Delay-Dependent Exponential Stability for Uncertain Neutral Stochastic Systems with Mixed Delays and Markovian Jumping Parameters

Abstract

This paper is mainly concerned with the globally exponential stability in mean square of uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. The mixed delays are comprised of the discrete interval time-varying delays and the distributed time delays. Taking the stochastic perturbation and Markovian jumping parameters into account, some delay-dependent sufficient conditions for the globally exponential stability in mean square of such systems can be obtained by constructing an appropriate Lyapunov-Krasovskii functional, which are given in the form of linear matrix inequalities (LMIs). The derived criteria are dependent on the upper bound and the lower bound of the time-varying delay and the distributed delay and are therefore less conservative. Two numerical examples are given to illustrate the effectiveness and applicability of our obtained results.

1. Introduction

It is well known that many dynamical systems not only depend on the present and past states but also involve the derivative with delays as well as the functional of the past history. Neutral delay differential equations are often used to describe the following systems [1]:𝑑[𝑥(𝑡)−𝐷𝑥(𝑡−𝜏)]((𝑡)(.𝑑𝑡=𝑓𝑡,𝑥,𝑥𝑡−𝜏)) (1.1) Many authors have considered the dynamical analysis of the neutral delay differential equations (see [2–7] and references therein). For example, Chen et al. in [3] and Wu et al. in [4, 5] have given some LMI-based conditions ensuring the stability analysis and the stabilization of neutral delay systems. Taking the environmental disturbances into account, the neutral stochastic delay differential equations can be given as follows:𝑑[𝑥(𝑡)−𝐷𝑥(𝑡−𝜏)]=𝑓(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝑡+𝑔(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏))𝑑𝐵(𝑡). (1.2) Some fundamental theories of neutral stochastic delay differential equations are introduced in [1, 8]. Since they can be extensively applied into many branches for the control field, the problem about the exponential stability and the asymptotical stability of the neutral stochastic delay systems has attracted many authors’ attention over the past few years, and many less conservative results of delay-dependent conditions ensuring the stabilization analysis and 𝐻_∞ filtering design for such systems have been reported in many works, see, for example, [9–14] and references therein. The methods used include the Razumikhin-type theorems [10], the Lyapunov functional [13], the fixed point theorem [14], and the linear matrix inequality [9, 11, 12]. For example, Huang and Mao in [9] and Chen et al. in [11] have given the exponential stability criteria of neutral stochastic delay systems. Some LMI-based sufficient conditions for the mean-square exponential stability analysis of stochastic systems of neutral type have been obtained by introducing an auxiliary vector in [12]. In practice, the parameter uncertainty is considered as one of the main sources leading to undesirable behavior (e.g., instability) of dynamical systems, especially when implementing neural networks in applications. The stability analysis of the uncertain neutral stochastic system has received considerable research attention, see, for example, [15–18], and the problem of the 𝐻_∞ filter design of the uncertain neutral stochastic delay systems has been discussed in [19, 20].

On the other hand, Markovian jump systems introduced by [21] are the hybrid systems with two components in the state. The first one refers to the mode that is described by a continuous-time finite-state Markovian process, and the second one refers to the state that is represented by a system of differential equations. The jump systems have the advantage of modeling the dynamic systems subject to abrupt variation in their structures, such as component failures or repairs, sudden environmental disturbance, changing subsystem interconnections, and operating in different points of a nonlinear plant [22]. The stability analysis and 𝐻_∞ filter design of stochastic delay systems with Markovian jumping parameters and delay systems with Markovian jumping parameters have been widely studied, see, for example, [23–38]. For example, in [24], Liu et al. have discussed the exponential stability of delayed recurrent neural networks with Markovian jumping parameters; Liu et al. in [24] and Wang et al. in [25] have considered some sufficient conditions for the exponential stability of stochastic neural networks with mixed time delays and Markovian switching; Mao in [13] has also given some sufficient conditions for the exponential stability of stochastic delay interval systems with Markovian switching. More recently, He and Liu in [39] and Balasubramaniam et al. in [40] have presented some LMI-based sufficient conditions for the exponential stability of uncertain neutral systems with Markovian jumping parameters. Although Kolmanovskii et al. in [17], and Mao et al. in [18] have derived the exponential stability of the neutral stochastic delay systems with Markovian jumping parameters, some sufficient conditions obtained by using the estimate method are not easily checked. Thus, the problem of the stability analysis of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters has not been fully investigated and there is still much room left for further consideration, which constitutes the motivation for the present research.

In this paper, the global exponential stability of a class of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters is discussed. The delays include the discrete and distributed time delays, and the jumping parameters are generated from a finite state Markov chain. By constructing an appropriate Lyapunov functional, some LMIs-based sufficient conditions ensuring the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters are obtained by using the stochastic analysis and some bounding technique. It is worth pointing out that compared with the earlier works in [17, 18], the obtained results given in the form of the linear matrix inequalities (LMIs) can be easily be solved by using the standard software packages. Two illustrative examples are exploited to demonstrate the effectiveness and applicability of the obtained results.

The content of the paper is arranged as follows. In Section 2, some necessary notations, definitions, and lemmas will be introduced. In Section 3, we mainly study the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters. Two illustrative numerical examples are given to show the power of our obtained results in Section 4.

Notations. Unless otherwise specified, for a real square matrix 𝐴, the matrix 𝐴>0 (𝐴≥0, 𝐴<0, 𝐴≤0) means that 𝐴 is a positive definite (positive semidefinite, negative definite, and negative semidefinite, resp.); 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum and minimum eigenvalues of the square matrix 𝐴, respectively. Let (Ω,ℑ,{ℑ_𝑡}_𝑡≥0,𝑃) be a probability space with a natural {ℑ_𝑡}_𝑡≥0 and let 𝐸{⋅} stand for the mathematical expectation operator with respect to this probability measure. If 𝐴 is a vector or matrix, its transpose is denoted by 𝐴^𝑇. |𝐵|=trace(𝐵^𝑇𝐵) denotes the Euclidean norm of a vector 𝐵 and its induced norm of a matrix 𝐵. Unless explicitly stated, matrices are assumed to have real entries and compatible dimensions. Let 𝜏>0 and 𝐶([−𝜏,0];𝑅^𝑛) be the family of all continuous 𝑅^𝑛-valued functions 𝜙 on the interval [−𝜏,0] with the norm ‖𝜙‖=sup{|𝜙(𝜃)|∶−𝜏≤𝜃≤0}. Denote by 𝐿²_ℑ0([−𝜏,0];𝑅^𝑛) the family of all ℑ₀-measurable 𝐶([−𝜏,0];𝑅^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0} such that sup_{𝜃∈[−𝜏,0]}𝐸|𝜉(𝜃)|²<+∞. 𝐵(𝑡)=[𝐵₁(𝑡),𝐵₂(𝑡),…,𝐵_𝑛(𝑡)]^𝑇 (𝑡≥0) is an 𝑛-dimensional standard Brownian motion defined on the completed probability space (Ω,ℑ,𝑃).

2. Problem Formulation

Let 𝑟(𝑡)(𝑡≥0) be a right-continuous Markov chain on the probability space taking values in a finite state space 𝑆={1,2,…,𝑁} with generator Γ={𝛾_𝑖𝑗}_𝑁×𝑁 given by𝑃𝑟()(𝑡)=𝛾𝑡+Δ=𝑗∣𝑟=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝛾_𝑖𝑖Δ+𝑜(Δ),if𝑖=𝑗, (2.1) where Δ>0. Here, 𝛾_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 while 𝛾_𝑖𝑖=−_𝑗≠𝑖𝛾_𝑖𝑗. (2.2) Now, we assume that the Markov chain 𝑟(⋅) is independent of the Brownian motion 𝐵(⋅). It is well known that almost every sample path of 𝑟(𝑡) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of 𝑅⁺.

