝑙𝑅Υ=^1𝑖𝑙𝑅_2𝑖𝑙𝑅_3𝑖𝑙𝑅_4𝑖𝑙. (42)
Let 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)=Υ⁻¹𝑥(𝑡). (43)
Then, for every 𝑖∈𝑆, system (20) is a restricted system equivalent to 𝑥^•₁(𝑡)=𝐴^∧_𝑖1𝑥₁(𝑡)+𝐴_𝑖𝑑1𝑥₁(𝑡−𝜏(𝑡))+𝐴_𝑖𝑑2𝑥₂((𝑡,−𝑡−𝜏))𝑥₂(𝑡)=𝐴^∧_𝑖13𝑥₁(𝑡)+𝐴_𝑖𝑑3𝑥₁(𝑡−𝜏(𝑡))+𝐴_𝑖𝑑4𝑥₂(𝑡−𝜏(𝑡)),𝑥(𝑡)=𝜑(𝑡)=𝑆⁻¹𝜙(𝑡),𝑡∈[−𝜏,0]. (44)
To prove the exponential stability in mean square, we define a new function as 𝑊𝑥_𝑡,𝑖,𝑡=𝑒^𝜀𝑡𝑉𝑥_𝑡,𝑖,𝑡, (45) where 𝜀>0 and then by Dynkin’s formula, we find that for every 𝑖∈𝑆, 𝐸𝑊𝑥_𝑡,𝑟_𝑡,𝑡≤𝐸𝑊𝑥₀,𝑟₀,0+𝐸^𝑡₀𝑒^𝜀𝑠𝜀𝑉𝑥_𝑠,𝑟_𝑠,𝑠−𝜆‖𝑥(𝑠)‖²𝑑𝑠. (46)
It can be seen from (23), (45), and (46) that if𝜀 is chosen small enough, a constant 𝑘>0 can be found such that for 𝑡>0, min_𝑖∈𝑆𝜆_min𝑃_𝑖1𝐸𝑥₁²≤𝐸𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)≤𝑘𝑒^−𝜀𝑡𝜙(𝑡)²_𝜏. (47)
Hence, for any 𝑡>0𝐸𝑥₁²≤𝛼𝑒^−𝜀𝑡𝜙(𝑡)²_𝜏, (48) where 𝛼=(min_𝑖∈𝑆{𝜆_min(𝑃_𝑖1)})⁻¹𝑘. To study the exponential stability in mean square of 𝑥₂(𝑡), we apply the similar analysis method of [12] and define 𝑒(𝑡)=𝐴^∧_𝑖13𝑥₁(𝑡)+𝐴_𝑖𝑑3𝑥₁(𝑡−𝜏(𝑡)). (49)
Then, according to (48), we have a constant 𝑚>0; it can be found that when 𝑡>0, 𝐸‖𝑒(𝑡)‖²≤𝑚𝑒^−𝜀𝑡𝜙(𝑡)²_𝜏. (50)
We construct a function as 𝐽(𝑡)=𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡)−𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡−𝜏(𝑡)). (51)
By premultiplying the second equation of (44) with 𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4, we obtain that 0=𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4𝑥₂(𝑡)+𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4𝑥₂(𝑡−𝜏(𝑡))+𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4𝑒(𝑡). (52)
