Stabilization of a Class of Stochastic Systems with Time Delays

Abstract

The problem of exponential stability is investigated for a class of stochastic time-delay systems. By using the decomposition technique and Lyapunov stability theory, two improved exponential stability criteria are derived. Finally, a numerical example is given to illustrate the effectiveness and the benefit of the proposed method.

1. Introduction

In fact, time delay constantly occurs in the real world, which results in instability of systems. Thus, the stability problem for time-delay systems has been studied for many years [1–8]. On the other hand, stochastic modelling has come to play an important role in many fields of science or industry. Stability analysis for stochastic systems has become increasingly meaningful. A number of results have appeared in the literature [9–19]. For instance, in [11], the author provided the criteria for the stability of a class of stochastic systems by Lyapunov theory.

It’s worth noting that, the grey systems can be established when parameters are evaluated by grey numbers (see [20]). Until now, there have been a few papers tackling the stability of the systems; some important and innovative results are obtained [20–23]. In [23], the authors provided the delay-dependent criteria for exponential robust stability in the forms of nonlinear matrix inequalities and linear matrix inequalities.

In this paper, we deal with the exponential stability for the time-delay grey stochastic systems. By using the method of [21–23] and Lyapunov stability theory, two improved criteria of mean-square exponential stability are proposed. At last, a numerical example is given to verify the criteria.

Notations. 𝑅^𝑛 denotes the 𝑛 dimensional Euclidean space, the superscript “𝑇” denotes matrix transposition, and the notation 𝑋≥𝑌, where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋−𝑌 is positive semidefinite. Let (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and 𝐹₀ contains all 𝑃-null sets). Let 𝑤(𝑡)=(𝑤₁(𝑡),…,𝑤_𝑚(𝑡))^𝑇 be an 𝑚-dimensional Brownian motion defined on the probability space. Let 𝜏>0 and 𝐶([−𝜏,0];𝑅^𝑛) be the family of continuous functions 𝜑 from [−𝜏,0] to 𝑅^𝑛. Let 𝐿²_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<∞.

2. Problem Formulation

Consider (𝑡)=[(𝑡)()(𝑡(𝑥(𝑡)())(𝑡)𝑥𝑑𝑥𝐴𝑥+𝐵𝑥𝑡−𝜏+𝐷𝑢)]𝑑𝑡+𝜎,𝑥𝑡−𝜏,𝑡𝑑𝑊,𝑡≥0,₀=𝜉,𝜉∈𝐿²_𝐹0[−𝜏,0];𝑅^𝑛,−𝜏≤𝑡≤0. (1)

Definition 1. If there is at least a grey matrix among matrices 𝐴,𝐵, and 𝐷 of system (1), then (1) is called grey stochastic time-delay system.
Hence (𝑡)=[𝐴(⊗)𝑥(𝑡)(⊗)𝑥()(𝑡(𝑥(𝑡)())(𝑡)𝑥𝑑𝑥+𝐵𝑡−𝜏+𝐷𝑢)]𝑑𝑡+𝜎,𝑥𝑡−𝜏,𝑡𝑑𝑊,𝑡≥0,₀=𝜉,𝜉∈𝐿²_𝐹0[−𝜏,0];𝑅^𝑛,−𝜏≤𝑡≤0, (2) where 𝐴(⊗) and 𝐵(⊗) are grey matrices and 𝐴(⊗)=(⊗^𝑎_𝑖𝑗), 𝐵(⊗)=(⊗^𝑏_𝑖𝑗).
Clearly, if matrices 𝐴(⊗) and 𝐵(⊗) are replaced by the deterministic matrices 𝐴 and 𝐵, the grey system (2) becomes system (1). The equations 𝐿_𝑎,𝑈_𝑎=𝐴⊗=𝑎_𝑖𝑗∶𝑎_𝑖𝑗≤𝑎_𝑖𝑗≤𝑎_𝑖𝑗,𝐿,𝑖,𝑗=1,2,…,𝑛_𝑏,𝑈_𝑏=𝐵⊗=𝑏_𝑖𝑗∶𝑏_𝑖𝑗≤𝑏_𝑖𝑗≤𝑏_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛 (3) are said to be the continuous matrix-covered sets of 𝐴(⊗) and 𝐵(⊗).

