Robust Stochastic Stability Analysis for Uncertain Neutral-Type Delayed Neural Networks Driven by Wiener Process

Abstract

The robust stochastic stability for a class of uncertain neutral-type delayed neural networks driven by Wiener process is investigated. By utilizing the Lyapunov-Krasovskii functional and inequality technique, some sufficient criteria are presented in terms of linear matrix inequality (LMI) to ensure the stability of the system. A numerical example is given to illustrate the applicability of the result.

1. Introduction

In the past few years, neural networks and their various generalizations have drawn much research attention owing to their promising potential applications in a variety of areas, such as robotics, aerospace, telecommunications, pattern recognition, image processing, associative memory, signal processing, and combinatorial optimization [1–3]. In such applications, it is of prime importance to ensure the asymptotic stability of the designed neural networks. Because of this, the stability of neural networks has been deeply investigated in the literature [4–14].

It is known that time delays and stochastic perturbations are commonly encountered in the implementation of neural networks, and may result in instability or oscillation. So it is essential to investigate the stability of delayed stochastic neural networks [15, 16]. Moreover, uncertainties are unavoidable in practical implementation of neural networks due to modeling errors and parameter fluctuation, which also cause instability and poor performance [15, 17, 18]. Therefore, it is significant to introduce such uncertainties into delayed stochastic neural networks.

On the other hand, because of the complicated dynamic properties of the neural cells in the real world, it is natural and important that systems will contain some information about the derivative of the past state. Practically, such phenomenon always appears in the study of automatic control, circuit analysis, chemical process simulation, and population dynamics, and so forth. Recently, there has been increasing interest in the study of delayed neural networks of neutral type, see [6–15, 18–24]. In [6, 8], the authors developed the global asymptotic stability of neutral-type neural networks with delays by utilizing the Lyapunov stability theory and LMI technique. In [9, 10], the global exponential stability of neutral-type neural networks with distributed delays is studied. However, the stochastic perturbations were not taken into account in those delayed neural networks [6–10].

In [23, 24], the authors discussed the robust stability for uncertain stochastic neural networks of neutral-type with time-varying delays. However, the distributed delays were not taken into account in the models. So far, there are only a few papers that not only deal with the stochastic stability analysis for delayed neural networks of neutral-type, but also consider the parameter uncertainties.

To the best of our knowledge, there are very few results on the stochastic stability analysis for uncertain neutral-type neural networks with both discrete and distributed delays driven by Wiener process. This motivates the research in this paper.

In this paper, a class of uncertain neutral-type delayed neural networks driven by Wiener process is considered. By constructing a suitable Lyapunov functional, some new stability criteria to guarantee the system to be stochastically asymptotically stable in the mean square are given, which are less conservative than some existing reports. The structure of the addressed system is more general than in the other papers. The criteria can be checked easily by the LMI control toolbox in MATLAB. Moreover, a numerical example is given to illustrate the effectiveness and improvement over some existing results.

2. Preliminaries

Notations 2. 𝐀<𝟎 denotes that 𝐀 is a negative definite matrix. The superscript “𝑇” stands for the transpose of a matrix. (Ω,𝐹,𝑃) denotes a complete probability space, E(⋅) stands for the mathematical expectation operator. ‖⋅‖ stands for the Euclidean norm. 𝐈 is the identity matrix of appropriate dimension, and the symmetric terms in a symmetric matrix are denoted by ∗.