Adding (52) to (51) yields that 𝐽(𝑡)=𝑥₂(𝑡)^𝑇𝑃_𝑖4+𝑃_𝑖4+𝑅_4𝑖𝑙𝑥₂(𝑡)+2𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4𝐴_𝑖𝑑4𝑥₂((𝑡−𝑡−𝜏))𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑅_4𝑖𝑙𝑥₂+2𝑥₂(𝑡)^𝑇𝑃^𝑇_𝑖4𝑒(𝑡)≤𝑥₂(𝑡)𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑃_𝑖4+𝑃^𝑇_𝑖4+𝑅_4𝑖𝑙𝑃^𝑇_𝑖4𝐴_𝑖𝑑4∗−𝑅_4𝑖𝑙×𝑥₂(𝑡)𝑥₂(𝑡−𝜏(𝑡))+𝜂₁𝑥₂(𝑡)^𝑇𝑥₂(𝑡)+𝜂⁻¹𝑒(𝑡)^𝑇𝑅_4𝑖𝑙𝑅^𝑇_4𝑖𝑙𝑒(𝑡), (53) where𝜂₁ is any positive scalar.
By pre- and postmultiplying ^{Σ11Σ1𝑖+𝑙}_{∗Σ𝑘+𝑙𝑘+𝑙}<0 byΥ00Υ𝑇 andΥ00Υ, respectively, we get a constant𝜂₂>0, such that 𝑃_𝑖4+𝑃^𝑇_𝑖4+𝑅_4𝑖𝑙𝑃^𝑇_𝑖4𝐴_𝑖𝑑4∗−𝑅_𝑖𝑑4𝜂≤−²𝐼000. (54)
On the other hand, since𝜂₁ can be chosen arbitrarily,𝜂₁ is chosen small enough, such that 𝜂₂−𝜂₁>0. Then, we can always find a scalar 𝜂₃>1, such that 𝑅_4𝑖𝑙−𝜂₂−𝜂₁𝐼≥𝜂₃𝑅_4𝑖𝑙. (55)
It follows from (51), (53), and (54) that 𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡)−𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡−𝜏(𝑡))≤−𝜂₂𝑥₂(𝑡)^𝑇𝑥₂(𝑡)+𝜂₁𝑥₂(𝑡)^𝑇𝑥₂(𝑡)+𝜂⁻¹𝑒(𝑡)^𝑇𝑃_𝑖4𝑃^𝑇_𝑖4𝑒(𝑡). (56)
It is clear that the above inequalities (55) and (56) imply that 𝜂₃𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡)≤𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙−𝜂₁−𝜂₂𝐼≤𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡−𝜏(𝑡))+𝜂⁻¹𝑒(𝑡)^𝑇𝑃_𝑖4𝑃^𝑇_𝑖4𝑒(𝑡). (57)
Thus, we get from (57) that 𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡)≤𝜂₃⁻¹𝑥₂(𝑡−𝜏(𝑡))^𝑇𝑅_4𝑖𝑙𝑥₂((𝑡+𝜂𝑡−𝜏))₁𝜂₃⁻¹𝑒(𝑡)^𝑇𝑃_𝑖4𝑃^𝑇_𝑖4𝑒(𝑡). (58)
Let𝑓(𝑡)=𝑥₂(𝑡)^𝑇𝑅_4𝑖𝑙𝑥₂(𝑡); from the above inequality and the fact (50) and (58), we have 𝐸𝑓(𝑡)≤𝜂₃⁻¹𝐸sup_{𝑡−𝜏≤𝑠≤𝑡}𝑓(𝑠)+𝛾𝑒^−𝜀𝑡≤𝜂₃⁻¹𝐸sup_{𝑡−𝜏≤𝑠≤𝑡}𝑓(𝑠)+𝛾𝑒^−𝛿𝑡, (59) where0<𝛿<min{𝜀,𝑑₂⁻¹ln𝜂₃}, 𝛾=(𝜂₁𝜂₃)⁻¹max_𝑖∈𝑆‖𝑃_𝑖4‖²‖𝜙(𝑡)‖²_𝜏.
Therefore, applying Lemma 4 to the above inequality yields that 𝐸𝑥₂(𝑡)²≤𝜆⁻¹_min𝑅_4𝑖𝑙𝜆_max𝑅_4𝑖𝑙𝑒^−𝛿𝑡𝑥₂(𝑡)²_𝜏+𝜆⁻¹_min𝑅_4𝑖𝑙𝛾𝑒^−𝛿𝑡1−𝜂₃⁻¹𝑒^𝛿𝑑2. (60)
We can find from (50) and the above inequality that the system (16) is exponentially stable in mean square.