In this paper, we will consider the following uncertain neutral stochastic systems with mixed delays and Markovian switching:𝑑[𝑥(𝑡)−𝐷(𝑟(𝑡))𝑥(𝑡−𝑑(𝑡))]=𝐴₁(𝑟(𝑡),𝑡)𝑥(𝑡)+𝐴₂(𝑟(𝑡),𝑡)𝑓(𝑥(𝑡))+𝐴₃(𝑟(𝑡),𝑡)𝑓(𝑥(𝑡−𝑑(𝑡)))+𝐴₄(𝑟(𝑡),𝑡)^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠(𝑡)((𝑡))𝑑𝑠𝑑𝑡+𝜎𝑡,𝑥,𝑥𝑡−𝑑)),^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠,𝑟(𝑡)𝑑𝐵(𝑡),𝑡≥0, (2.3) with the initial value 𝑥₀=𝜑∈𝐿²_ℑ0([−𝜏,0],𝑅^𝑛) (𝜏=max{𝑑₂,𝜎}), where 𝑥(𝑡)∈𝑅^𝑛 is the system state vector associated with the neurons and 𝑑(𝑡) and 𝜎(𝑡) are the time-varying delays. Here, we assume that the Markov chain 𝑟(⋅) is independent of the Brownian motion 𝐵(𝑡)(𝑡≥0). 𝑓∶𝑅^𝑛→𝑅^𝑛 is neuron activation function, and the noise perturbation 𝜎∶𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑛×𝑆→𝑅^𝑛×𝑛 is the noise intensity matrix. When 𝑟(𝑡)=𝑖(𝑖∈𝑆), 𝐷(𝑟(𝑡)), 𝐴₁(𝑟(𝑡),𝑡), 𝐴₂(𝑟(𝑡),𝑡), 𝐴₃(𝑟(𝑡),𝑡), and 𝐴₄(𝑟(𝑡),𝑡) are, respectively, denoted as 𝐷_𝑖, 𝐴_1𝑖(𝑡), 𝐴_2𝑖(𝑡), 𝐴_3𝑖(𝑡), and 𝐴_4𝑖(𝑡), and 𝐷_𝑖 (𝑖∈𝑆) are known matrices with |𝐷_𝑖|<1(𝑖∈𝑆). 𝐴_1𝑖(𝑡), 𝐴_2𝑖(𝑡), 𝐴_3𝑖(𝑡), and 𝐴_4𝑖(𝑡) are matrix functions with time-varying uncertainties, that is,𝐴_1𝑖(𝑡)=𝐴_1𝑖+Δ𝐴_1𝑖(𝑡),𝐴_2𝑖(𝑡)=𝐴_2𝑖+Δ𝐴_2𝑖(𝑡),𝐴_3𝑖(𝑡)=𝐴_3𝑖+Δ𝐴_3𝑖(𝑡),𝐴_4𝑖(𝑡)=𝐴_4𝑖+Δ𝐴_4𝑖(𝑡), (2.4) where 𝐴_1𝑖, 𝐴_2𝑖, 𝐴_3𝑖, and 𝐴_4𝑖 (𝑖∈𝑆) are known real constant matrices and Δ𝐴_1𝑖(𝑡), Δ𝐴_2𝑖(𝑡), Δ𝐴_3𝑖(𝑡), and Δ𝐴_4𝑖(𝑡) (𝑖∈𝑆) are unknown matrices representing time-varying parameter uncertainties in system model. We assume that the uncertainties are norm-bounded and can be described asΔ𝐴_1𝑖(𝑡)Δ𝐴_2𝑖(𝑡)Δ𝐴_3𝑖(𝑡)Δ𝐴_4𝑖(𝑡)=𝑀_𝑖𝐹_𝑖(𝑡)𝑁_1𝑖𝑁_2𝑖𝑁_3𝑖𝑁_4𝑖,𝐹^𝑇_𝑖(𝑡)𝐹_𝑖(𝑡)≤𝐼,𝑖∈𝑆, (2.5) where 𝑀_𝑖,𝑁_1𝑖,𝑁_2𝑖,𝑁_3𝑖,and𝑁_4𝑖(𝑖∈𝑆) are known real matrices and 𝐹_𝑖(𝑡)(𝑖∈𝑆) is unknown real and possibly time-varying matrix for any given 𝑡. It is assumed that the elements of 𝐹_𝑖(𝑡)(𝑖∈𝑆) are Lebesgue measurable. When 𝐹_𝑖(𝑡)=0(𝑖∈𝑆), systems (2.3) have the following nominal case:𝑑𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))=𝐴_1𝑖𝑥(𝑡)+𝐴_2𝑖𝑓(𝑥(𝑡))+𝐴_3𝑖𝑓(𝑥(𝑡−𝑑(𝑡)))+𝐴_4𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠(𝑡)((𝑡,))𝑑𝑠𝑑𝑡+𝜎𝑡,𝑥,𝑥𝑡−𝑑))^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠,𝑖𝑑𝐵(𝑡),𝑡≥0. (2.6)

In order to obtain our results, we need some assumptions as follows.

Assumption 2.1. The neuron activation functions 𝑓(⋅) in (2.3) (or (2.6)) are bounded and satisfy the following Lipschitz condition: 𝑓(𝑥)−𝑓𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛, (2.7) where 𝐿∈𝑅^𝑛×𝑛 is known constant matrix and 𝑓(0)=0.

Assumption 2.2. The noise perturbation 𝜎 satisfies the following condition: trace𝜎^𝑇𝑡,𝑥(𝑡),𝑥(𝑡−𝑑(𝑡)),^𝑡_{𝑡−𝜎(𝑡)}𝑥(𝑠)𝑑𝑠,𝑖𝜎𝑡,𝑥(𝑡),𝑥(𝑡−𝑑(𝑡)),^𝑡_{𝑡−𝜎(𝑡)}𝑥(𝑠)𝑑𝑠,𝑖≤𝑥^𝑇(𝑡)𝑅^𝑇_1𝑖𝑅_1𝑖𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑(𝑡))𝑅^𝑇_2𝑖𝑅_2𝑖𝑥(𝑡−𝑑(𝑡))+^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝑅^𝑇_3𝑖𝑅_3𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑥(𝑠)𝑑𝑠, (2.8) where 𝑅_1𝑖, 𝑅_2𝑖, and 𝑅_3𝑖 are known constant matrices with appropriate dimensions and 𝜎(𝑡,0,0,0,𝑖)=0 (𝑖∈𝑆).

Remark 2.3. Under Assumptions 2.1 and 2.2, it is easily shown that the system (2.3) with uncertainties (2.4) admits a unique trivial solution when the initial data 𝜉=0. The readers can refer to [41].

Assumption 2.4. 𝑑(𝑡) and 𝜎(𝑡) are two time-varying continuous functions that satisfy 0≤𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇<1,0≤𝜎(𝑡)≤𝜎,𝜎̇(𝑡)≤𝜈<1, (2.9) where 𝑑₁ and 𝑑₂ are the lower and upper bounds of the time delay 𝑑(𝑡), respectively.
We present the definitions and three useful lemmas as follows.

Definition 2.5. The neutral stochastic systems with mixed delays and Markovian jump parameters (2.6) is said to be exponentially stable in mean square if there exist a pair of positive scalar 𝛼>0 and 𝑙>0 such that every solution 𝑥(𝑡,𝜉,𝑖) of systems (2.6) satisfies 𝐸𝑥(𝑡,𝜉,𝑖)²≤𝑙sup_{𝑠∈[−𝜏,0]}𝐸𝜑(𝑠)²𝑒^−𝛼𝑡,∀𝑡≥0, (2.10) for any 𝜑∈𝐿²_ℑ0([−𝜏,0],𝑅^𝑛).