Consider the following class of uncertain neutral-type delayed neural networks driven by Wiener process: d[𝐱(𝑡)−𝐂𝐱(𝑡−ℎ(𝑡))]=+−𝐇𝐀(𝑡)𝐱(𝑡)+𝐁(𝑡)𝐟(𝐱(𝑡−𝜏(𝑡)))+𝐃(𝑡)^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠d𝑡₀𝐱𝑡(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡),₀+𝑠=𝜑(𝑠),𝑠∈𝑡₀−𝜌,𝑡₀,(2.1) where 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇 is the neuron state vector, 𝐀(𝑡)=𝐀+Δ𝐀(𝑡), 𝐁(𝑡)=𝐁+Δ𝐁(𝑡), 𝐃(𝑡)=𝐃+Δ𝐃(𝑡), 𝐇₀(𝑡)=𝐇₀+Δ𝐇₀(𝑡), 𝐇₁(𝑡)=𝐇₁+Δ𝐇₁(𝑡), 𝐀=diag(𝑎_𝑖)_𝑛×𝑛 is a positive diagonal matrix, 𝐁,𝐂,𝐃∈𝐑^𝑛×𝑛 are the connection weight matrices, 𝐇₀,𝐇₁∈𝐑^𝑛×𝑛 are known real constant matrices, Δ𝐀(𝑡),Δ𝐁(𝑡),Δ𝐃(𝑡),Δ𝐇₀(𝑡),Δ𝐇₁(𝑡) represent the time-varying parameter uncertain terms. 𝐟(𝐱)=(𝑓₁(𝑥₁),𝑓₂(𝑥₂),…,𝑓_𝑛(𝑥_𝑛))^𝑇 is the neuron activation function with 𝐟(0)=0. 𝐰(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Wiener process defined on a complete probability space (Ω,𝐹,𝑃). 𝑟(𝑡),𝜏(𝑡),ℎ(𝑡) are nonnegative, bounded, and differentiable time varying delays satisfying0<𝜏(𝑡)≤0<𝑟(𝑡)≤𝑟<∞,0<ℎ(𝑡)≤𝜏<∞,𝜏.(𝑡)≤𝜂₁<∞,ℎ<∞,ℎ̇(𝑡)≤𝜂₂<∞.(2.2)

The admissible parameter uncertain terms are assumed to be the following form:Δ𝐀(𝑡),Δ𝐁(𝑡),Δ𝐃(𝑡),Δ𝐇₀(𝑡),Δ𝐇₁(𝑡)=𝐔𝐅(𝑡)𝐌₁,𝐌₂,𝐌₃,𝐌₄,𝐌₅,(2.3) where 𝐔,𝐌_𝑖, 𝑖=1,…,5 are known real constant matrices, 𝐅(𝑡) is the time-varying uncertain matrix satisfying 𝐅^𝑇𝐅≤𝐈.(2.4)

Suppose that 𝐟(⋅) is bounded and satisfies the following condition:‖𝐟(𝐱)‖≤‖𝐆𝐱‖,(2.5) where 𝐆∈𝐑^𝑛×𝑛 is a known constant matrix.

Assume that the initial value 𝜑∶[−𝜌,0]→𝐑^𝑛 is 𝐹₀-measurable and continuously differentiable, we introduce the following norm:‖𝜑‖^2𝜌=maxsup_{−𝛼≤𝑠≤0}E||𝜑_𝑖(𝑠)||²,sup_{−ℎ≤𝑠≤0}E||𝜑^𝑖(𝑠)||²<∞,(2.6) where 𝜌=max{𝜏,ℎ,𝑟}, 𝛼=max{𝜏,𝑟}.

Under the above assumptions, it is easy to verify that there exists a unique equilibrium point of system (2.1) (see [25]).

Definition 2.1. The equilibrium point of (2.1) is said to be globally robustly stochastically asymptotically stable in the mean square, if the following condition holds: lim_𝑡→+∞E||𝐱𝑡,𝑡₀,𝜑||²=0,𝑡≥𝑡₀,(2.7) where 𝐱(𝑡,𝑡₀,𝜑) is any solution of model (2.1) with initial value 𝜑.

Lemma 2.2 (Schur complement [26]). Given constant matrices Ω₁,Ω₂,Ω₃ with appropriate dimensions, where Ω^𝑇1=Ω₁ and Ω^𝑇2=Ω₂>0, then 𝛀₁+𝛀^𝑇3𝛀²⁻¹𝛀₃<0,(2.8) if and only if ⎛⎜⎜⎝𝛀₁𝛀^𝑇3∗−𝛀₂⎞⎟⎟⎠<0,or⎛⎜⎜⎝−𝛀₂𝛀₃∗𝛀₁⎞⎟⎟⎠<0.(2.9)

