Robust Adaptive Exponential Synchronization of Stochastic Perturbed Chaotic Delayed Neural Networks with Parametric Uncertainties

Abstract

This paper investigates the robust adaptive exponential synchronization in mean square of stochastic perturbed chaotic delayed neural networks with nonidentical parametric uncertainties. A robust adaptive feedback controller is proposed based on Gronwally’s inequality, drive-response concept, and adaptive feedback control technique with the update laws of nonidentical parametric uncertainties as well as linear matrix inequality (LMI) approach. The sufficient conditions for robust adaptive exponential synchronization in mean square of uncoupled uncertain stochastic chaotic delayed neural networks are derived in terms of linear matrix inequalities (LMIs). The effect of nonidentical uncertain parameter uncertainties is suppressed by the designed robust adaptive feedback controller rapidly. A numerical example is provided to validate the effectiveness of the proposed method.

1. Introduction

Synchronization of chaotic delayed neural networks has been an intensive topic because of its promising connections with many disciplines, such as image encryption [1], image processing [2], harmonic oscillation generation [3], and secure communications [4–6]. The diverse control schemes have been proposed for the synchronization of chaotic delayed neural networks, for example, adaptive control (see, [7–13]), slide mode control [14, 15], coupling control [16], feedback control [17–21], impulsive control [22–25], and sampled-data control [26–28].

The existence of random uncertainties such as stochastic noise in the electrical circuits design of neural networks possesses an important source in what may change or destroy the synchronization. Therefore, the stochastic effects must be taken into consideration for the synchronization problem of chaotic delayed neural networks. Some works on the synchronization of stochastic perturbed chaotic neural networks have been reported in the literature; see [8, 9, 17, 29, 30] and the references therein. The authors in [29] were concerned with the problem of exponential synchronization for stochastic jumping chaotic neural networks (SJCNNs) with mixed delays and sector nonlinearities by employed Lyapunov-Krasovskii functional and free-weighting matrix method and proposed a delay-dependent feedback controller with sector nonlinearities to achieve the synchronization in mean square in terms of linear matrix inequalities (LMIs). In [9], Zhu and Cao have derived some novel sufficient conditions achieving complete synchronization of unidirectionally coupled stochastic delayed neural networks by utilizing LaSalle invariant principle of stochastic differential delay equations and the stochastic analysis as well as the adaptive feedback control technique and LMI approach. Li et al. in [30] investigated the synchronization problem of a class of chaotic neural networks with time-varying delays and unbounded distributed delays under stochastic perturbations via Lyapunov-Krasovskii functional, drive-response concept, output coupling with delay feedback, and LMI approach, some sufficient conditions in terms of LMIs ensuring the exponential synchronization of the addressed neural networks are derived.

On the other hand, besides stochastic noise, it is well known that the effects of parametric uncertainties which may also destroy the stability of the controlled system cannot be ignored in many applications. However, all of the abovementioned works mainly focus on the stochastic perturbed chaotic delayed neural networks without parametric uncertainties. According to the best of our knowledge, there are still few results about the synchronization of stochastic perturbed chaotic delayed neural networks with nonidentical parametric uncertainties. This is the motivation of our research in the present paper.

In this paper, the main aim is to design a robust adaptive feedback controller with the update laws of nonidentical parametric uncertainties and find some sufficient conditions in order to guarantee exponential synchronization in mean square for uncoupled chaotic delayed neural networks with stochastic perturbation and parametric uncertainties. Based on Gronwally’s inequality, drive-response concept, adaptive feedback control technique, and linear matrix inequality (LMI) approach, several sufficient conditions in the form of linear matrix inequalities (LMIs) are derived to ensure exponential synchronization in mean square for uncoupled uncertain stochastic chaotic delayed neural networks. In addition, the existence of the desired controller can be validated by MATLAB LMI toolbox efficiently. The significant difference from previous results is that the nonidentical parametric uncertainties are entered into both the connection weight matrix and the delayed connection weight matrix in the drive-response systems. The task for compensating the nonidentical parametric uncertainties can be realized by the designed robust adaptive feedback controller rapidly. Moreover, we have pointed out that the LaSalle invariance principle for stochastic differential delay equation [31, Corollary 3.1] cannot be applied for the stability analysis of stochastic delayed systems without a trivial solution 𝑒(𝑡;0)=0. Finally, a numerical example and its simulation are given to illustrate the usefulness of the given method.