Definition 2 (see [20]). System (2) is said to be robustly exponentially stable in the mean square, if, for all 𝜉∈𝐿²_𝐹0([−𝜏,0];𝑅^𝑛) and arbitrary matrices 𝐴(⊗)∈[𝐿_𝑎,𝑈_𝑎], 𝐵(⊗)∈[𝐿_𝑏,𝑈_𝑏], there exist scalars 𝑟>0 and 𝐶>0, such that 𝐸𝑥(𝑡,𝜉)²≤𝐶𝑒^−𝑟𝑡sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²,𝑡≥0. (4) In addition, the following assumptions are made on the system (2). (H1)𝐻∶𝑅^𝑛×𝑅^𝑛×𝑅₊→𝑅^𝑛×𝑛. (H2)Supposing there exist scalars 𝛼≥0, 𝛽≥0, such that, for (𝑥,𝑦,𝑡)∈𝐻∶𝑅^𝑛×𝑅^𝑛×𝑅₊, the inequality holds, Trace[𝜎^𝑇(𝑥,𝑦,𝑡)𝜎(𝑥,𝑦,𝑡)]≤𝛼|𝑥|²+𝛽|𝑦|².
Before giving the main results, we first present Lemmas 3 and 4, which are important for the proof of main theorems.

Lemma 3 (see [20]). For arbitrary whitened matrix 𝐴(⊗)∈[𝐿_𝑎,𝑈_𝑎], it follows that(i)𝐴(⊗)=𝐿_𝑎+Δ𝐴(ii)0≤Δ𝐴≤𝑈_𝑎−𝐿_𝑎(iii)‖𝐴(⊗)‖=‖𝐿_𝑎‖+‖𝑈_𝑎−𝐿_𝑎‖,where 𝐿_𝑎=(𝑎_𝑖𝑗)_𝑚×𝑛, 𝑈_𝑎=(𝑎_𝑖𝑗)_𝑚×𝑛, Δ𝐴=(𝛿_𝑖𝑗𝑟̂_𝑖𝑗)_𝑚×𝑛, 𝛿_𝑖𝑗=𝑎_𝑖𝑗−𝑎_𝑖𝑗≥0.

Lemma 4 (see [18]). Let 𝑁 be a real matrix of appropriate dimensions; for any vectors 𝑥,𝑦∈𝑅^𝑛, one has 2𝑥^𝑇𝑁𝑦≤𝜀𝑥^𝑇𝑥+𝜀⁻¹𝑦^𝑇𝑁^𝑇𝑁𝑦.

3. Proof of the Main Theorem

In this section, we discuss the exponential stability for system (2); two improved criteria for robust exponential stability in mean square are proposed.

Theorem 5. System (2) is exponentially robustly stable in mean square. If there exist positive scalars 𝜀₁, 𝜀₂, and 𝜀₃, such that 𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝑘₁+𝑘₂<0 (5) here 𝑘₁=1+𝜀₁+𝜀₁⁻¹𝑈_𝑎−𝐿_𝑎²+𝜀₂𝑘+𝛼,₂=𝜀₂⁻¹1+𝜀₃𝜆_max𝐿^𝑇_𝑏𝐿_𝑏+𝜀₂⁻¹1+𝜀₃⁻¹𝑈_𝑏−𝐿_𝑏²+𝛽. (6) Then, for all 𝜉∈𝐿²_𝐹0([−𝜏,0];𝑅^𝑛), the following inequality holds: 𝐸𝑥(𝑡,𝜉)²≤1+𝑘₂𝜏𝑒^𝑟𝜏𝑒^−𝑟𝑡sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²,𝑡≥0, (7) where 𝑟 is the unique positive solution of the following equation: 𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝑘₁+𝑘₂𝑒^𝑟𝜏=0. (8)