Lemma 2.3 (see [26]). Given matrices 𝐃,𝐄, and 𝐅 with 𝐅^𝑇𝐅≤𝐈 and a scalar 𝜀>0, then 𝐃𝐅𝐄+𝐄^𝑇𝐅^𝑇𝐃^𝑇≤𝜀𝐃𝐃^𝑇+𝜀⁻¹𝐄^𝑇𝐄.(2.10)

Lemma 2.4 (see [27]). For any constant matrix 𝐌∈𝐑^𝑛×𝑛, 𝐌=𝐌^𝑇>0, a scalar 𝛾>0, vector function 𝑥(𝑡)∶[0,𝛾]→𝐑^𝑛 such that the integrations are well defined, then ^𝛾0𝑥(𝑠)d𝑠^𝑇𝐌^𝛾0𝑥(𝑠)d𝑠≤𝛾^𝛾0𝑥^𝑇(𝑠)𝐌𝑥(𝑠)d𝑠.(2.11)

3. Main Results

Theorem 3.1. System (2.1) is globally robustly stochastically asymptotically stable in the mean square, if there exist symmetric positive definite matrices 𝐏,𝐐,𝐑,𝐒,𝐔₁,𝐔₂ and positive scalars 𝛿,𝜀₁,𝜀₂>0 such that LMI holds: Λ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Γ¹∗𝐀^𝑇Γ𝐏₂𝐂𝐏𝐁−𝜀₁𝐌^𝑇1𝐌₂𝐏𝐃−𝜀₁𝐌^𝑇1𝐌₃𝜀₂𝐌^𝑇4𝐌₅𝐇^𝑇0𝐏−𝐏𝐔0∗∗−𝐂Γ^𝑇₃𝐏𝐁−𝐂^𝑇𝐏𝐃00−𝐂^𝑇𝐏𝐔0∗∗∗𝜀₁𝐌Γ^𝑇2₄𝐌₃0000∗∗∗∗Γ0₅000∗∗∗∗∗∗∗∗∗∗𝐇−∗^𝑇1𝐏𝐏−𝜀00₁𝐏0𝐔∗∗∗∗∗∗∗𝐈−𝜀0₂𝐈⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,(3.1) where Γ₁=−𝐏𝐀−𝐀^𝑇𝐏+𝐐+𝐑+𝑟𝐆^𝑇𝐒𝐆+𝜀₁𝐌^𝑇1𝐌₁+𝜀₂𝐌^𝑇4𝐌₄, Γ₂=−𝐔₁−(1−𝜂₂)𝐑, Γ₃=−𝛿𝐈+𝜀₁𝐌^𝑇2𝐌₂, Γ₄=−𝑟⁻¹𝐒+𝜀₁𝐌^𝑇3𝐌₃, Γ₅=−𝐔₂−(1−𝜂₁)𝐐+𝛿𝐆^𝑇𝐆+𝜀₂𝐌^𝑇5𝐌₅.

