Sufficient Conditions on the Exponential Stability of Neutral Stochastic Differential Equations with Time-Varying Delays

Abstract

The exponential stability is investigated for neutral stochastic differential equations with time-varying delays. Based on the Lyapunov stability theory and linear matrix inequalities (LMIs) technique, some delay-dependent criteria are established to guarantee the exponential stability in almost sure sense. Finally a numerical example is provided to illustrate the feasibility of the result.

1. Introduction

Neutral differential equations are well-known models from many areas of science and engineering, wherein, quite often the future state of such systems depends not only on the present state but also involves derivatives with delays. Deterministic neutral differential equations were originally introduced by Hale and Meyer [1] and discussed in Hale and Lunel (see [2]) and Kolmanovskii et al. (for details, see also [3, 4]), among others. Motivated by chemical engineering systems as well as theory of aeroelasticity, stochastic neutral delay systems have been intensively studied over recent year [5–9]. Mao initiated the study of exponential stability of neutral stochastic differential delay equations in [5], while [9] incorporated Razumikhinis approach in neutral stochastic functional differential equations to investigate the stability problem. It is pointed out in Section 5 [10] that the conditions imposed in [5, 9] make the theory not applicable to the delay equation.

More recently, Luo et al. [6] proposed new criteria on exponential stability of neutral stochastic delay differential equations. In [11, 12], Milošević investigated the almost sure exponential stability of a class of highly nonlinear neutral stochastic differential equations with time-dependent delay, and some sufficient conditions were given for the considered systems. However, when the exponential stability of the neutral system with time-delay is considered, one always assumes that the derivative of the delay function is less than 1 (e.g., [6]). Meanwhile, the delay-independent conditions in [6, 10] are restricted when the delay is small. On the other hand, some results are proposed on stochastic Markovian jumping systems (e.g., [13–20]) and finite-time problems of stochastic systems (e.g., [18–22]), which can provide some useful methods and techniques for the neutral stochastic systems. This paper aims to develop the exponential stability in almost sure sense of the neutral stochastic differential equations with time-varying delays. Under the weaker assumptions that the derivative of time delay is less than some constant, sufficient conditions for the exponential stability are given in terms of linear matrix inequality (LMI) based on Lyapunov stability theory, which can be checked easily by MATLAB LMI Toolbox.

The paper is organized as follows. In the remainder of this section we recall some preliminaries, mainly from [5]. In Section 3 we state the main results on exponential stability. Section 4 will provide numerical examples to illustrate the feasibility and effectiveness of the results, and the conclusion will be made in Section 5.

2. Preliminaries

Throughout this paper, unless otherwise specified, let {Ω,ℱ,𝑃} be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., right continuous and 𝐹₀ containing all 𝑃-null sets). Let 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑚(𝑡))^𝑛 be 𝑚-dimensional Wiener process defined on the probability space. Let |⋅| denote the Euclidean norm in ℝ^𝑛. 𝐴^𝑇 stands for the transpose of the vector or matrix 𝐴. If 𝐴 is a matrix, its trace norm is denoted by |𝐴|=𝐴^𝑇𝐴. 𝑎∨𝑏 denotes max{𝑎,𝑏}. 𝜆_max(⋅),𝜆_min(⋅) are maximum eigenvalue and minimum eigenvalue, respectively.