Proof. First, define Lyapunov-Krasovskii functional as follows: 𝑉(𝑥(𝑡),𝑡)=𝑒^𝑟𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+⁰_−𝜏𝑒^{𝑟(𝑡+𝜃)}𝑥^𝑇(𝑡+𝜃)𝑥(𝑡+𝜃)𝑑𝜃. (9) Then, we have 𝐿𝑉(𝑥(𝑡),𝑡)=(𝑟+1)𝑒^𝑟𝑡𝑥^𝑇(𝑡)𝑥(𝑡)−𝑒^{𝑟(𝑡−𝜏)}𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏)+𝑒^𝑟𝑡2𝑥^𝑇(𝑡)𝐴⊗𝑥(𝑡)+2𝑥^𝑇(𝑡)𝐵⊗𝑥()𝜎𝑡−𝜏+Trace^𝑇(𝑥(𝑡)())(𝑥(𝑡)().,𝑥𝑡−𝜏,𝑡×𝜎,𝑥𝑡−𝜏,𝑡) (10) Using Lemmas 3 and 4, we derive 2𝑥^𝑇(𝑡)𝐴⊗𝑥(𝑡)≤𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎𝑥^𝑇(𝑡)𝑥(𝑡)+𝜀₁𝑥^𝑇(𝑡)𝑥(𝑡)+𝜀₁⁻¹𝑈_𝑎−𝐿_𝑎²𝑥^𝑇(𝑡)𝑥(𝑡),2𝑥^𝑇(𝑡)𝐵⊗𝑥(𝑡−𝜏)≤𝜀₂𝑥^𝑇(𝑡)𝑥(𝑡)+𝜀₂⁻¹1+𝜀₃𝜆_max×𝐿^𝑇_𝑏𝐿_𝑏𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏)+𝜀₂⁻¹1+𝜀₃⁻¹𝑈_𝑏−𝐿_𝑏²𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏). (11) By assumption (H2), we can obtain Trace𝜎^𝑇(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)𝜎(𝑥(𝑡),𝑥(𝑡−𝜏),𝑡)≤𝛼𝑥^𝑇(𝑡)𝑥(𝑡)+𝛽𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏). (12) Substituting (11)–(12) into (10), we see that (𝑥(𝑡))≤𝐿𝑉,𝑡1+𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝜀₁+𝜀₁⁻¹𝑈_𝑎−𝐿_𝑎²+𝜀₂+𝛼𝑒^𝑟𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+𝜀₂⁻¹1+𝜀₃𝜆_max𝐿^𝑇_𝑏𝐿_𝑏+𝜀₂⁻¹1+𝜀₃⁻¹𝑈_𝑏−𝐿_𝑏²+𝛽×𝑒^𝑟𝑡𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏). (13) Using Itô’s differential formula and integrating both sides, we obtain 𝐸𝑉(𝑥(𝑡),𝑡)≤sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+1+𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝜀₁+𝜀₁⁻¹𝑈_𝑎−𝐿_𝑎²+𝜀₂×+𝛼_𝑡0𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²+𝜀𝑑𝑠₂₋₁1+𝜀₃𝜆_max𝐿^𝑇_𝑏𝐿_𝑏+𝜀₂⁻¹1+𝜀₃⁻¹𝑈_𝑏−𝐿_𝑏²+𝛽×^𝑡₀𝑒^𝑟𝑠𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠. (14) Moreover, we have ^𝑡₀𝑒^𝑟𝑠𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠≤𝑒^𝑟𝜏𝑡₀𝑒^𝑟𝑢𝐸|𝑥(𝑢)|²𝑑𝑢+𝜏𝑒^𝑟𝜏sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)². (15) Combining (14) with (15) and noting the definitions of 𝑘₁, 𝑘₂, we see that (𝑥(𝑡))≤𝐸𝑉,𝑡1+𝑘₂𝜏𝑒^𝑟𝜏sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝑘₁+𝑘₂𝑒^𝑟𝜏×^𝑡₀𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²𝑑𝑠. (16) Furthermore, 𝑓(𝑟)=𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝑘₁+𝑘₂𝑒^𝑟𝜏,then𝑓^′(𝑟)=1+𝑘₂𝜏𝑒^𝑟𝜏. (17) Since 𝑓^′(𝑟)>0, 𝑓(0)=𝜆_max(𝐿_𝑎+𝐿^𝑇_𝑎)+𝑘₁+𝑘₂ and 𝑓(+∞)=+∞.
When (5) holds, (8) must have a unique solution 𝑟>0.
Hence, we have 𝐸𝑥(𝑡,𝜉)²≤1+𝑘₂𝜏𝑒^𝑟𝜏𝑒^−𝑟𝑡sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²,𝑡≥0. (18) This completes the proof of Theorem 5.