Definition 2.6. The uncertain neutral stochastic systems with mixed delays and Markovian jump parameters (2.3) are said to be exponentially stable in mean square if (2.10) holds for all admissible uncertainties (2.5).

Lemma 2.7 (see [8]). For any vectors 𝑎,𝑏∈𝑅^𝑛, the inequality ±2𝑎^𝑇𝑏≤𝑎^𝑇𝑋𝑎+𝑏^T𝑋⁻¹𝑏 (2.11) holds, in which 𝑋 is any 𝑛×𝑛matrix with 𝑋>0.

Lemma 2.8 (see [8]). Let 𝑋∈𝑅^𝑛×𝑛; then 𝜆_min(𝑋)𝑎^𝑇𝑏≤𝑎^𝑇𝑋𝑏≤𝜆_max(𝑋)𝑎^𝑇𝑏 (2.12) for any 𝑎∈𝑅^𝑛 if 𝑋 is a symmetric matrix.

Lemma 2.9 (see [42] Schur complement). For a given matrix 𝑆𝑆=¹¹𝑆₁₂𝑆^𝑇₁₂𝑆₂₂ (2.13) with 𝑆₁₁=𝑆^𝑇₁₁, 𝑆₂₂=𝑆^𝑇₂₂, the following conditions are equivalent:(1)𝑆<0,(2)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0,(3)𝑆₁₁<0, 𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0.

Lemma 2.10 (see [42]). Let 𝑈, 𝑉, 𝑊, and 𝑀 be real matrices of appropriate dimensions with 𝑀 satisfying 𝑀=𝑀^𝑇; then 𝑀+𝑈𝑉𝑊+𝑊^𝑇𝑉^𝑇𝑈^𝑇<0,∀𝑉^𝑇𝑉≤𝐼, (2.14) if and only if there exist a scalar 𝜀>0 such that 𝑀+𝜀⁻¹𝑈𝑈^𝑇+𝜀𝑊^𝑇𝑊<0. (2.15)

Lemma 2.11 (see [43]). For any positive symmetric constant matrix 𝑀∈𝑅^𝑛×𝑛 and a scalar 𝛾>0, a vector function 𝜔∶[0,𝛾]→𝑅^𝑛 such that the integrations concerned are well defined, and then the following inequality holds: ^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝜔(𝑠)𝑑𝑠≤𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (2.16)

3. Main Results

Theorem 3.1. Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar 𝜅∈(0,1), the neutral stochastic systems with mixed delays and Markovian switching (2.6) are exponentially stable in mean square if there exist 𝜆_𝑖>0 (𝑖∈𝑆) and some positive definite matrices 𝑃_𝑖>0 (𝑖∈𝑆) and 𝑄_𝑙>0 (𝑙=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: for 𝑖∈𝑆, 𝑃_𝑖≤𝜆_𝑖Ω𝐼,_𝑖=Ω_𝑖11Ω_𝑖120𝑃_𝑖𝐴_2𝑖𝑃_𝑖𝐴_3𝑖𝑃_𝑖𝐴_4𝑖000∗Ω_𝑖220000Ω_𝑖27Ω_𝑖28Ω_𝑖29∗∗Ω_𝑖33000000∗∗∗−𝑄₅00000∗∗∗∗−𝑄₆0000∗∗∗∗∗−𝑄₇000∗∗∗∗∗∗−𝑄₈00∗∗∗∗∗∗∗−𝑄₉0∗∗∗∗∗∗∗∗−𝑄₁₀<0, (3.1)(3.2) where ∗ denotes the entries that are readily inferred bysymmetry of a symmetric matrix and Ω_𝑖11=𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑅^𝑇_1𝑖𝑅_1𝑖+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑄₁+d₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄,Ω_𝑖12=𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷_𝑖,Ω_𝑖22=𝐿^𝑇𝑄₆+𝑄₉𝐿+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖+𝜆_𝑖𝑅^𝑇_2𝑖𝑅_2𝑖−(1−𝑢)𝑄₁,Ω_𝑖33=𝐿^𝑇𝑄₇+𝑄₁₀𝐿+𝜆_𝑖𝑅^𝑇_3𝑖𝑅_3𝑖−𝜅(1−𝑣)𝑄₅,Ω_𝑖27=𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖,Ω_𝑖28=𝐷^𝑇_i𝑃_𝑖𝐴_3𝑖,Ω_𝑖29=𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖. (3.3)

Proof. Denote by 𝐶^2,1(𝑅⁺×𝑅^𝑛×𝑆;𝑅^𝑛) the family of all nonnegative functions 𝑉(𝑡,𝑥,𝑖) on 𝑅⁺×𝑅^𝑛×𝑆 that are once differentiable with respect to the first variable 𝑡 and twice differentiable with respect to the second variable 𝑥. To obtain the stability conditions, we consider the following Lyapunov functional: 𝑉(𝑡,𝑥(𝑡),𝑖)=𝑉₁(𝑡,𝑥(𝑡),𝑖)+𝑉₂(𝑡,𝑥(𝑡),𝑖)+𝑉₃(𝑡,𝑥(𝑡),𝑖), (3.4) where 𝑉₁(𝑡,𝑥(𝑡),𝑖)=[𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))]^𝑇𝑃_𝑖𝑥(𝑡)−𝐷_𝑖𝑥((𝑡,𝑉𝑡−𝑑))₂(𝑡,𝑥(𝑡),𝑖)=^𝑡_{𝑡−𝑑(𝑡)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝑑(𝑡)}^𝑡_𝑠𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)𝑉𝑑𝜃𝑑𝑠,₃(𝑡,𝑥(𝑡),𝑖)=^−𝑑1_−𝑑2^𝑡_𝑡+𝑠^𝑡_𝜃𝑥^𝑇(𝑢)𝑄₃𝑥(𝑢)𝑑𝑢𝑑𝜃𝑑𝑠+𝜎⁰_{−𝜎(𝑡)}^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₄𝑥(𝜃)𝑑𝜃𝑑𝑠. (3.5)
The weak infinitesimal operator 𝔏𝑉 [17] along (2.6) from 𝑅⁺×𝑅^𝑛×𝑆 to 𝑅 is given by𝔏𝑉(𝑡,𝑥(𝑡),𝑖)=𝔏𝑉₁(𝑡,𝑥(𝑡),𝑖)+𝔏𝑉₂(𝑡,𝑥(𝑡),𝑖)+𝔏𝑉₃(𝑡,𝑥(𝑡),𝑖), (3.6) where 𝔏𝑉₁(𝑡,𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_2𝑖𝑓(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_3𝑖𝑓(𝑥(𝑡−𝑑(𝑡)))+2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_4𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠+2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_1𝑖𝑥(𝑡)−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑓(𝑥(𝑡))−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑓(𝑥(𝑡−𝑑(𝑡)))−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠+𝑥^𝑇(𝑡)−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑥^𝑇(𝑡−𝑑(𝑡))−^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑(𝑡))^𝑁_𝑗=1𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖𝑥((𝑡𝜎𝑡−𝑑))+trace^𝑇𝑡,𝑥(𝑡),𝑥(𝑡−𝑑(𝑡)),^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠𝑉_1𝑥𝑥𝜎𝑡,𝑥(𝑡),𝑥(𝑡−𝑑(𝑡)),^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠. (3.7) By Lemma 2.7, we have 𝑥^𝑇(𝑡)𝑃_𝑖𝐴_2𝑖𝑓(𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_2𝑖𝑄₅⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡))𝑄₅𝑓(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_2𝑖𝑄₅⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐿^𝑇𝑄₅𝐿𝑥(𝑡),2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_3𝑖𝑓(𝑥(𝑡−𝑑(𝑡)),𝑖)≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_3𝑖𝑄₆⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)),𝑖)𝑄₆𝑓(𝑥(𝑡−𝑑(𝑡)),𝑖)≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_3𝑖𝑄₆⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇𝑄₆𝐿𝑥(𝑡−𝑑(𝑡)),−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑓(𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑄₈⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑓^𝑇(𝑥(𝑡))𝑄₈𝑓(𝑥(𝑡))≤𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑄₈⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑥^𝑇(𝑡)𝐿^𝑇𝑄₈𝐿𝑥(𝑡),−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑓(𝑥(𝑡−𝑑(𝑡)),𝑖)≤𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑄₉⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑓^𝑇(𝑥(𝑡−𝑑(𝑡)))𝑄₉𝑓(𝑥(𝑡−𝑑(𝑡)))≤𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑄₉⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑥^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇𝑄₉𝐿𝑥(𝑡−𝑑(𝑡)). (3.8)(3.9)(3.10)(3.11) From Lemmas 2.7 and 2.11, it implies that 2𝑥^𝑇(𝑡)𝑃_𝑖𝐴_4𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_4𝑖𝑄₇⁻¹𝐴^𝑇_4𝑖𝑃_𝑖𝑥(𝑡)+^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑄₇^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_4𝑖𝑄₇⁻¹𝐴^𝑇_4𝑖𝑃_𝑖𝑥(𝑡)+^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝐿^𝑇𝑄₇𝐿^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠,−2𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖𝑄⁻¹₁₀𝐴^𝑇_4𝑖𝑃_𝑖𝐷_𝑖𝑥((𝑡+𝑡−𝑑))_{𝑡(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠))𝑑𝑠^𝑇𝑄₁₀^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠≤𝑥^𝑇(𝑡−𝑑(𝑡))𝑃_𝑖𝐴_4𝑖𝑄⁻¹₁₀𝐴^𝑇_4𝑖𝑃_𝑖𝑥((𝑡+𝑡−𝑑))_{𝑡(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝐿^𝑇𝑄₁₀𝐿^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠. (3.12)(3.13) Substituting (3.8)–(3.13) into (3.7) and from Assumption 2.2, it follows that 𝔏𝑉₁(𝑡,𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑅^𝑇_1𝑖𝑅_1𝑖+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑃_𝑖𝐴_2𝑖𝑄₅⁻¹𝐴^𝑇_2𝑖𝑃_𝑖+𝑃_𝑖𝐴_3𝑖(𝑡)𝑄₆⁻¹𝐴^−𝑇_3𝑖𝑃_𝑖+𝑃_𝑖𝐴_4𝑖𝑄₇⁻¹𝐴^𝑇_4𝑖𝑃_𝑖𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑥^𝑇((𝑡×𝐷𝑡−𝑑))_𝑇𝑖𝑃_𝑖𝐴_1𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝑥(𝑡)((𝑡×𝐷+𝑥𝑡−𝑑))_𝑇𝑖𝑃_𝑖𝐴_2𝑖𝑄₈⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑄₉⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖𝑄⁻¹₁₀𝐴^𝑇_4𝑖𝑃_𝑖𝐷_𝑖+𝐿^𝑇𝑄₆+𝑄₉𝐿+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖+𝜆_𝑖𝑅^𝑇_2𝑖𝑅_2𝑖𝑥((𝑡+𝑡−𝑑))_{𝑡(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝐿^𝑇𝑄₇+𝑄₁₀𝐿+𝜆_𝑖𝑅^𝑇_3𝑖𝑅_3𝑖^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠. (3.14) On the other hand, we can obtain 𝔏𝑉₂(𝑡,𝑥(𝑡),𝑖)≤𝑥^𝑇𝑄₁+𝑑₂𝑄₂𝑥(𝑡)−1−𝜇𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₁𝑥((𝑡−𝑡−𝑑))1−𝜇^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠,𝔏𝑉₃(𝑡,𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡1)2𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄𝑥(𝑡)−^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₃𝑥(𝜃)()𝑑𝜃𝑑𝑠−𝜎1−𝜈^𝑡_{𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)𝑑𝑠≤𝑥^𝑇(𝑡1)2𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄𝑥(𝑡)−^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₃𝑥(𝜃)()𝑑𝜃𝑑𝑠−𝜎1−𝜅)(1−𝜈^𝑡_{𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)(𝑑𝑠−𝜅1−𝜈)^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝑄₅^t_{𝑡−𝜎(𝑡)}𝑥(𝑠)𝑑𝑠. (3.15)(3.16) Substituting (3.14)–(3.16) into (3.6), we have 𝔏𝑉(𝑡,𝑥(𝑡),𝑖)≤𝑥^𝑇(𝑡)𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑅^𝑇_1𝑖𝑅_1𝑖+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑃_𝑖𝐴_2𝑖𝑄₅⁻¹𝐴^𝑇_2𝑖𝑃_𝑖+𝑃_𝑖𝐴_3𝑖(𝑡)𝑄₆⁻¹𝐴^𝑇_3𝑖𝑃_𝑖+𝑃_𝑖𝐴_4𝑖𝑄₇⁻¹𝐴^𝑇_4𝑖𝑃_𝑖+𝑄₁+𝑑₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝑥(𝑡−𝑑(𝑡))𝐷^𝑇_𝑖𝑃_𝑖𝐴^𝑇_1𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^T_𝑖𝑃_𝑗𝑥(𝑡)+𝑥^𝑇(𝑡−𝑑(𝑡))𝐿^𝑇𝑄₆+𝑄₉𝐿+𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑄₈⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑄₉⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖𝑄⁻¹₁₀𝐴^𝑇_4𝑖𝑃_𝑖𝐷_𝑖+𝜆_𝑖𝑅^𝑇_2𝑖𝑅_2𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖−1−𝜇𝑄₁𝑥((𝑡+𝑡−𝑑))_{𝑡(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠^𝑇𝐿^𝑇𝑄₇+𝑄₁₀𝐿+𝜆_𝑖𝑅^𝑇_3𝑖𝑅_3𝑖−𝜅(1−𝜈)𝑄₄^𝑡_{𝑡−𝜎(𝑡)}𝑥(𝑠)−𝑑𝑠1−𝜇^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠−^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₃𝑥(𝜃)()𝑑𝜃𝑑𝑠−𝜎1−𝜅)(1−𝜈^𝑡_{𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝑄₅𝑥(𝑠)𝑑𝑠≤𝜉^𝑇(𝑡)Π_𝑖𝜉(𝑡)−1−𝜇^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠−^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₃𝑥(𝜃)()𝑑𝜃𝑑𝑠−𝜎1−𝜅)(1−𝜈^𝑡_{𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠, (3.17) where 𝜉(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))∫^𝑡_{𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝑑𝑠]^𝑇 and, for 𝑖∈𝑆, Π_𝑖=Π₁₁Π₁₂0∗Π₂₂0∗∗Π₃₃,Π₁₁=𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑅^𝑇_1𝑖𝑅_1𝑖+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑃_𝑖𝐴_2𝑖𝑄₅⁻¹𝐴^𝑇_2𝑖𝑃_𝑖+𝑃_𝑖𝐴_3𝑖(𝑡)𝑄₆⁻¹𝐴^𝑇_3𝑖𝑃_𝑖+𝑃_𝑖𝐴_4𝑖𝑄₇⁻¹𝐴^𝑇_4𝑖𝑃_𝑖+𝑄₁+𝑑₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄,Π₁₂=𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝑃_𝑗𝐷_𝑖,Π₂₂=𝐿^𝑇𝑄₆+𝑄₉𝐿+𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖𝑄₈⁻¹𝐴^𝑇_2𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖𝑄₉⁻¹𝐴^𝑇_3𝑖𝑃_𝑖𝐷_𝑖+𝐷^𝑇_𝑖𝑃_i𝐴_4𝑖𝑄⁻¹₁₀𝐴^𝑇_4𝑖𝑃_𝑖𝐷_𝑖+𝜆_𝑖𝑅^𝑇_2𝑖𝑅_2𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖−1−𝜇𝑄₁,Π₃₃=𝐿^𝑇𝑄₇+𝑄₁₀𝐿+𝜆_𝑖𝑅^𝑇_3𝑖𝑅_3𝑖−𝜅(1−𝜈)𝑄₄. (3.18) In view of (3.2), we have Π_𝑖<0, for 𝑖∈𝑆. By the Lyapunov functional 𝑉(𝑡,𝑥(𝑡),𝑖), 𝑉(𝑡,𝑥(𝑡),𝑖)=𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))^𝑇𝑃_𝑖𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_{𝑡(𝑡)𝑡−𝑑𝑡𝑠}𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)𝑑𝜃𝑑𝑠+^−𝑑1_−𝑑2^𝑡_𝑡+𝑠^𝑡_𝜃𝑥^𝑇(𝑢)𝑄₃𝑥(𝑢)𝑑𝑢𝑑𝜃𝑑𝑠+𝜎⁰_{−𝜎(𝑡)}^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑄₄𝑥(𝜃)𝑑𝜃𝑑𝑠≤𝜉^𝑇(𝑡)Π^′_𝑖𝜉(𝑡)+^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₁+𝑑₂𝑄₂𝑥(𝑠)𝑑𝑠+𝜎^2𝑡_{(𝑡)𝑡−𝜎}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠+^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝑑₂𝑄₃𝑥(𝜃)𝑑𝜃𝑑𝑠, (3.19) where Π^′_𝑖=𝑃_𝑖−𝑃_𝑖𝐷_𝑖0−𝐷^𝑇_𝑖𝑃_𝑖𝐷^𝑇_𝑖𝑃_𝑖𝐷_𝑖0000. (3.20)
Letting 𝛼>0, for system (2.6), we can define another operator 𝔏[𝑒^𝛼𝑡𝑉(𝑡,𝑥(𝑡),𝑖)]∶𝑅⁺×𝑅^𝑛×𝑆→𝑅 as follows:𝔏𝑒^𝛼𝑡𝑉(𝑡,𝑥(𝑡),𝑖)=𝑒^𝛼𝑡[𝛼𝑉(𝑡,𝑥(𝑡),𝑖)+𝔏𝑉(𝑡,𝑥(𝑡),𝑖)]≤𝑒^𝛼𝑡𝜉^𝑇(𝑡)𝛼Π^′_i+Π_𝑖𝜉(𝑡)+^𝑡_{(𝑡)𝑡−𝑑}𝑥^𝑇(𝑠)𝛼𝑄₁+𝑑₂𝑄₂−1−𝜇𝑄₂𝑥(𝑠)𝑑𝑠+^−𝑑1_−𝑑2^𝑡_𝑡+𝑠𝑥^𝑇(𝜃)𝛼𝑑₂−1𝑄₃𝑥(𝜃)+𝑑𝜃𝑑𝑠_{𝑡𝑡−𝜎(𝑡)}𝑥^𝑇(𝑠)𝛼𝜎²−𝜎(1−𝜅)(1−𝜈)𝑄₄𝑑𝑠. (3.21)
Now, we can choose 𝛼>0 sufficiently small such that𝛼𝜆_maxΠ^′_𝑖+𝜆_maxΠ_𝑖𝛼𝑄<0,₁+𝑑₂𝑄₂−1−𝜇𝑄₂<0,𝛼𝑑₂−1<0,𝛼𝜎²−𝜎(1−𝜅)(1−𝜈)<0, (3.22) for 𝑖∈𝑆.
By the weak infinitesimal operator along (2.6), it is obtained from (3.21) and (3.22) that𝑒^𝛼𝑡𝐸𝑉(𝑡,𝑥(𝑡),𝑖)≤𝐸𝑉(0,𝑥(0),𝑖)+^𝑡₀𝐸𝔏𝑒^𝛼𝑠𝑉((𝑠)((0)).𝑠,𝑥,𝑖)𝑑𝑠≤𝐸𝑉0,𝑥,𝑖 (3.23)
Using the definition of the Lyapunov functional (3.4) again, we have𝐸𝑉(0,𝑥(0),𝑖)=𝐸𝑥(0)−𝐷_𝑖𝑥(−𝑑(0))^𝑇𝑃_𝑖𝑥(0)−𝐷_𝑖𝑥(−𝑑(0))+⁰_−𝑑(0)𝐸𝑥^𝑇(s)𝑄₁𝑥(𝑠)𝑑𝑠+⁰_−𝑑(0)⁰_𝑠𝐸𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)𝑑𝜃𝑑𝑠+^−𝑑1_−𝑑2⁰_𝑠⁰_𝜃𝐸𝑥^𝑇(𝑢)𝑄₃𝑥(𝑢)𝑑𝑢𝑑𝜃𝑑𝑠+𝜎⁰_−𝜎(0)⁰_𝑠𝑥^𝑇(𝜃)𝑄₄𝑥(𝜃)≤𝑑𝜃𝑑𝑠2𝜆_maxΠ_𝑖+𝑑₂𝜆_max𝑄₁+𝜏²𝜆_max𝑄₂+𝑑₂−𝑑₁𝜏²𝜆_max𝑄₃+𝜎³𝜆_max𝑄₄×sup_{[]𝜃∈−𝜏,0}𝐸𝜑(𝜃)²≜𝑀_𝑖, (3.24) for 𝑖∈𝑆.
Thus, from (3.23) and (3.24), it follows that𝐸|𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))|²≤1Θ𝑀𝑒^−𝛼𝑡, (3.25) where Θ=min_𝑖∈𝑆{𝜆_min(𝑃_𝑖)} and 𝑀=max_𝑖∈𝑆{𝑀_𝑖}.
From |𝐷_𝑖|<1 (𝑖∈𝑆), we obtain that there exist a positive scalar 𝑙>0 such that 𝑙=max_𝑖∈𝑆|𝐷_𝑖|<1. So, for all 𝜀∈(0,min{𝛼,−(2/𝜏)log|𝑙|}) and any 𝜃>0, by using the elementary inequality, we derive𝑒^𝜀𝑡𝐸|𝑥(𝑡)|²=𝑒^𝜀𝑡𝐸𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))+𝐷_𝑖𝑥(𝑡−𝑑(𝑡))²≤(1+𝜃)𝑒^𝜀𝑡𝐸𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))²+11+𝜃𝑒^𝜀𝑡𝐸𝐷_𝑖𝑥(𝑡−𝑑(𝑡))²≤1+𝜃Θ𝑀𝑒^{−(𝛼−𝜀)𝑡}+11+𝜃𝑒^𝜀𝑡𝐸𝐷_𝑖𝑥(𝑡−𝑑(𝑡))²≤1+𝜃Θ𝑀+𝑙²11+𝜃𝑒^𝜀𝜏𝑒^{𝜀(𝑡−𝜏(𝑡))}𝐸|𝑥(𝑡−𝑑(𝑡))|². (3.26) From for all 𝜀∈(0,min{𝛼,−(2/𝜏)log|𝑙|}), we have 𝑙²𝑒^𝜀𝜏<1. Thus, we can choose 𝜃 sufficiently large such that Δ≜𝑙²11+𝜃𝑒^𝜀𝜏<1. (3.27)
So,𝑒^𝜀𝑡𝐸|𝑥(𝑡)|²≤1+𝜃Θ𝑀+Δ𝑒^{𝜀(𝑡−𝑑(𝑡))}𝐸|𝑥(𝑡−𝑑(𝑡))|². (3.28)
For all 𝑇>0, from (3.28), it follows thatsup_{0≤𝑡≤𝑇}𝑒^𝜀𝑡𝐸|𝑥(𝑡)|²≤1+𝜃Θ𝑀+Δ𝐸𝜑²+Δsup_{0≤𝑡≤𝑇}𝑒^𝜀𝑡𝐸|𝑥(𝑡)|², (3.29)
that is,sup_{0≤𝑡≤𝑇}𝑒^𝜀𝑡𝐸|𝑥(𝑡)|²≤((1+𝜃)/Θ)𝑀+Δ𝐸𝜑².1−Δ (3.30)
When 𝑇→+∞, it follows from (3.30) thatsup_0≤𝑡<∞𝑒^𝜀𝑡𝐸|𝑥(𝑡)|²≤((1+𝜃)/Θ)𝑀+Δ𝐸𝜑².1−Δ (3.31) So, we can obtain 𝐸|𝑥(𝑡)|²≤((1+𝜃)/Θ)𝑀+Δ𝐸𝜑²𝑒1−Δ^−𝜀𝑡. (3.32) The proof of this theorem is completed.