Proof. Using Lemma 2.2, the matrix Λ<0 implies that ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Γ∗₁𝐀^𝑇Γ𝐏₂𝐂𝐏𝐁−𝜀₁𝐌^𝑇1𝐌₂𝐏𝐃−𝜀₁𝐌^𝑇1𝐌₃𝜀₂𝐌^𝑇4𝐌₅𝐇^𝑇0𝐏∗∗−𝐂Γ^𝑇₃𝐏𝐁−𝐂^𝑇𝐏𝐃00∗∗∗𝜀₁𝐌Γ^𝑇2₄𝐌₃00∗∗∗∗Γ0₅0∗+∗⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏_𝑇𝐔∗0∗∗𝐇−^𝑇1𝐏𝐏⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠0000𝐏𝐔𝐏0000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎝𝜀₁₋₁∗𝐈𝜀0₂₋₁𝐈⎞⎟⎟⎠⎛⎜⎜⎝𝐔^𝑇0𝐏−𝐔0^𝑇𝐏𝐂000000𝐔0_𝑇=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Φ₁𝐏⎞⎟⎟⎠∗𝐀^𝑇Γ𝐏₂𝐂𝐏𝐁𝐏𝐃0𝐇^𝑇0𝐏∗∗∗∗−𝐂−∗^𝑇𝛿𝐏𝐈𝐁−−𝐂0^𝑇𝐏𝐃0000∗∗∗𝑟∗⁻¹𝐒Φ0₂0∗∗∗∗∗𝐇−^𝑇1𝐏𝐏⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+𝜀¹⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌^𝑇1𝐌0_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌^𝑇1𝐌0_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀₂⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀¹⁻¹⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏_𝑇𝐔0000𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏0000^𝑇𝐔𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀²⁻¹⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇<0,(3.2) where Φ₁=−𝐏𝐀−𝐀^𝑇𝐏+𝐐+𝐑+𝑟𝐆^𝑇𝐒𝐆,  Φ₂=−𝐔₂−(1−𝜂₁)𝐐+𝛿𝐆^𝑇𝐆.
From (2.3), (2.4), using Lemma 2.3, we have ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐏Δ𝐀(𝑡)−∗Δ𝐀^𝑇(𝑡)𝐏Δ𝐀^𝑇0(𝑡)𝐏𝐂−𝐂𝐏_𝑇Δ𝐁(𝑡)𝐏Δ𝐃(𝑡)0Δ𝐇^𝑇0(𝑡)𝐏∗∗∗∗∗∗𝐏∗∗0Δ𝐁(𝑡)−𝐂^𝑇𝐏∗00Δ𝐃(𝑡)0000Δ𝐇_𝑇1000=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌∗_𝑇1∗∗∗∗0(𝑡)𝐏⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐌0_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐅^𝑇(𝑡)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏^𝑇𝐔0000𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏^𝑇𝐔0000𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐅(𝑡)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌𝐌0^𝑇1_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐅^𝑇(𝑡)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐅(𝑡)⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇≤𝜀₁⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌^𝑇1𝐌0_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐌^𝑇1𝐌0_𝑇2𝐌^𝑇300⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀¹⁻¹⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏_𝑇𝐔0000𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−𝐂𝐏0000^𝑇𝐔𝐏𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀₂⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐌^𝑇4𝐌000_𝑇50⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇+𝜀²⁻¹⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐏00000𝐔⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇.(3.3) Together with (3.2), we get ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Ψ∗𝐀^𝑇(Γ𝑡₂)𝐏𝐂𝐏𝐁(𝑡)𝐏𝐃(𝑡)0𝐇^𝑇0(𝑡)𝐏∗∗∗∗−𝐂−^𝑇∗𝐏𝛿𝐁𝐈(𝑡)−𝐂−^𝑇𝐏0𝐃(𝑡)0000∗∗∗𝑟∗⁻¹𝐒Φ0₂0∗∗∗∗∗𝐇^𝑇1−(𝐏𝑡)𝐏⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠<0,(3.4) where Ψ=−𝐏𝐀(𝑡)−𝐀^𝑇(𝑡)𝐏+𝐐+𝐑+𝑟𝐆^𝑇𝐒𝐆.
Utilizing Lemma 2.2 again, we obtain Σ=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝Ψ𝐀^𝑇∗(Γ𝑡₂)𝐏𝐂𝐏𝐁(𝑡)𝐏𝐃(𝑡)0∗∗∗∗−𝐂−^𝑇∗𝐏𝛿𝐁𝐈(𝑡)−𝐂−^𝑇𝐏0𝐃(𝑡)00∗∗∗𝑟∗⁻¹𝐒Φ0₂⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐇𝐇^𝑇0000(𝑡)_𝑇1(𝑡)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠𝐏⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐇^𝑇0𝐇000(𝑡)_𝑇1(𝑡)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠^𝑇<0.(3.5)
Constructing a positive definite Lyapunov-Krasovskii functional as follows: 𝑉(𝑡,𝐱(𝑡))=+𝐲^𝑇(𝑡)𝐏𝐲(𝑡)+^{𝑡𝑡−𝜏(𝑡)}𝐱^𝑇(𝑠)𝐐𝐱(𝑠)d𝑠+^{𝑡𝑡−ℎ(𝑡)}𝐱^𝑇(𝑠)𝐑𝐱(𝑠)d𝑠_{0−𝑟(𝑡)}^{𝑡𝑡+𝜃}+𝐟^𝑇(𝐱(𝑠))𝐒𝐟(𝐱(𝑠))d𝑠d𝜃+^𝑇𝑡𝐱^𝑇(𝑠−ℎ(𝑠))𝐔₁𝐱(𝑠−ℎ(𝑠))d𝑠_𝑇𝑡𝐱^𝑇(𝑠−𝜏(𝑠))𝐔₂𝐱(𝑠−𝜏(𝑠))d𝑠,(3.6) where 𝑦(𝑡)=𝐱(𝑡)−𝐂𝐱(𝑡−ℎ(𝑡)), 𝑇>𝑡 is a constant.
By Ito’s differential formula, we get d𝑉(𝑡,𝐱(𝑡))≤+2𝐱𝐲^𝑇_𝑇(𝑡)𝐏−𝐀(𝑡)𝐱(𝑡)+𝐁(𝑡)𝐟(𝐱(𝑡−𝜏(𝑡)))+𝐃(𝑡)^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠−1−(𝑡)ℎ𝐐̇(𝐱𝑡)(𝑡𝐱)_𝑇−1−𝜏.(𝑡)𝐱^𝑇(𝑡−𝜏(𝑡))𝐐𝐱(𝑡−𝜏(𝑡))+𝐱^𝑇(𝑡)𝐑𝐱(𝑡)−(𝑡−ℎ(𝑡))𝐑𝐱(𝑡−ℎ(𝑡))+𝑟𝐟^𝑇(𝐱(𝑡))𝐒𝐟(𝐱(𝑡))_{𝑡𝑡−𝑟(𝑡)}−𝐱_𝑇𝐟^𝑇(𝐱(𝑠))𝐒𝐟(𝐱(𝑠))d𝑠+𝐇(𝑡−ℎ(𝑡))𝐔₁𝐱(𝑡−ℎ(𝑡))−𝐱^𝑇(𝑡−𝜏(𝑡))𝐔₂𝐱(𝑡−𝜏(𝑡))₀+2𝐲_𝑇(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))^𝑇𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝑡≤2[𝐱((𝑡𝑡))𝐏−𝐂𝐇𝐱₀((𝑡𝑡)−𝐱(ℎ𝑡)(𝑡+))]𝐇_1𝑇(𝑡)𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡)+×𝐏𝐱_𝑇−𝐀(𝑡)𝐱(𝑡)+𝐁(𝑡)𝐟(𝐱(𝑡−𝜏(𝑡)))+𝐃(𝑡)^