By a similar approach in [22], whitened system of (2) can be written as 𝑑𝑥(𝑡)=𝐴⊗+𝐿_𝑏𝑥(𝑡)−𝐿_𝑏(𝑥(𝑡)()()(𝑡(𝑥(𝑡)())(𝑡).−𝑥𝑡−𝜏+Δ𝐵𝑥𝑡−𝜏+𝐷𝑢)𝑑𝑡+𝜎,𝑥𝑡−𝜏,𝑡𝑑𝑊 (19) Since 𝑥(𝑡)()=−𝑥𝑡−𝜏_{𝑡𝑡+𝜃}𝐴⊗𝑥(𝑠)⊗𝑥()(𝑠++𝐵𝑠−𝜏+𝐷𝑢)𝑑𝑠_{𝑡𝑡+𝜃}𝜎(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)𝑑𝑊(𝑠),𝑡≥𝜏, (20) then, we can introduce 𝐻()=𝑥,𝑡_{𝑡𝑡+𝜃}𝐴⊗𝑥(𝑠)⊗𝑥(++𝐵𝑠−𝜏)𝑑𝑠_{𝑡𝑡+𝜃}𝜎(𝑥(𝑠)())(𝑠)𝑥(𝑡)(),𝑥𝑠−𝜏,𝑠𝑑𝑊,𝑡≥𝜏−𝑥𝑡+𝜃,0≤𝑡≤𝜏. (21) Combining (19) and (21) together, whitened system of (2) can be rewritten as 𝑑𝑥(𝑡)=𝐴⊗+𝐿_𝑏𝑥(𝑡)−𝐿_𝑏𝐻()()(𝑡(𝑥(𝑡)())(𝑡)𝑥𝑥,𝑡+Δ𝐵𝑥𝑡−𝜏+𝐷𝑢)𝑑𝑡+𝜎,𝑥𝑡−𝜏,𝑡𝑑𝑊,𝑡≥0,₀=𝜉,𝜉∈𝐿²_𝐹0[−𝜏,0];𝑅^𝑛,−𝜏≤𝑡≤0. (22)

Lemma 6. For all 𝜉∈𝐿²_𝐹0([−𝜏,0];𝑅^𝑛) and 𝑡>0, the following inequality holds: ^𝑡₀𝑒^𝑟𝑠𝐸|𝐻(𝑥(𝑠),𝑠)|²≤𝑘𝑑𝑠₁𝜏𝑒^𝑟𝜏+𝑘₂𝜏𝑒^{2𝑟𝜏𝑡}₀𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²𝑑𝑠+𝑘₂𝜏²𝑒^2𝑟𝜏sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+sup_{−𝜏≤𝜃≤0}𝜓(𝜃), (23) where 𝑙₁=4𝜏𝐿_𝑎+𝑈_𝑎−𝐿_𝑎²𝑙+2𝛼,₂=4𝜏𝐿_𝑏+𝑈_𝑏−𝐿_𝑏²𝜓(𝜃)=+2𝛽,_𝜏0𝑒^𝑟𝑠𝐸|𝑥(𝑠)−𝑥(𝑠−𝜏)|²𝑑𝑠. (24)

Proof. First, by the definition of 𝐻(𝑥,𝑡), we can derive 𝐸|𝐻(𝑥(𝑠),𝑠)|²≤2𝐸^𝑡_𝑡+𝜃𝐴⊗𝑥(𝑠)+𝐵⊗𝑥(𝑠−𝜏)𝑑𝑠²+2𝐸^𝑡_𝑡+𝜃𝜎(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)𝑑𝑊(𝑠)². (25) Clearly, 𝐸^𝑡_𝑡+𝜃𝐴⊗𝑥(𝑠)+𝐵⊗𝑥(𝑠−𝜏)𝑑𝑠²≤2𝜏𝐿_𝑎+𝑈_𝑎−𝐿_𝑎^2𝑡_𝑡+𝜃𝐸|𝑥(𝑠)|²𝐿𝑑𝑠+2𝜏_𝑏+𝑈_𝑏−𝐿_𝑏^2𝑡_𝑡+𝜃𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠. (26) By assumption (H2), we see that 𝐸^𝑡_𝑡+𝜃𝜎(𝑥(𝑠),𝑥(𝑠−𝜏),𝑠)𝑑𝑊(𝑠)²≤𝛼^𝑡_𝑡+𝜃𝐸|𝑥(𝑠)|²𝑑𝑠+𝛽^𝑡_𝑡+𝜃𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠. (27) By (25)–(27) and noting the definitions of 𝑘₁, 𝑘₂, we can obtain 𝐸|𝐻(𝑡,𝑥(𝑡))|²≤𝑘₁^𝑡_𝑡+𝜃𝐸|𝑥(𝑠)|²𝑑𝑠+𝑘₂^𝑡_𝑡+𝜃𝐸|𝑥(𝑠−𝜏)|²𝑑𝑠,𝑡≥𝜏. (28) By (28) and integrating both sides, we have ^𝑡₀𝑒^𝑟𝑠𝐸|𝐻(𝑥(𝑠),𝑠)|²𝑑𝑠≤sup_{−𝜏≤𝜃≤0}𝜓(𝜃)+𝑘₁^𝑡_𝜏𝑒^𝑟𝑠𝑠_𝑠+𝜃𝐸|𝑥(𝑢)|²𝑑𝑢𝑑𝑠+𝑘₂^𝑡_𝜏𝑒^𝑟𝑠𝑠_𝑠+𝜃𝐸|𝑥(𝑢−𝜏)|²𝑑𝑢𝑑𝑠. (29) Moreover, we can obtain ^𝑡_𝜏𝑒^𝑟𝑠𝑠_𝑠+𝜃𝐸|𝑥(𝑢−𝜏)|²𝑑𝑢𝑑𝑠≤𝜏𝑒^{2𝑟𝜏𝑡}₀𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²𝑑𝑠+𝜏²𝑒^2𝑟𝜏sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²,_𝑡𝜏𝑒^𝑟𝑠𝑠_𝑠+𝜃𝐸|𝑥(𝑢)|²𝑑𝑢𝑑𝑠≤𝜏𝑒^𝑟𝜏𝑡₀𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²𝑑𝑠. (30) Substituting (30) into (29) and noting the definitions of 𝑙₁, 𝑙₂, and 𝜓(𝜃), (23) holds. The proof of Lemma 6 is completed.