Theorem 3.2. Suppose that Assumptions 2.1–2.4 hold and for any given positive scalar 𝜅∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are robustly exponentially stable in mean square if there exist 𝜆_𝑖>0 (𝑖∈𝑆), 𝜀_1𝑖>0, 𝜀_2𝑖>0 (𝑖∈𝑆), and some positive definite matrices 𝑃_𝑖>0 (𝑖∈𝑆) and 𝑄_𝑙>0 (𝑙=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: 𝑃_𝑖<𝜆_𝑖Ω𝐼,𝑖∈𝑆,_𝑖=Ω₁₁Ω₁₂0Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈Ω₁₉Ω₁₁₀0∗Ω₂₂0000Ω₂₇Ω₂₈Ω₂₉0Ω₂₁₁∗∗Ω₃₃00000000∗∗∗Ω₄₄Ω₄₅Ω₄₆00000∗∗∗∗Ω₅₅Ω₅₆00000∗∗∗∗∗Ω₆₆00000∗∗∗∗∗∗Ω₇₇Ω₇₈Ω₇₉00∗∗∗∗∗∗∗Ω₈₈Ω₈₉00∗∗∗∗∗∗∗∗Ω₉₉00∗∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼<0,𝑖∈𝑆, (3.33) where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and Ω₁₁=𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑅^𝑇_1𝑖𝑅_1𝑖+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑄₁+𝑑₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄+𝜀_1𝑖+𝜀_2𝑖𝑁^𝑇_1𝑖𝑁_1𝑖,Ω₁₂=𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷_𝑖,Ω₁₄=𝑃_𝑖𝐴_2𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_2𝑖,Ω₁₅=𝑃_𝑖𝐴_3𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_3𝑖,Ω₁₁₀=𝑃_𝑖𝑀_𝑖,Ω₁₆=𝑃_𝑖𝐴_4𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_4𝑖,Ω₁₇=𝜀_2𝑖𝑁^𝑇_1𝑖𝑁_2𝑖,Ω₁₈=𝜀_2𝑖𝑁^𝑇_1𝑖𝑁_3𝑖,Ω₁₉=𝜀_2𝑖𝑁^𝑇_1𝑖𝑁_4𝑖,Ω₂₂=𝐿^𝑇𝑄₆+𝑄₉𝐿+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖+𝜆_𝑖𝑅^𝑇_2𝑖𝑅_2𝑖−1−𝜇𝑄₁,Ω₂₇=𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖,Ω₂₈=𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖,Ω₂₉=𝐷^𝑇_𝑖𝑃_𝑖𝐴_4𝑖,Ω₂₁₁=𝐷^𝑇_𝑖𝑃_𝑖𝑀_𝑖,Ω₃₃=𝐿^𝑇𝑄₇+𝑄₁₀𝐿+𝜆_𝑖𝑅^𝑇_3𝑖𝑅_3𝑖−𝜅(1−𝜐)𝑄₄,Ω₄₄=−𝑄₅+𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_2𝑖,Ω₄₅=𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_3𝑖,Ω₄₆=𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_4𝑖,Ω₅₅=−𝑄₆+𝜀_1𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Ω₅₆=𝜀_1𝑖𝑁^𝑇_3𝑖𝑁_4𝑖,Ω₆₆=−𝑄₇+𝜀_1𝑖𝑁^𝑇_4𝑖𝑁_4𝑖,Ω₇₇=−𝑄₈+𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_2𝑖,Ω₇₈=𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_3𝑖,Ω₇₉=𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_4𝑖,Ω₈₈=−𝑄₉+𝜀_2𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Ω₈₉=𝜀_2𝑖𝑁^𝑇_3𝑖𝑁_4𝑖,Ω₉₉=−𝑄₁₀+𝜀_2𝑖𝑁^𝑇_4𝑖𝑁_4𝑖. (3.34)