Remark 8. It should be pointed out that sometimes too many free variables cannot reduce the conservatism of the obtained results. In the theorem, the system is exponentially stable in mean square without resorting to the free-weighting matrices method, which is in contrast with [6], where the free-weighting matrices method was used.

Remark 9. In the proof of the theorem, the delay interval [0;𝜏] is divided into 𝑁 segments of equal length 𝜏/𝑁, such that the information of delayed states 𝑥(𝑡−𝑖𝜏/𝑁), 𝑖=1,2,…,𝑁, are all taken into account. It is clear that the Lyapunov function defined in the theorem is more general than the ones in [7].

4. Numerical Examples

In this section, we provide a numerical example to verify the effectiveness of the proposed method. Consider systems (1) in 𝑅² with two regines 𝑟_𝑡∈𝑆={1,2}. The system parameters are described as follows: 𝐸=1000,𝐴₁=,𝐴−0.800.6−2.0₂=−1.80.50.5−1.8,𝐵₁=,𝐵1.71.32.52.1₂=−0.5−0.30−0.2,𝐶₁=,𝐶0.2000.2₂=0.4000.4,𝐷_𝑞=,𝐷1001_−𝑞=(𝑠)=(𝑠)(𝑠)0000,𝜑3cos−sin−3cos²(𝑠),𝜋_𝑖𝑗2×2=−404010−10,𝐷⁻_𝑞=,𝜋0000_𝑖𝑗2×2=−404010−10,𝐻₁=,𝐻1001₂=1001,𝜏=1,𝜇=3. (61)

As 𝑙=1,𝑘=1, and (1)𝑟_𝑡=1, by calculating we have 𝑃₁=,𝑃13.81761.31471.314727.9707₂=,𝐾2.3514−0.2388−0.238827.9707₁=,𝑄−9.9593−6.2795−6.2795−12.1579₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅1.9317−0.2833−0.28331.9317₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=1.07240.26000.26001.0724,𝑁₁=,𝑁0.95220.26050.26050.9522₂=,𝑇−2.48980.28950.2895−2.4898₁=2.84381.54691.54698.7138,𝑇₂=;5.7161005.7161 (62)(2)𝑟_𝑡=2, by calculating we have 𝑃₁=0.95220.26050.26050.9522,𝑃₂=,𝐾9.04782.22192.221936.3742₂=,𝑄0.1904−0.1109−0.11090.3192₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅0.1793−0.7876−0.78760.1793₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=1.07240.26000.26001.0724,𝑁₁=,𝑁0.95220.26050.26050.9522₂=−2.48980.28950.2895−2.4898,𝑇₁=,𝑇2.83011.54991.54998.7230₂=.5.7161005.7161 (63)

As 𝑙=1,𝑘=2, and(1)𝑟_𝑡=1, by calculating we have 𝑃₁=0.95220.26050.26050.9522,𝑃₂=,𝐾8.48781.29351.293532.5716₁=,𝑄−9.9593−6.2795−6.2795−12.1579₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅1.9317−0.2833−0.28331.9317₁₁=,𝑄2.3514−0.2388−0.23882.3514₁₂₁=,𝑄0.2586−0.3256−0.32560.2586₂₂₁=,𝑀0.1904−0.0521−0.05210.1904₁₂=1.07240.26000.26001.0724,𝑅₁₂=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₂₁=,𝑀0.1332−0.0926−0.09260.1332₂₂=0.2025−0.2003−0.20030.2025,𝑁₁=,𝑇0.95220.26050.26050.9522₁=2.84381.54691.54698.7138,𝑇₂=;5.7161005.7161 (64)(2)𝑟_𝑡=2, by calculating we have 𝑃₁=80.58985.29545.2954229.2114,𝑃₂=,𝐾2.4898−0.2895−0.28952.4898₂=,𝑄0.1904−0.1109−0.11090.3192₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅1.9317−0.2833−0.28331.9317₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=,𝑀1.07240.26000.26001.0724₂₁=,𝑀0.1332−0.0926−0.09260.1332₂₂=0.2025−0.2003−0.20030.2025,𝑁₁=,𝑁0.95220.26050.26050.9522₂=−2.48980.28950.2895−2.4898,𝑇₁=,𝑇2.84381.54691.54698.7138₂=.5.7167005.7161 (65)