By (22) and Lemma 6, another criterion for system (2) will be given.

Theorem 7. System (2) is exponentially robustly stable in mean square. If there exist positive scalars 𝜀₁, 𝜀₂, and 𝜀₃, such that 𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏+𝑚₁+𝑚₂+𝑚₃<0, (31) here 𝑚₁=1+𝜀₁+𝜀₁⁻¹𝑈_𝑎−𝐿_𝑎²+𝜀₂+𝜀₃𝑚+𝛼,₂=𝜀₃⁻¹𝑈_𝑏−𝐿_𝑏²𝑚+𝛽,₃=𝜀₂⁻¹𝑙₁+𝑙₂𝜏𝜆_max𝐿^𝑇_𝑏𝐿_𝑏. (32) Then, for all 𝜉∈𝐿²_𝐹0([−𝜏,0];𝑅^𝑛), the following inequality holds: 𝐸𝑥(𝑡,𝜉)²≤1+𝑚₂𝜏𝑒^𝑟𝜏+𝜀₂⁻¹𝑙₂𝜏²𝑒^2𝑟𝜏𝜆_max𝐿^𝑇_𝑏𝐿_𝑏×sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+𝜀₂⁻¹𝜆_max𝐿^𝑇_𝑏𝐿_𝑏sup_{−𝜏≤𝜃≤0}𝜓(𝜃)𝑒^−𝑟𝑡, (33) where 𝑟 is the unique positive solution of the following equation: 𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏+𝑚₁+𝑚₂𝑒^𝑟𝜏+𝜀₂⁻¹𝑙₁𝜏𝑒^𝑟𝜏+𝑙₂𝜏𝑒^2𝑟𝜏𝜆_max𝐿^𝑇_𝑏𝐿_𝑏=0. (34)