Consider the following 𝑛-dimensional neutral stochastic differential delay equations with time-varying delays: d[𝑥(𝑡)(𝑥((𝑡(𝑥(𝑡)((𝑡)(𝑥(𝑡)((𝑡)(𝑡)𝑥(𝑡)(𝑡)−𝐺𝑡−𝛿)))]=𝑓,𝑥𝑡−𝛿)),𝑡d𝑡+𝑔,𝑥𝑡−𝛿)),𝑡d𝑤=𝜉∈𝐶^𝑏_ℱ0[−𝜏,0],ℝ^𝑛,𝑡∈[−𝜏,0], (1) where 𝑓∶ℝ^𝑛×ℝ^𝑛×ℝ₊→ℝ^𝑛, 𝑔∶ℝ^𝑛×ℝ^𝑛×ℝ₊→ℝ^𝑛×𝑚, and 𝐺∈𝐶(ℝ^𝑛,ℝ^𝑛). The functions 𝛿(𝑡)∶ℝ₊→[0,𝜏] are continuously differentiable such that 0≤𝛿(𝑡)≤𝛿,𝛿̇(𝑡)≤𝛿. Let 𝐶([−𝜏,0],ℝ^𝑛) denote the family of continuous functions 𝜙 from [−𝜏,0] to ℝ^𝑛 with the norm ‖𝜙‖=sup_{−𝜏≤𝜃≤0}|𝜙(𝜃)|. Let 𝐶^𝑏_ℱ0([−𝜏,0],ℝ^𝑛) be the family of all ℱ₀-measurable 𝐶([−𝜏,0],ℝ^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶𝜏≤𝜃≤0} such that sup_{−𝜏≤𝜃≤0}𝐸|𝜉(𝜃)|²<∞. To guarantee the existence and uniqueness of the solution, we first list the following hypotheses.(H₁)Both the functionals 𝑓 and 𝑔 satisfy the uniform Lipschitz conditions. That is, there is a diagonal positive matrix 𝐿=diag{𝐿₁,𝐿₂,…,𝐿_𝑛} such that 𝑓𝑥,𝑦,𝑡−𝑓𝑥,𝑦,𝑡∨𝑔𝑥,𝑦,𝑡−𝑔𝑥,≤|𝐿(𝑦,𝑡𝑥−𝑥)|+𝐿𝑦−𝑦 (2) for all 𝑡≥0 and those 𝑥,𝑦,𝑥,𝑦∈ℝ^𝑛.(H₂)There is a constant 𝑘∈(0,1) such that for all 𝜙₁,𝜙₂∈𝐶^𝑏_ℱ0([−𝜏,0],ℝ^𝑛)𝐺𝜙₁−𝐺𝜙₂²≤𝑘sup_{−𝜏≤𝜃≤0}𝜙₁(𝜃)−𝜙₂(𝜃)². (3)It is well known (see, e.g., [3]) that under hypotheses H₁, H₂ (1) has a unique continuous solution on 𝑡≥−𝜏.

To obtain sufficient conditions on almost sure exponential stability, the following lemmas and definition are given.

Lemma 1 (see [23]). For any positive definite constant matrix 𝑀∈ℝ^𝑛×𝑛, scalar 𝑟>0, and vector function 𝑓(⋅)∶[0,𝑟]→ℝ^𝑛 such that the integrations in the following are well defined, then the following inequality holds: ^𝑟₀𝑓(𝑠)𝑑𝑠^𝑇𝑀^𝑟₀𝑓(𝑠)𝑑𝑠≤𝑟^𝑟₀𝑓^𝑇(𝑠)𝑀𝑓(𝑠)𝑑𝑠. (4) The following semimartingale convergence theorem will play an important role in the later parts.

Lemma 2 (see [24]). Let 𝐴(𝑡) and 𝑈(𝑡) be two continuous adapted increasing processed on 𝑡≥0 with 𝐴(0)=𝑈(0)=0 a.s. Let 𝑀(𝑡) be a real-valued continuous local martingale with 𝑀(0)=0 a.s. Let 𝜍 be a nonnegative ℱ₀-measurable random variable. Define 𝑋(𝑡)=𝜍+𝐴(𝑡)−𝑈(𝑡)+𝑀(𝑡)𝑓𝑜𝑟𝑡≥0. (5) If 𝑋(𝑡) is nonnegative, then lim_𝑡→∞𝐴(𝑡)<∞⊂lim_𝑡→∞𝑋(𝑡)∩<∞lim_𝑡→∞𝑈(𝑡)<∞𝑎.𝑠., (6) where 𝐵⊂𝐷 a.s. means 𝑃(𝐵∩𝐷)=0. In particular, if lim_𝑡→∞𝐴(𝑡)<∞ a.s., then for almost all 𝑤∈Ω, lim_𝑡→∞𝑋(𝑡)<∞,lim_𝑡→∞𝑈(𝑡)<∞; (7) that is, both 𝑋(𝑡) and 𝑈(𝑡) converge to finite random variables.

Definition 3 (see [25]). The equilibrium of solution {𝑥(𝑡),𝑡≥0} of (1) is said to be almost sure exponentially stable if there exists a constant 𝜀>0 such that limsup_𝑡→∞1𝑡log|𝑥(𝑡)|≤−𝜀a.s. (8) for any bounded initial condition 𝜉.