Proof. Replacing 𝐴_1𝑖, 𝐴_2𝑖, 𝐴_3𝑖, and 𝐴_4𝑖 in (3.2) with 𝐴_1𝑖+Δ𝐴_1𝑖(𝑡), 𝐴_2𝑖+Δ𝐴_2𝑖(𝑡), 𝐴_3𝑖+Δ𝐴_3𝑖(𝑡), and 𝐴_4𝑖+Δ𝐴_4𝑖(𝑡), Δ𝐴_1𝑖(𝑡), Δ𝐴_2𝑖(𝑡), Δ𝐴_3𝑖(𝑡), Δ𝐴_4𝑖(𝑡) are described in (2.4) and (2.5), in view of Lemma 2.9 and Lemma 2.10, we obtain Π_𝑖(𝑡)=Π_𝑖+Ξ^𝑇_𝑖𝐹_𝑖(𝑡)𝐸_1𝑖+𝐸^𝑇_1𝑖𝐹^𝑇_𝑖(𝑡)Ξ_𝑖+Ψ^𝑇_𝑖𝐹_𝑖(𝑡)𝐸_2𝑖+𝐸^𝑇_2𝑖𝐹^𝑇_𝑖(𝑡)Ψ_𝑖=Π_𝑖+𝜀⁻¹_1𝑖Ξ_𝑖Ξ^𝑇_𝑖+𝜀_1𝑖𝐸^𝑇_1𝑖𝐸_1𝑖+𝜀⁻¹_2𝑖Ψ_𝑖Ψ^𝑇_𝑖+𝜀_2𝑖𝐸^𝑇_2𝑖𝐸_2𝑖=Ω_𝑖<0, (3.35) where Ξ_𝑖=[𝑀^𝑇_𝑖𝑃_𝑖00000000]^𝑇,Ψ_𝑖=[0𝑀^𝑇_𝑖𝑃_𝑖𝐷_𝑖0000000]^𝑇,𝐸_1𝑖=[𝑁^𝑇_1𝑖00𝑁^𝑇_2𝑖𝑁^𝑇_3𝑖𝑁^𝑇_4𝑖000]^𝑇,𝐸_2𝑖=[𝑁^𝑇_1𝑖00000𝑁^𝑇_2𝑖𝑁^𝑇_3𝑖𝑁^𝑇_4𝑖]^𝑇, (3.36) where 𝜀_1𝑖>0 and 𝜀_2𝑖>0, for any 𝑖∈𝑆. The proof of the remainder can be easily finished by following a similar line as in the proof of Theorem 3.1. The proof is completed.

Remark 3.3. The delay-dependent sufficient conditions ensuring the robustly exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian switching (2.3) are provided in Theorem 3.2. It should be pointed out that such conditions are given in the form of LMIs, which could be easily solved by using the standard software packages. Besides, the criteria derived are dependent upon both the upper and lower bound of the time-varying delay and the distributed delay, which are less conservative.

Remark 3.4. Besides, by the Borel-Cantelli Lemma, we can also obtain the almost surely exponential stability of systems (2.3). Here, for the sake of brevity, we omit it and the readers can refer to [17]. Besides, we can easily come to a conclusion that the uncertain neutral stochastic delay systems with mixed delays and Markovian jumping parameters (2.3) are asymptotically stable in mean square from the conditions |𝐷_𝑖|<1 (𝑖∈𝑆). Thus, the results given in [15] are generalized.