Proof. Similar to the proof process of Theorem 5, we can derive 𝐿𝑉(𝑥(𝑡),𝑡)≤𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏+𝑚₁×𝑒^𝑟𝑡𝑥^𝑇(𝑡)𝑥(𝑡)+𝑚₂𝑒^𝑟𝑡𝑥^𝑇(𝑡−𝜏)𝑥(𝑡−𝜏)+𝜀₂⁻¹𝜆_max𝐿^𝑇_𝑏𝐿_𝑏𝑒^𝑟𝑡|𝐻(𝑥(𝑡),𝑡)|². (35) Using Itô’s differential formula and integrating and by the definition of 𝐻(𝑥,𝑡), we obtain 𝐸𝑉(𝑥(𝑡),𝑡)≤𝑟+𝜆_max𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏+𝑚₁+𝑚₂𝑒^𝑟𝑡+𝜀₂⁻¹𝑙₁𝜏𝑒^𝑟𝜏+𝑙₂𝜏𝑒^2𝑟𝜏×𝜆_max𝐿^𝑇_𝑏𝐿_𝑏×^𝑡₀𝑒^𝑟𝑠𝐸|𝑥(𝑠)|²+𝑑𝑠1+𝑚₂𝜏𝑒^𝑟𝜏+𝜀₂⁻¹𝑙₂𝜏²𝑒^2𝑟𝜏𝜆_max𝐿^𝑇_𝑏𝐿_𝑏sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+𝜀₂⁻¹𝜆_max𝐿^𝑇_𝑏𝐿_𝑏sup_{−𝜏≤𝜃≤0}𝜓(𝜃). (36) If 𝑔(𝑟)=𝑟+𝜆_max(𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏)+𝑚₁+𝑚₂𝑒^𝑟𝜏+𝜀₂⁻¹(𝑙₁𝜏𝑒^𝑟𝜏+𝑙₂𝜏𝑒^2𝑟𝜏)𝜆_max(𝐿^𝑇_𝑏𝐿_𝑏), then 𝑔^′(𝑟)=1+𝑚₂𝜏𝑒^𝑟𝜏+𝜀₂⁻¹(𝑙₁𝜏²𝑒^𝑟𝜏+2𝑙₂𝜏²𝑒^2𝑟𝜏)𝜆_max(𝐿^𝑇_𝑏𝐿_𝑏).
Because 𝑔^′(𝑟)>0, 𝑔(0)=𝜆_max(𝐿_𝑎+𝐿^𝑇_𝑎+𝐿_𝑏+𝐿^𝑇_𝑏)+𝑚₁+𝑚₂+𝑚₃ and 𝑔(+∞)=+∞.
If (31) holds, (34) must have a unique solution 𝑟>0.
Therefore, we can easily get𝐸𝑉(𝑥(𝑡),𝑡)≤1+𝑚₂𝜏𝑒^𝑟𝜏+𝜀₂⁻¹𝑙₂𝜏²𝑒^2𝑟𝜏𝜆_max𝐿^𝑇_𝑏𝐿_𝑏×sup_{−𝜏≤𝜃≤0}𝐸𝜉(𝜃)²+𝜀₂⁻¹𝜆_max𝐿^𝑇_𝑏𝐿_𝑏sup_{−𝜏≤𝜃≤0}𝜓(𝜃), (37) or equivalently (33) holds; the proof of Theorem 7 is completed.

4. Examples

Consider a stochastic time-delay system (𝑡)=[𝐴(⊗)𝑥(𝑡)(⊗)𝑥((𝑥(𝑡)())(𝑡),𝑥𝑑𝑥+𝐵𝑡−0.5)]𝑑𝑡+𝜎,𝑥𝑡−0.5,𝑡𝑑𝑊₀=𝜉,𝜉∈𝐿²_𝐹0[−0.5,0];𝑅²,−0.5≤𝑡≤0, (38) where 𝐿_𝑎=−3.350.220.23−3.34;𝑈_𝑎=𝐿−3.150.320.31−3.45_𝑏=−1.150.200.23−1.16;𝑈_𝑏=.−1.120.220.31−1.09 (39)

Respectively, 𝐿_𝑎, 𝑈_𝑎, 𝐿_𝑏, and 𝑈_𝑏 are the lower bound and upper bound matrices of 𝐴(⊗) and 𝐵(⊗).

In addition, 𝜎(𝑥(𝑡)())=1,𝑥𝑡−0.5,𝑡2𝑥₁(𝑡)sin𝑥₂()1𝑡−0.52𝑥₂(𝑡)sin𝑥₁().𝑡−0.5 (40) Clearly, Trace𝜎^𝑇(𝑥(𝑡),𝑥(𝑡−0.5),𝑡)𝜎(𝑥(𝑡),𝑥(𝑡−0.5),𝑡)≤0.25𝑥²(𝑡). (41) By using the method of [21], we can obtain that 𝑟=1.7513 or 𝑟=1.8162, which indicates that the system (38) is exponentially stable in mean square.

5. Conclusion

In this paper, we have investigated a class of grey stochastic systems with time delay; by constructing a suitable Lyapunov-Krasovskii functional combined with Itô’s differential formula, two improved exponential stability criteria are derived. The criteria obtained in this paper are so conveniently verified that the results in this paper should be proved to be very useful in applications.

Conflict of Interests

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

Acknowledgments

The research is supported by the National Natural Science Foundation of China (no. 11301009) and the Natural Science Foundation of Henan Province (no. 0511013800).