3. Main Results

Theorem 4. Let hypotheses H₁, H₂ hold. System (1) is almost sure exponentially stable, if there exists positive definite matrix such that the following LMI holdsΞΩ=¹¹𝑁^𝑇₂−𝑁^𝑇₁𝑁^𝑇₃+𝑈^𝑇𝑁^𝑇₄−𝑁^𝑇₁+𝑁^𝑇₅0∗Ξ₂₂−𝑁^𝑇₃−𝑁^𝑇₄−𝑁^𝑇₂−𝑁^𝑇₅0∗∗Ξ₃₃0−𝑁^𝑇₃𝑈∗∗∗Ξ₄₄−𝑁^𝑇₄01∗∗∗∗−𝛿𝑅−2𝑁^𝑇₅0∗∗∗∗∗−𝜀𝐼<0, (9)where Ξ₁₁=𝛽𝑃+𝑒^𝛽𝛿𝑄+𝑃+2𝑁^𝑇₁,Ξ₂₂=−1−𝛿𝑒^𝛽𝛿∨−1−𝛿𝑄−2𝑁₂,Ξ₃₃=−2𝑈+𝑒^𝛽𝛿𝑆+2𝜆_max(𝑃)𝐿^𝑇𝐿+2𝛽𝜆_max(𝑅)𝑒^𝛽𝛿−1𝐿₁,Ξ₄₄=−1−𝛿𝑆+2𝜆_max(𝑃)𝐿^𝑇𝐿+2𝛽𝜆_max(𝑅)𝑒^𝛽𝛿−1𝐿₁+𝜀𝑘𝐼. (10)