Notation. Let ℝ denote the set of real numbers, let ℝ⁺ denote the set of all nonnegative real numbers, and ℝ^𝑛 and ℝ^𝑛×𝑚 denote the 𝑛-dimensional and 𝑛×𝑚 dimensional real spaces equipped with the Euclidean norm; ‖𝑧‖ is the Euclidean norm of the vector 𝑧. ℕ denotes the set of positive integers. For any matrix 𝑋∈ℝ^𝑛×𝑛, 𝑋>0 denotes that 𝑋 is a symmetric and positive definite matrix. If 𝑋₁ and 𝑋₂ are symmetric matrices, then 𝑋₁≤𝑋₂ means that 𝑋₁−𝑋₂ is a negative semidefinite matrix. 𝑋^𝑇 and 𝑋⁻¹ mean the transpose of 𝑋 and the inverse of a square matrix. 𝐼 denotes the identity matrix with appropriate dimensions. Let 𝜏>0 and 𝐶([−𝜏,0];ℝ^𝑛) denote the family of all continuous ℝ^𝑛-valued functions 𝜉(𝜃) on [−𝜏,0] with the norm ‖𝜉‖=sup_{−𝜏≤𝜃≤0}|𝜉(𝜃)|. Let 𝐶^𝑏_ℱ0([−𝜏,0];ℝ^𝑛)(𝑡≥0) denote the family of all ℱ₀ measurable bounded 𝐶([−𝜏,0];ℝ^𝑛)-valued random variables 𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0}, such that ∫⁰_−𝜏𝐸|𝜉(𝑠)|²𝑑𝑠<∞, where 𝐸{⋅} stands for the correspondent expectation operator with respect to the given probability measure 𝒫. Let 𝐶^1,2(ℝ⁺×ℝ^𝑛;ℝ⁺) denote the family of all nonnegative functions 𝑉(𝑡,𝑥) on ℝ⁺×ℝ^𝑛 which are continuously twice differentiable in 𝑥 and differentiable in 𝑡. The notation ⋆ always denotes the symmetric block in one symmetric matrix.

2. Model Description and Preliminaries

Consider the following chaotic delayed neural networks with time-varying parametric uncertainties: 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴+Δ𝐴₁(𝑡)𝑓(𝑥(𝑡+))𝐵+Δ𝐵₁(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐽𝑑𝑡, (1) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the neuron state vector, 𝐶 is a positive diagonal matrix, 𝐴=(𝑎_𝑖𝑗)∈ℝ^𝑛×𝑛 and 𝐵=(𝑏_𝑖𝑗)∈ℝ^𝑛×𝑛 are the connection weight matrix and the delayed connection weight matrix, respectively, Δ𝐴₁(𝑡),Δ𝐵₁(𝑡) are the time-varying parametric uncertainties, 𝐽 is the constant input vector, 𝑓(𝑥(⋅))=(𝑓₁(𝑥₁(⋅)),𝑓₂(𝑥₂(⋅)),…,𝑓_𝑛(𝑥_𝑛(⋅)))^𝑇 is the nonlinear neuron activation function which describes the behavior in which the neurons respond to each other, and the time delay 𝜏(𝑡) is bounded and continuously differentiable function; that is, there exist two positive constants 𝜏 and 𝜇, such that 0≤𝜏(𝑡)≤𝜏 and 0≤𝜏̇(𝑡)≤𝜇<1.

For drive system (1), we construct the following response system with nonidentical parametric uncertainties and stochastic perturbation: 𝑑𝑦(𝑡)=−𝐶𝑦(𝑡)+𝐴+Δ𝐴₂(𝑡)𝑓𝑦(𝑡)+𝐵+Δ𝐵₂(𝑡)𝑓𝑦((𝑡(𝑡)(𝑡)(𝑡)((𝑡((𝑡(𝑡),𝑡−𝜏))+𝑢+𝐽𝑑𝑡+𝜎𝑡,𝑦−𝑥,𝑦𝑡−𝜏))−𝑥𝑡−𝜏))𝑑𝜔 (2) where 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇 is the neuron state vector of the response system; 𝜔(𝑡)=(𝜔₁(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brown motion defined on a complete probability space (Ω,ℱ,𝒫) with a natural filtration {ℱ_𝑡}_𝑡≥0 (i.e., ℱ_𝑡=𝜎{𝜔(𝑠)∶0≤𝑠≤𝑡}), which satisfies 𝐸{𝑑𝜔(𝑡)}=0 and 𝐸{[𝑑𝜔(𝑡)]²}=𝑑𝑡; 𝜎∶ℝ⁺×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛×𝑛 is the noise intensity matrix; 𝑢(𝑡) is the robust adaptive controller which will be designed to compensate the nonidentical parametric uncertainties.

Let 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡) be the synchronization error; then we can obtain the error system between systems (1) and (2) as follows: 𝑑𝑒(𝑡)=−𝐶𝑒(𝑡)+𝐴+Δ𝐴₂(𝑡)ℎ(𝑒(𝑡+))𝐵+Δ𝐵₂(𝑡)ℎ(𝑒((𝑡+𝑡−𝜏)))Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)𝑓(𝑥(𝑡+))Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)𝑓(𝑥((𝑡(𝑡((𝑡)((𝑡(𝑡),𝑡−𝜏)))+𝑢)𝑑𝑡+𝜎𝑡,𝑒,𝑒𝑡−𝜏)))𝑑𝜔 (3) where ℎ(𝑒(⋅))=𝑓(𝑒(⋅)+𝑥(⋅))−𝑓(𝑥(⋅)).