As 𝑙=2,𝑘=1, and (1)𝑟_𝑡=1, by calculating we have 𝑃₁=2.4898−0.2895−0.28952.1898,𝑃₂=,𝐾16.53212.82702.827018.5944₁=,𝑄−9.9593−6.2795−6.2795−12.1579₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅1.9317−0.2833−0.28331.9317₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=1.07240.26000.26001.0724,𝑁₁=,𝑁0.95220.26050.26050.9522₂=−2.48980.28950.2895−2.4898,𝑇₁=,𝑇2.84381.54691.54698.7138₂=.5.7161005.7161 (66)(2)𝑟_𝑡=2, by calculating we have 𝑃₁=0.95220.26050.26050.9522,𝑃₂=,𝐾7.42111.43221.432229.4614₂=,𝑄0.1904−0.1109−0.11090.3192₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑅0.1793−0.7876−0.78760.1793₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=1.07240.26000.26001.0724,𝑁₁=,𝑁0.95220.26050.26050.9522₂=−2.48980.28950.2895−2.4898,𝑇₁=,𝑇2.84381.54691.54698.7138₂=.5.7161005.7161 (67)

As 𝑙=2,𝑘=2, and (1)𝑟_𝑡=1, by calculating we have 𝑃₁=,𝑃497.535930.449830.4498728.9455₂=,𝑄2.4898−0.2895−0.28952.4898₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑄1.9317−0.2833−0.28331.9317₁₂₁=,𝑅0.2586−0.3256−0.32560.2586₁₁=,𝑄2.3514−0.2388−0.23882.3514₂₂₁=,𝑅0.1904−0.0521−0.05210.1904₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=,𝑀1.07240.26000.26001.0724₂₁=,𝑀0.1332−0.0926−0.09260.1332₂₂=,𝑁0.2025−0.2003−0.20030.2025₁=,𝑁0.95220.26050.26050.9522₂=,𝑇−2.48980.28950.2895−2.4898₁=2.84381.54691.54698.7138,𝑇₂=.5.7161005.7161 (68)(2)𝑟_𝑡=2, by calculating we have 𝑃₁=,𝑃0.95220.26050.26050.9522₂=,𝐾36.1907−7.0282−7.0282101.6875₂=,𝑄0.1904−0.1109−0.11090.3192₁₁₁=,𝑄2.0672−0.1326−0.13262.0672₁₁₂=,𝑀0.1793−0.7876−0.78760.1793₁₂=,𝑄1.07240.26000.26001.0724₁₂₁=,𝑄0.2586−0.3256−0.32560.2586₂₂₁=,𝑅0.1904−0.0521−0.05210.1904₁₁=,𝑅2.3514−0.2388−0.23882.3514₂₁=,𝑀0.8620−1.2919−1.29190.8620₁₁=,𝑀3.5860−0.2001−0.20013.5860₁₂=,𝑀1.07240.26000.26001.0724₂₁=,𝑁0.1332−0.0926−0.09260.1332₁=,𝑁0.95220.26050.26050.9522₂=,𝑇−2.48980.28950.2895−2.4898₁=2.84381.54691.54698.7138,𝑇₂=.5.7161005.7161 (69)

So we can conclude that the condition of the theorem is satisfied; the system (16) is exponentially stable in mean square.

5. Conclusions

In this paper, we have investigated the exponential stability in mean square of singular Markovian jump systems with singular Markovian jump systems with saturating actuators and time-varying delay. To guarantee the singular Markovian jump system to be regular, impulse free, and exponentially stable in mean square. Memoryless feedback control has been designed. By exploiting Lyapunov-Krasovskii theory, a new sufficient condition of exponentially stable in mean square is obtained. Finally, the numerical example shows that the approach proposed is effective. To reduce conservatism of the results, our future work focuses on reducing the constraint by an appropriate reduction method.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant no. 11172086 and by the Natural Science Foundation of Hefei University under Grant no. 13KY03ZD.