Proof. To confirm that the stochastic neutral differential equation (1) is mean-square exponentially stable with decay rate 𝛽, we define a Lyapunov-Krasovskii functional 𝑉(𝑥(𝑡),𝑡) as follows: 𝑉(𝑥(𝑡),𝑡)=𝑒^𝛽𝑡𝜌^𝑇(𝑡)(𝑡)+𝑃𝜌_{𝑡(𝑡)𝑡−𝛿}𝑒^{𝛽(𝑠+𝛿)}𝜌^𝑇(𝑠)(𝑠)+𝑄𝜌d𝑠_{0−𝛿𝑡𝑡+𝜃}𝑒^{𝛽(𝑠−𝜃)}𝑓^𝑇𝑥(𝑠)(𝑠)𝑥(𝑠)(𝑠)+,𝑦,𝑠×𝑅𝑓,𝑦,𝑠d𝑠d𝜃_{𝑡(𝑡)𝑡−𝛿}𝑒^{𝛽(𝑠+𝛿)}𝑥^𝑇(𝑠)𝑆𝑥(𝑠)d𝑠. (11) For simplicity, let 𝑦(𝑡)=𝑥(𝑡−𝛿(𝑡)), 𝜌(𝑡)=𝑥(𝑡)−𝐺(𝑦(𝑡)). By generalizing Ito’s formula, we have that 𝐸𝑉𝜌(𝑡),𝑡=𝐸𝑉𝜌(0),0+^𝑡₀ℒ𝑉𝑥(𝑠),𝑦(𝑠),𝑠d𝑠. (12) Then, the derivative of 𝑉(𝜌(𝑡),𝑡) along the solution of (1) gives 𝐿𝑉𝑥(𝑡),𝑦(𝑡),𝑡=𝛽𝑒^𝛽𝑡𝜌^𝑇(𝑡)𝑃𝜌(𝑡)+2𝑒^𝛽𝑡𝜌(𝑡)^𝑇𝑃𝑓𝑥(𝑡),𝑦(𝑡),𝑡+𝑒^𝛽𝑡𝑔^𝑇𝑥(𝑡),𝑦(𝑡),𝑡𝑃𝑔𝑥(𝑡),𝑦(𝑡),𝑡+𝑒^{𝛽(𝑡+𝛿)}𝜌(𝑡)𝑄𝜌(𝑡)−𝑒^{𝛽(𝑡−𝛿(𝑡))+𝛿}𝜌^𝑇((𝑡((𝑡𝛿̇(𝑡)+1𝑡−𝛿))𝑄𝜌𝑡−𝛿))1−𝛽𝑒^𝛽𝑡𝑒^𝛽𝛿−1𝑓^𝑇𝑥(𝑡),𝑦(𝑡),𝑡𝑅𝑓𝑥(𝑡),𝑦(𝑡),𝑡−𝑒^𝛽𝑡𝑡_𝑡−𝛿𝑓^𝑇𝑥(𝑠),𝑦(𝑠),𝑠𝑅𝑓𝑥(𝑠),𝑦(𝑠),𝑠d𝑠+𝑒^{𝛽(𝑡+𝛿)}𝑥(𝑡)𝑆𝑥(𝑡)−𝑒^{𝛽(𝑡−𝛿(𝑡))+𝛿}×𝑥^𝑇(𝑡−𝛿(𝑡))𝑆𝑥(𝑡−𝛿(𝑡))1−𝛿̇(𝑡). (13) Note that, from system (1) and Newton-Leibniz formula, we have 𝑀=𝜌(𝑡)−𝜌(𝑡−𝛿(𝑡))−^𝑡_{(𝑡)𝑡−𝛿}𝑓𝑥(𝑠)(𝑠)−,𝑦,𝑠d𝑠_{𝑡(𝑡)𝑡−𝛿}𝑔𝑥(𝑠),𝑦(𝑠),𝑠d𝑤(𝑠)=0. (14) By calculation, it is clear that 𝑓^𝑇𝑥(𝑡),𝑦(𝑡),𝑡𝑅𝑓𝑥(𝑡),𝑦(𝑡),𝑡≤𝜆_max(𝑅)𝑓^𝑇𝑥(𝑡),𝑦(𝑡),𝑡𝑓𝑥(𝑡),𝑦(𝑡),𝑡≤𝜆_max(𝑅)𝑓𝑥(𝑡),𝑦(𝑡),𝑡²≤2𝜆_max(𝑅)𝑥^𝑇(𝑡)𝐿^𝑇𝐿𝑥(𝑡)+𝑥^𝑇(𝑡−𝛿(𝑡))𝐿^𝑇𝐿𝑦(𝑡), (15) and then by which, we have 2𝜌^𝑇(𝑡)𝑃𝑓𝑥(𝑡),𝑦(𝑡),𝑡≤𝜌^𝑇(𝑡)𝑃^𝑇𝜌(𝑡)+𝑓^𝑇𝑥(𝑡),𝑦(𝑡),𝑡𝑃𝑓𝑥(𝑡),𝑦(𝑡),𝑡≤𝜌^𝑇(𝑡)𝑃^𝑇𝜌(𝑡)+2𝜆_max(𝑃)𝑥^𝑇(𝑡)𝐿^𝑇𝐿𝑥(𝑡)+𝑦^𝑇(𝑡)𝐿^𝑇𝐿𝑦(𝑡),−𝑒^{𝛽(𝑡−𝛿(𝑡))+𝛿}𝜌^𝑇(𝑡)𝑄𝜌(𝑡)1−𝛿̇(𝑡)≤−1−𝛿𝑒^𝛽𝑡𝜌^𝑇(𝑡)𝑄𝜌(𝑡).−𝑒^{𝛽(𝑡−𝛿(𝑡))+𝛿}𝑥^𝑇(𝑡)𝑆𝑥(𝑡)1−𝛿̇(𝑡)≤−1−𝛿𝑒^𝛽𝑡𝑥^𝑇(𝑡)𝑆𝑥(𝑡). (16) Moreover, by Lemma 2, one can get −^𝑡_𝑡−𝛿𝑓^𝑇𝑥(𝑠),𝑦(𝑠),𝑠𝑅𝑓𝑥(𝑠),𝑦(𝑠),𝑠d𝑠≤−^𝑡_{𝑡−𝛿(𝑡)}𝑓^𝑇𝑥(𝑠)(𝑠)𝑥(𝑠)(𝑠)1,𝑦,𝑠𝑅𝑓,𝑦,𝑠d𝑠≤−𝛿^𝑡_{𝑡−𝛿(𝑡)}𝑓(𝑥(𝑠),𝑦(𝑠),𝑠)d𝑠^𝑇×𝑅^𝑡_{𝑡−𝛿(𝑡)}𝑓𝑥(𝑠),𝑦(𝑠),𝑠d𝑠. (17) Letting 𝐿₁=𝐿^𝑇𝐿 and substituting (14)–(17) into (13) yield 