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠−1−(𝑡)𝜂𝐐₂𝐱(𝑡)−1−𝜂₁𝐱^𝑇(𝑡−𝜏(𝑡))𝐐𝐱(𝑡−𝜏(𝑡))+𝐱^𝑇(𝑡)𝐑𝐱(𝑡)−𝐱^𝑇(𝑡−ℎ(𝑡))𝐑𝐱(𝑡−ℎ(𝑡))+𝑟𝐟^𝑇(𝐱(𝑡))𝐒𝐟(𝐱(𝑡))_{𝑡𝑡−𝑟(𝑡)}−𝐱_𝑇𝐟^𝑇(𝐱(𝑠))𝐒𝐟(𝐱(𝑠))d𝑠+𝐇(𝑡−ℎ(𝑡))𝐔₁𝐱(𝑡−ℎ(𝑡))−𝐱^𝑇(𝑡−𝜏(𝑡))𝐔₂𝐱(𝑡−𝜏(𝑡))₀+2𝐲_𝑇(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))^𝑇𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝑡(𝑡)𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡).(3.7) From (2.5), for a scalar 𝛿>0, we have −𝛿𝐟^𝑇(𝐱(𝑡−𝜏(𝑡)))𝐟(𝐱(𝑡−𝜏(𝑡)))−𝐱^𝑇(𝑡−𝜏(𝑡))𝐆^𝑇𝐆𝐱(𝑡−𝜏(𝑡))≥0.(3.8) Using Lemma 2.4, we have ^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠^𝑇𝑟⁻¹𝐒^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠≤^{𝑡𝑡−𝑟(𝑡)}𝐟^𝑇(𝐱(𝑠))𝐒𝐟(𝐱(𝑠))d𝑠.(3.9) Together (3.8), (3.9) with d𝑉(𝑡,𝐱(𝑡)), we obtain d𝑉(𝑡,𝐱(𝑡))≤+𝐱𝐱^𝑇_𝑇(𝑡)−𝐏𝐀(𝑡)−𝐀^𝑇(𝑡)𝐏+𝐐+𝐑+𝑟𝐆^𝑇𝐒𝐆𝐱(𝑡)+𝐱^𝑇(𝑡)𝐀^𝑇(𝑡)𝐏𝐂𝐱(𝑡−ℎ(𝑡))+𝐟_𝑇(𝑡−ℎ(𝑡))𝐂^𝑇𝐏𝐀(𝑡)𝐱(𝑡)+𝐱^𝑇(𝑡)𝐏𝐁(𝑡)𝐟(𝐱(𝑡−𝜏(𝑡)))+𝐱_𝑇(𝐱(𝑡−𝜏(𝑡)))𝐁^𝑇(𝑡)𝐏𝐱(𝑡)+𝐱_𝑇(𝑡)𝐏𝐃(𝑡)^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠+^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠^𝑇𝐃^𝑇(𝑡)𝐏𝐱(𝑡)−𝐟_𝑇(𝑡−ℎ(𝑡))−𝐔₁−1−𝜂₂𝐑𝐱(𝑡−ℎ(𝑡))−𝐱^𝑇(𝑡−ℎ(𝑡))𝐂^𝑇𝐏𝐁(𝑡)𝐟(𝐱(𝑡−𝜏(𝑡)))−(𝐱(𝑡−𝜏(𝑡)))𝐁^𝑇(𝑡)𝐏𝐂𝐱(𝑡−ℎ(𝑡))−𝐱^𝑇(𝑡−ℎ(𝑡))𝐂^𝑇𝐏𝐃(𝑡)^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠_{𝑡𝑡−𝑟(𝑡)}×−𝐔₂𝐟(𝐱(𝑠))d𝑠^𝑇𝐃^𝑇(𝑡)𝐏𝐂𝐱(𝑡−ℎ(𝑡))+𝐱^𝑇(𝑡−𝜏(𝑡))−−1−𝜂₁𝐐𝐱(𝑡−𝜏(𝑡))_{𝑡𝑡−𝑟(𝑡)}+𝐇𝐟(𝐱(𝑠))d𝑠^𝑇𝑟⁻¹𝐒^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠₀+2𝐲_𝑇(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))^𝑇𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝑡(𝑡)𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡).(3.10) That is, d𝑉(𝑡,𝐱(𝑡))≤𝜉^𝑇(𝑡)Σ𝜉(𝑡)d𝑡+2𝐲^𝑇(𝑡)𝐏𝐇₀(𝑡)𝐱(𝑡)+𝐇₁(𝑡)𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡),(3.11) where 𝜉^𝑇(𝑡)=(𝐱^𝑇(𝑡),𝐱^𝑇(𝑡−ℎ(𝑡)),𝐟^𝑇(𝐱(𝑡−𝜏(𝑡))),(∫^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠)^𝑇,𝐱^𝑇(𝑡−𝜏(𝑡))), and the matrix Σ is given in (3.5).
Taking the mathematical expectation, we get Ed𝑉(𝑡,𝐱(𝑡))d𝑡≤E𝜉^𝑇(𝑡)Σ𝜉(𝑡)≤𝜆_max(Σ)E‖𝐱(𝑡)‖².(3.12) From (3.5), we know Σ<0, that is, 𝜆_max(Σ)<0. By Lyapunov-Krasovskii stability theorems, the system (2.1) is globally robustly asymptotically stable. The proof is completed.