Case 1. Consider the problem of delay-dependent robust exponential stability for a special case of the uncertain neutral stochastic systems with mixed delays and Markovian switching in (2.6), that is, 𝑑𝑥(𝑡)−𝐷_𝑖𝑥((𝑡=𝐴𝑡−𝑑))_1𝑖(𝑡)𝑥(𝑡)+𝐴_2𝑖(𝑡)𝑓(𝑥(𝑡))+𝐴_3𝑖(𝑡)𝑓(𝑥(𝑡−𝑑(𝑡)))+𝐴_4𝑖(𝑡)^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠+𝐴))𝑑𝑠𝑑𝑡_5𝑖(𝑡)𝑥(𝑡)+𝐴_6𝑖(𝑡)𝑥(𝑡−𝑑(𝑡))+𝐴_7𝑖(𝑡)^𝑡_{(𝑡)𝑡−𝜎}𝑥(𝑠)𝑑𝑠𝑑𝐵(𝑡),𝑡≥0, (3.37) where 𝐴_1𝑖(𝑡), 𝐴_2𝑖(𝑡), 𝐴_3𝑖(𝑡), and 𝐴_4𝑖(𝑡) are given in (2.3), while 𝐴_5𝑖(𝑡), 𝐴_6𝑖(𝑡), and 𝐴_7𝑖(𝑡) are of the following form: 𝐴_5𝑖(𝑡)𝐴_6𝑖(𝑡)𝐴_7𝑖(𝑡)=𝐴_5𝑖𝐴_6𝑖𝐴_7𝑖+𝑀_𝑖𝐹_𝑖(𝑡)𝑁_5𝑖𝑁_6𝑖𝑁_7𝑖, (3.38) where 𝐴_5𝑖, 𝐴_6𝑖, 𝐴_7𝑖, 𝑀_𝑖, 𝑁_5𝑖, 𝑁_6𝑖, and 𝑁_7𝑖 (𝑖∈𝑆) are known constant matrices. For systems (3.37), we can also obtain the robust exponential stability. The proof can be easily established by following a similar line as in the proof of Theorem 3.1 and then is omitted here.

Theorem 3.5. Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar 𝜅∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.37) are robustly exponentially stable in mean square if there exist 𝜆_𝑖>0, 𝜀_1𝑖>0, 𝜀_2𝑖>0, 𝜀_3𝑖>0 (𝑖∈𝑆) and some positive definite matrices 𝑃_𝑖>0 (𝑖∈𝑆) and 𝑄_𝑖>0 (𝑖=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: 𝑃_𝑖≤𝜆_𝑖Γ𝐼,𝑖∈𝑆,₁₁Γ₁₂Γ₁₃Γ₁₄Γ₁₅Γ₁₆000𝐴^𝑇_5𝑖𝑃_𝑖𝑃_𝑖𝑀_𝑖00∗Γ₂₂Γ₂₃000Γ₂₇Γ₂₈Γ₂₉𝐴^𝑇_6𝑖𝑃_𝑖0𝐷^𝑇_𝑖𝑃_𝑖𝑀_𝑖0∗∗Γ₃₃000000𝐴^T_7𝑖𝑃_𝑖000∗∗∗Γ₄₄Γ₄₅Γ₄₆0000000∗∗∗∗Γ₅₅Γ₅₆0000000∗∗∗∗∗Γ₆₆0000000∗∗∗∗∗∗Γ₇₇Γ₇₈Γ₇₉0000∗∗∗∗∗∗∗Γ₈₈Γ₈₉0000∗∗∗∗∗∗∗∗Γ₉₉0000∗∗∗∗∗∗∗∗∗−𝑃_𝑖00𝑃_𝑖𝑀_𝑖∗∗∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼00∗∗∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼0∗∗∗∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼<0, (3.39) where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and Γ₁₁=𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑄₁+𝑑₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₅+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_1𝑖+𝜀_3𝑖𝑁^𝑇_5𝑖𝑁_5𝑖,Γ₁₂=𝐴^𝑇_1𝑖𝑃_𝑖𝐷_𝑖−^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗𝐷^𝑇_𝑖+𝜀_3𝑖𝑁^𝑇_5𝑖𝑁_6𝑖,Γ₁₃=𝜀_3𝑖𝑁^𝑇_5𝑖𝑁_7𝑖,Γ₁₄=𝑃_𝑖𝐴^𝑇_2𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_2𝑖,Γ₁₅=𝑃_𝑖𝐴_3𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_3𝑖,Γ₁₆=𝑃_𝑖𝐴_4𝑖+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_4𝑖,Γ₂₂=𝐿^𝑇𝑄₆+𝑄₉𝐿+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖−1−𝜇𝑄₁+𝜀_3𝑖𝑁^𝑇_6𝑖𝑁_6𝑖,Γ₂₃=𝜀_3𝑖𝑁^𝑇_6𝑖𝑁_7𝑖,Γ₂₇=𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖,Γ₂₈=𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖,Γ₂₉=𝐷^𝑇_𝑖𝑃_𝑖𝐴_3𝑖,Γ₃₃=𝐿^𝑇𝑄₇+𝑄₁₀𝐿−𝜅(1−𝜈)𝑄₅+𝜀_3𝑖𝑁^𝑇_7i𝑁_7𝑖,Γ₄₄=−𝑄₅+𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_2𝑖,Γ₄₅=𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_3𝑖,Γ₄₆=𝜀_1𝑖𝑁^𝑇_2𝑖𝑁_4𝑖,Γ₅₅=−𝑄₆+𝜀_1𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Γ₅₆=𝜀_1𝑖𝑁^𝑇_3𝑖𝑁_4𝑖,Γ₆₆=−𝑄₇+𝜀_1𝑖𝑁^𝑇_4𝑖𝑁_4𝑖,Γ₇₇=−𝑄₈+𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_2𝑖,Γ₇₈=𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_3𝑖,Γ₇₉=𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_4𝑖,Γ₈₈=−𝑄₉+𝜀_2𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Γ₈₉=𝜀_2𝑖𝑁^𝑇_3𝑖𝑁_4𝑖,Γ₉₉=𝜀_2𝑖𝑁^𝑇_4𝑖𝑁_4𝑖. (3.40)

Case 2. When 𝜎=0, the uncertain neutral stochastic systems (2.3) are described as

𝑑𝑥(𝑡)−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))=𝐴_1𝑖(𝑡)𝑥(𝑡)+𝐴_2𝑖(𝑡)𝑓(𝑥(𝑡))+𝐴_3𝑖(𝑡)𝑓(𝑥(𝑡−𝑑(𝑡)))+𝐴_4𝑖(𝑡)^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠))𝑑𝑠𝑑𝑡,𝑡≥0. (3.41)