𝐿𝑉𝑥(𝑡),𝑦(𝑡),𝑡≤𝑒^𝛽𝑡𝛽𝜌^𝑇(𝑡)𝑃𝜌(𝑡)+𝜌^𝑇(𝑡)𝑃^𝑇𝜌(𝑡)+2𝜆_max(𝑃)𝑥^𝑇(𝑡)𝐿₁𝑥(𝑡)+2𝜆_max(𝑃)𝑦^𝑇(𝑡)𝐿₁𝑦(𝑡)+−1−𝛿𝑒^𝛽𝛿∨−1−𝛿𝜌^𝑇(𝑡)𝑄𝜌(𝑡)+𝑒^𝛽𝛿𝜌^𝑇(𝑡)𝑄₁𝜌(𝑡)+2𝜆_max1𝛽𝑒^𝛽𝛿×𝑥−1_𝑇(𝑡)𝐿₁𝑥(𝑡)+𝑦^𝑇(𝑡)𝐿₁𝑦(𝑡)−1𝛿^𝑡_{𝑡−𝛿(𝑡)}𝑓(𝑥(𝑠),𝑦(𝑠),𝑠)d𝑠^𝑇×𝑅^𝑡_{𝑡−𝛿(𝑡)}𝑓𝑥(𝑠),𝑦(𝑠),𝑠d𝑠. (18) Furthermore, from (14), it follows that 𝐴=2𝜂^𝑇𝑁^𝑇₁,𝑁^𝑇₂,𝑁^𝑇₃,𝑁^𝑇₄,𝑁^𝑇₅^𝑇𝑀=0,𝐵=2𝑥^𝑇(𝑡)𝑈𝜌(𝑡)−𝑥(𝑡)+𝐺𝑦(𝑡)=0, (19) where 𝜂=(𝜌^𝑇, 𝜌^𝑇(𝑡−𝛿(𝑡)), 𝑥^𝑇(𝑡), 𝑦^𝑇(𝑡), (∫^𝑡_{𝑡−𝛿(𝑡)}𝑓(𝑥,𝑦,𝑠)d𝑠)^𝑇)^𝑇, and 𝑁_𝑖(1≤𝑖≤5), 𝑈 are matrices with compatible dimensions.
It can be shown that ℒ𝑉𝑥^𝑡₀(𝑠),𝑦(𝑠),𝑠d𝑠+𝑒^𝛽𝑡(𝐴+𝐵)+𝑒^𝛽𝑡𝑀^𝑇𝑃₁𝑀(𝑡)+^𝑡_{𝑡−𝛿(𝑡)}𝑔𝑥,𝑦,𝑠d𝑠≤𝑒^𝛽𝑡𝛽𝜌^𝑇(𝑡)𝑃𝜌(𝑡)+𝜌^𝑇(𝑡)𝑃^𝑇𝜌(𝑡)+𝑥^𝑇(𝑡)𝐿₁𝑥(𝑡)+𝑒^𝛽𝛿𝜌^𝑇(𝑡)𝑄𝜌(𝑡)+𝑦^𝑇(𝑡)𝐿₁𝑦(𝑡)+2𝜆_max(𝑃)𝑥^𝑇(𝑡)𝐿₁𝑥(𝑡)+2𝜆_max𝑦^𝑇(𝑡)𝐿₁𝑦(𝑡)−1−𝛿𝜌^𝑇(𝑡)𝑄𝜌(𝑡)+𝑒^𝛽𝛿𝑥^𝑇(𝑡)(𝑡)−𝑆𝑥1−𝛿𝑥^𝑇(𝑡)𝑆𝑥(𝑡)+2𝜆_max1𝛽𝑒^𝛽𝛿×𝑥−1_𝑇(𝑡)𝐿₁𝑥(𝑡)+𝑦^𝑇(𝑡)𝐿₁𝑦(𝑡)−1𝛿^𝑡_{𝑡−𝛿(𝑡)}𝑓(𝑥(𝑠),𝑦(𝑠),𝑠)d𝑠^𝑇×𝑅^𝑡_{𝑡−𝛿(𝑡)}𝑓𝑥(𝑠),𝑦(𝑠),𝑠d𝑠≤𝑒^𝛽𝑡𝜂^𝑇Ω𝜂+𝜀⁻¹𝑥^𝑇(𝑡)𝑈𝑈^𝑇𝑥(𝑡), (20) where Ω is defined as ΞΩ=¹¹𝑁^𝑇₂−𝑁^𝑇₁𝑁^𝑇₃+𝑈^𝑇𝑁^𝑇₄−𝑁^𝑇₁+𝑁^𝑇₅∗Ξ₂₂−𝑁^𝑇₃−𝑁^𝑇₄−𝑁^𝑇₂−𝑁^𝑇₅∗∗Ξ₃₃0−𝑁^𝑇₃∗∗∗Ξ₄₄−𝑁^𝑇₄1∗∗∗∗−𝛿𝑅−2𝑁^𝑇₅. (21) By Schur complement, we know that 𝜂^𝑇Ω𝜂+𝜀⁻¹𝑥^𝑇(𝑡)𝑈𝑈^𝑇𝑥(𝑡)<0. On the other hand, it follows that 𝑉𝜌(𝑡),𝑡=𝑉𝜌(0),0+^𝑡₀𝑥(𝑠)(𝑠)+ℒ𝑉,𝑦,𝑠d𝑠_𝑡02𝑒^𝛽𝑠𝑥^𝑠(𝑡)𝑔𝑥(𝑠),𝑦(𝑠),𝑠d𝑤(𝑠). (22) Note that 𝜉 is bounded and 𝑉,𝐺 are continuous; then 𝑉(𝜌(0)) must be nonnegative bounded. Moreover, ℒ𝑉(𝑥,𝑦,𝑡)≤0 can be obtained directly: 𝑉𝜌(𝑡)𝜌(0)+,𝑡≤𝑉,0_𝑡02𝑒^𝛽𝑡𝑥^𝑇(𝑠)𝑔𝑥(𝑠),𝑦(𝑠),𝑠d𝑤(𝑠). (23) By applying Lemma 2 to (23), one sees that lim_𝑡→∞sup𝑉𝜌(𝑡),𝑡<∞; (24) hence there exists a positive random variable 𝜁 satisfying sup_0≤𝑡<∞𝑒^𝛽𝑡𝑥(𝑡)−𝐺𝑦(𝑡)²≤𝜁. (25) Since, for any 𝜀₃∈(0,1)𝑥(𝑡)−𝐺𝑦(𝑡)²≥1−𝜀₃⁻¹|𝑥(𝑡)|²−𝜀₃−1𝐺𝑦(𝑡)², (26) we must have sup_{0≤𝑡≤𝑇}𝑒^𝛽𝑡|𝑥(𝑡)|²𝑘≤𝜁+²𝜀₃sup_{0≤𝑡≤𝑇}𝑒^𝛽𝑡𝑦(𝑡)²≤𝜁+𝑘²𝑒^𝛽𝜏𝜉²+𝑘²𝜀₃𝑒^𝛽𝜏sup_{0≤𝑡≤𝑇}𝑒^𝛽𝑡|𝑥(𝑡)|². (27) From the above inequality (26), it yields the desired result limsup_𝑡→∞1𝑡|𝑥(𝑡𝛽log)|≤−2. (28) That completes the proof.