Remark 3.2. To the best of our knowledge, few authors have considered the stochastically asymptotic stability for uncertain neutral-type neural networks driven by Wiener process. We can find recent papers [18, 22–24]. However, it is assumed in [18] that the system is a linear model and all delays are constants. In [22], it is assumed that the time-varying delays satisfying 𝜏.(𝑡)≤𝜌_𝜏<1, ℎ̇(𝑡)≤𝜌_ℎ<1, in this paper, we relax it to 𝜏.(𝑡)≤𝜌_𝜏<∞,ℎ̇(𝑡)≤𝜌_ℎ<∞. In [23, 24], the authors discussed the robust stability for uncertain stochastic neural networks of neutral-type with time-varying delays. However, the distributed delays were not taken into account in the models. Hence, our results in this paper have wider adaptive range.

Remark 3.3. Suppose that 𝐂=0,𝐃(𝑡)=0 (i.e., without neutral-type and distributed delays), then the system (2.1) becomes the one investigated in [15].

Remark 3.4. In [17], the authors studied the global stability for uncertain stochastic neural networks with time-varying delay by Lyapunov functional method and LMI technique. However, the neutral term and distributed delays were not taken into account in the models. Therefore, our developed results in this paper are more general than those reported in [17].

Remark 3.5. It should be noted that the condition is given as linear matrix inequalities LMIs, therefore, by using the MATLAB LMI Toolbox, it is straightforward to check the feasibility of the condition.

4. Numerical Example

Consider the following uncertain neutral-type delayed neural networks:d[𝐱(𝑡)−𝐂𝐱(𝑡−ℎ(𝑡))]=+−𝐃𝐀++𝐔𝐔𝐅𝐅((𝑡𝑡))𝐌𝐌₁₃𝐱(𝑡)+𝐁+𝐔𝐅(𝑡)𝐌₂𝐟(𝐱(𝑡−𝜏(𝑡)))+𝐔𝐅(𝑡)𝐌₄^{𝑡𝑡−𝑟(𝑡)}𝐟(𝐱(𝑠))d𝑠d𝑡𝐱(𝑡)+𝐔𝐅(𝑡)𝐌₅𝐱(𝑡−𝜏(𝑡))d𝐰(𝑡),(4.1) where 𝑛=2, 𝑓_𝑖(𝑥_𝑖)=sin𝑥_𝑖, 𝑖=1,2, 𝜂₁=0.7, 𝜂₂=0.5, 0<𝑟(𝑡)≤𝑟=3, 𝐅^𝑇(𝑡)𝐅(𝑡)≤𝐈.

The constant matrices are𝐃=𝐀⎛⎜⎜⎝−=00.⎛⎜⎜⎝0.3004203⎞⎟⎟⎠00..,0035⎞⎟⎟⎠,𝐁=𝐔⎛⎜⎜⎝00=.0.24⎛⎜⎜⎝0000..15..106800⎞⎟⎟⎠..53,⎞⎟⎟⎠,𝐂𝐌=₁⎛⎜⎜⎝00.200.2⎞⎟⎟⎠,𝐌=⎛⎜⎜⎝00.600.6₂⎞⎟⎟⎠,=𝐌₃=𝐌₄=⎛⎜⎜⎝00.200.2⎞⎟⎟⎠,𝐌₅=⎛⎜⎜⎝00.400.4⎞⎟⎟⎠,𝐆=⎛⎜⎜⎝1001⎞⎟⎟⎠.(4.2) By using the MATLAB LMI Control Toolbox, we obtain the feasible solution as follows: 𝛿=2.0876, 𝜀₁=5.0486,  𝜀₂=8.0446,𝐏𝐒==⎛⎜⎜⎝⎛⎜⎜⎝−−6300..8.4.78492168545473−−6100..6.9.78052118542873⎞⎟⎟⎠⎞⎟⎟⎠,,𝐐𝐔=₁⎛⎜⎜⎝−90.9.2377912−100..42378982⎞⎟⎟⎠,𝐑=⎛⎜⎜⎝−82.0.3190346−52.4.3999316⎞⎟⎟⎠,=⎛⎜⎜⎝10..1318181901..31808690⎞⎟⎟⎠,𝐔₂=⎛⎜⎜⎝−20.4.6159631−20.6.6257601⎞⎟⎟⎠.(4.3) That is the system (4.1) is globally robustly stochastically asymptotically stable in the mean square.

5. Conclusion

In this paper, the stochastically asymptotic stability problem has been studied for a class of uncertain neutral-type delayed neural networks driven by Wiener process by utilizing the Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach. A numerical example is given to illustrate the applicability of the result.

Acknowledgment

This paper was fully supported by the National Natural Science Foundation of China (no. 10771199 and no. 10871117).