Theorem 3.6. Suppose that Assumptions 2.1 and 2.2 hold and for any given positive scalar 𝜅∈(0,1), the uncertain neutral stochastic systems with mixed delays and Markovian switching (3.41) are robustly exponentially stable in mean square if there exist 𝜆_𝑖>0 (𝑖∈𝑆), 𝜀_𝑖>0 (𝑖∈𝑆) and some positive definite matrices 𝑃_𝑖>0 (𝑖∈𝑆) and 𝑄_𝑖>0 (𝑖=1,2,…,10) such that the following linear matrix inequalities (LMIs) are satisfied: for 𝑖∈𝑆, Ω_𝑖=Ω₁₁Ω₁₂0Ω₁₄Ω₁₅Ω₁₆Ω₁₇Ω₁₈Ω₁₉Ω₁₁₀0∗Ω₂₂0000Ω₂₇Ω₂₈Ω₂₉0Ω₂₁₁∗∗Ω₃₃00000000∗∗∗Ω₄₄Ω₄₅Ω₄₆00000∗∗∗∗Ω₅₅Ω₅₆00000∗∗∗∗∗Ω₆₆00000∗∗∗∗∗∗Ω₇₇Ω₇₈Ω₇₉00∗∗∗∗∗∗∗Ω₈₈Ω₈₉00∗∗∗∗∗∗∗∗Ω₉₉00∗∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼<0,𝑖∈𝑆, (3.42) where ∗ denotes the entries that are readily inferred by symmetry of a symmetric matrix and Ω₁₁=𝑃_𝑖𝐴_1𝑖+𝐴^𝑇_1𝑖𝑃_𝑖+^𝑁_𝑗=1𝛾_𝑖𝑗𝑃_𝑗+𝐿^𝑇𝑄₅+𝑄₈𝐿+𝑄₁+𝑑₂𝑄₂+12𝑑²₂−𝑑²₁𝑄₃+𝜎²𝑄₄+𝜀_1𝑖+𝜀_2𝑖𝑁^𝑇_1𝑖𝑁_1𝑖,Ω₂₂=𝐿^𝑇𝑄₆+𝑄₉𝐿+^𝑁_𝑗=1𝛾_𝑖𝑗𝐷^𝑇_𝑖𝑃_𝑗𝐷_𝑖−1−𝜇𝑄₁,Ω₂₇=𝐷^𝑇_𝑖𝑃_𝑖𝐴_2𝑖,Ω₃₃=𝐿^𝑇𝑄₇+𝑄₁₀𝐿−𝜅(1−𝜐)𝑄₄, (3.43) and Ω₁₂, Ω₁₄, Ω₁₅, Ω₁₆, Ω₁₇, Ω₁₈, Ω₁₉, Ω₂₇, Ω₂₈, Ω₂₉, Ω₃₃, Ω₄₄, Ω₄₅, Ω₄₆, Ω₅₅, Ω₅₆, Ω₆₆, Ω₇₇, Ω₇₈, Ω₇₉, Ω₈₈, Ω₈₉, and Ω₉₉ are given in Theorem 3.2.

Remark 3.7. The sufficient conditions ensuring the exponential stability in mean square of the uncertain neutral delay systems with Markovian jumping parameters in [39, 40] are dependent upon the upper bound of the exponential stability rate. But, Theorem 3.6 can remove this restrictive conditions. So, we can improve and generalize the results in [39, 40].

4. Numerical Examples

In this section, two examples are provided to illustrate the feasibility and applicability of our obtained results.

Example 4.1. Consider the case of 2D Brownian motion, and 𝑟(𝑡) is right-continuous Markovian chain taking values in 𝑆={1,2} with its generator Γ=⁻³³₄₋₄. And the parameters in systems (2.3) are given as follows: 𝐴₁₁=−1.500−1.8,𝐴₁₂=−1.800−1.8,𝐴₂₁=0.30.2−0.6−0.8,𝐴₂₂=,𝐴0.6−0.5−0.4−0.3₃₁=−0.7−0.50.40.6,𝐴₃₂=0.70.5−0.30.3,𝐴₄₁=0.20.3−0.10.2,𝐴₄₂=,𝐷0.30.2−0.20.4₁=−0.20.01.00.1,𝐷₂=0.10.00.70.2,𝑀₁=0.10.20.00.1,𝑀₂=,𝑁0.20.10.10.2₁₁=𝑁₁₂=𝑁₂₁=𝑁₂₂=𝑁₃₁=𝑁₃₂=𝑁₄₁=𝑁₄₂=0.10.1. (4.1) The delays 𝑑(𝑡)=0.45sin(𝑡)+0.3, 𝜎(𝑡)=0.4cos(𝑡)+0.4, and it is easily obtained that 𝑑₁=0.15, 𝑑₂=0.75, 𝜇=0.45, 𝜈=0.4, and 𝜎=0.8. Assume that the activation function 𝑓 satisfies Assumption 2.1 with the Lipschitz coefficient matrix 𝐿=^0.20.0_0.00.2 and the function 𝜎 satisfies Assumption 2.2 with the matrices 𝑅_𝑖𝑗 (𝑖=1,2; 𝑗=1,2,3) given by 𝑅₁₁=0.150.00.200.35,𝑅₁₂=0.50.10.280.20,𝑅₂₁=0.30.00.20.1,𝑅₂₂=,𝑅0.10.00.20.1₃₁=0.10.20.10.2,𝑅₃₂=.0.10.10.20.1 (4.2) With the parameters above, when 𝜅=0.9999, by using Matlab LMI Toolbox, according to Theorem 3.2, we solve LMIs (3.33) and obtain there feasible solutions as follows: 𝑃₁=70.093012.228512.228521.8248,𝑃₂=74.154915.487315.487320.7284,𝑄₁=,𝑄134.466225.422325.422320.5296₂=1.8693−0.0065−0.00650.6213,𝑄₃=4.8828−0.0153−0.01531.6598,𝑄₄=,𝑄16.7672−0.0153−0.01531.6598₅=117.1703−17.8696−17.8696119.1560,𝑄₆=114.706167.043867.043891.0274,𝑄₇=,𝑄58.328423.081123.081170.1310₈=57.018112.728012.728058.0073,𝑄₉=41.692114.691914.691940.8478,𝑄₁₀=,𝜆40.5029−0.2295−0.229533.2920₁=75.3750,𝜆₂=80.6197,𝜀₁₁=66.1002,𝜀₁₂=97.9532,𝜀₂₁=47.7433,𝜀₂₂=56.9671. (4.3)

Example 4.2. Let 𝑟(𝑡) be right-continuous Markovian chain taking values in 𝑆={1,2} with its generator Γ=⁻²²₁₋₁. Consider the following uncertain neutral systems with mixed delays and Markovian switching: for any 𝑖∈𝑆, 𝑑𝑥(𝑡)𝑑𝑡−𝐷_𝑖𝑥(𝑡−𝑑(𝑡))=𝐴_1𝑖(𝑡)𝑥(𝑡)+𝐴_2𝑖(𝑡)𝑥(𝑡−𝑑(𝑡)),𝑡≥0, (4.4) where 𝐴₁₁=𝐴₁₂=−20.00.0−3,𝐴₂₁=𝐴₂₂=−10−1−1,𝐷₁=,𝐷0.3000.3₂=0.10.00.00.1,𝑀₁=0.10.10.20.1,𝑀₂=,𝑁0.2−0.10.10.1₁₁=0.10.0,𝑁₁₂=0.00.1,𝑁₂₁=−0.00.1,𝑁₂₂=0.10.1. (4.5) Applying Matlab toolbox, by Theorem 3.6, for 𝜇=0, it can be obtained that the uncertain neutral systems with delays and Markovian jumping parameters (4.4) are robustly exponentially stable in mean square with the delay 𝑑(𝑡) satisfying 0≤𝑑(𝑡)≤6.0×10¹⁷.

5. Conclusions

In this paper, the exponential stability in mean square of the uncertain neutral stochastic systems with mixed delays and Markovian jumping parameters has been considered. The mixed delays consist of the discrete delay and the distributed delay. The LMI-based conditions ensuring the exponential stability in mean square of such systems are obtained by constructing an appropriate Lyapunov functional, which are dependent upon the upper bound and lower bound of the discrete time delays and distributed delays. It is worth pointing out that, compared with the previous works [17, 18], the stability criteria in this paper can be easily checked by using some standard numerical packages. Two illustrative examples are provided to show the effectiveness and applicability of the proposed results.

Acknowledgments

The author would like to thank the anonymous referees for their very helpful comments and suggestions that greatly improved this paper and the editors for their careful reading of this paper. This work is supported by the National Natural Science Foundation of China under Grant no. 11126278 and Grant no. 11101202, the Natural Science Foundation of Jiangxi Province under Grant no. 20114BAB211001 and Grant no. 2009GQS0018, and the Youth Foundation of Jiangxi Provincial Educations of China under Grant no. GJJ11045 and Grant no. GJJ10051.