4. Example

In this section, a numerical example will be given to illustrate that the proposed method is effective.

Example 1. Consider the following system: 𝑑𝑥₁(𝑡)−0.1𝑥₂(𝑡−𝛿(𝑡))=−𝑥₁(𝑡)𝑥₂(𝑡−𝛿(𝑡))d𝑡+𝑥₁(𝑡)sin²((𝑡(𝑡),𝑑𝑥𝑡−𝛿))d𝜔₂(𝑡)−0.1𝑥₁(𝑡−𝛿(𝑡))=−𝑥₂(𝑡)𝑥₁(𝑡−𝛿(𝑡))d𝑡+𝑥₂(𝑡)cos²(𝑡−𝛿(𝑡))d𝜔(𝑡), (29) where the delay function is defined as 𝛿(𝑡)=(1/4)sin(𝑡),𝑡>0. It is obvious that (29) satisfies the assumptions H₁ and H₂, and here 𝐿=𝐼, 𝑘=0.1. Moreover, since 𝛿̇=(1/4)cos(𝑡), then 𝛿=1/4.

According to Theorem 4 and employing MATLAB LMI Toolbox, it is relatively easy to deduce that the neutral stochastic differential equation (29) is almost sure exponentially stable.

Remark 5. Comparing with some existing sufficient criteria for neural stochastic differential equations (e.g., [6, 11, 12]), the obtained result is given in terms of linear matrix inequality (LMI), which can be easily checked by MATLAB LMI Toolbox.

5. Conclusion

The exponential stability is investigated for a class of neutral stochastic differential equations with time-varying delays. In order to overcome the difficulties, we introduce suitable Lyapunov functionals and employ linear matrix inequalities (LMIs) technique, and then a delay-dependent criteria are given to check the almost sure exponential stability of the concerned equations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.