Throughout this paper, we assume that(H1)[32] each neuron activation function 𝑓_𝑖∶ℝ→ℝ is continuous with 𝑓_𝑖(0)=0(𝑖=1,2,…,𝑛), and there exist positive scalars 𝑙_𝑖, such that 𝑓_𝑖(𝑎)−𝑓_𝑖(𝑏)≤𝑙_𝑖|𝑎−𝑏|, (4) for any 𝑎, 𝑏∈ℝ, 𝑎≠𝑏, and 𝐿=diag(𝑙₁,𝑙₂,…,𝑙_𝑛);(H2)the noise matrix 𝜎(⋅) is local Lipschitz continuous and satisfies the linear growth condition as well, and 𝜎(𝑡,0,0)=0. Moreover, there exist two real positive 𝑛×𝑛 matrices 𝑄₁, 𝑄₂, such that trace𝜎^𝑇(𝑡,𝑢,𝑣)𝜎(𝑡,𝑢,𝑣)≤𝑢^𝑇𝑄₁𝑢+𝑣^𝑇𝑄₂𝑣, (5) for all 𝑢, 𝑣∈ℝ^𝑛;(H3)the parametric uncertainties Δ𝐴_𝑗(𝑡), Δ𝐵_𝑗(𝑡), 𝑗=1,2 are in terms of Δ𝐴_𝑗(𝑡)=𝐸_𝑗𝐹_𝑗(𝑡)𝐻_𝑗,Δ𝐵_𝑗(𝑡)=𝐸^′_𝑗𝐹^′_𝑗(𝑡)𝐻^′_𝑗, (6) where 𝐸_𝑗, 𝐸^′_𝑗, 𝐻_𝑗, and 𝐻^′_𝑗 are known real constant matrices with appropriate dimensions and 𝐹_𝑗(𝑡) and 𝐹^′_𝑗(𝑡) are the uncertain matrices which are Lebesque measurable in 𝑡 and satisfy 𝐹^𝑇_𝑗(𝑡)𝐹_𝑗(𝑡)≤𝐼 and 𝐹_𝑗^′𝑇(𝑡)𝐹^′_𝑗(𝑡)≤𝐼;(H4)there exist unknown constants 𝛼>0, 𝛽>0, such that Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)≤𝛼,Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)≤𝛽. (7)

Remark 1. Considering Hypothesis (H1) and the boundedness of the chaotic signals, one can get that there exist positive constants 𝑙=max_{1≤𝑖≤𝑛}𝑙_𝑖 and ‖𝑥‖≤𝝌, such that 𝑓(𝑥(𝑡))≤𝑙𝝌,𝑓(𝑥(𝑡−𝜏(𝑡)))≤𝑙𝝌. (8)

Remark 2. Let 𝑒(𝑡;𝜉) denote the state trajectory from the initial data 𝑒(𝜃)=𝜉(𝜃) on −𝜏≤𝜃≤0 in 𝐶²_ℱ0([−𝜏,0];ℝ^𝑛). Although 𝜎(𝑡,0,0)=0 and 𝑓_𝑖(0)=0, system (3) would not admit a unique solution or trivial solution 𝑒(𝑡;0)=0. Thus, Hypothesis (H2) in [31] is not satisfied. Therefore, the LaSalle invariance principle for stochastic differential delay equation [31, Corollary 3.1] cannot be applied for the stability analysis of the error system (3) and other stochastic delayed systems without a trivial solution 𝑒(𝑡;0)=0, which implies that the methods proposed in [26, 33, 34] are debatable. In order to release the restriction of the nonexistence of a trivial solution, we turn to utilize Gronwally’s inequality to ensure the rigorousness of mathematical proof.

The following definition and lemmas are useful for future derivations.

Definition 3. If the function 𝑉∈𝐶^1,2(ℝ⁺×ℝ^𝑛;ℝ⁺), then an operator 𝔏𝑉 from ℝ⁺×ℝ^𝑛 along the trajectory of the error system (3) is defined as 𝔏𝑉(𝑡,𝑒(𝑡))=𝑉_𝑡(𝑡,𝑒(𝑡))+𝑉_𝑒((𝑡×(𝑡)+𝑡,𝑒))−𝐶𝑒𝐴+Δ𝐴₂(𝑡)ℎ(𝑒(𝑡+))𝐵+Δ𝐵₂(𝑡ℎ(𝑒((𝑡+)𝑡−𝜏)))Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)𝑓(𝑥(𝑡+))Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)𝑓(𝑥((𝑡(𝑡+1𝑡−𝜏)))+𝑢)2trace𝜎^𝑇𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡))𝑉_𝑒𝑒×(𝑡,𝑒(t))𝜎^𝑇(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)), (9) where 𝑉_𝑡((𝑡=((𝑡𝑡,𝑒))𝜕𝑉𝑡,𝑒)),𝑉𝜕𝑡_𝑒((𝑡=((𝑡𝑡,𝑒))𝜕𝑉𝑡,𝑒))𝜕𝑒₁((𝑡,…,𝜕𝑉𝑡,𝑒))𝜕𝑒_𝑛,𝑉_𝑒𝑒((𝑡=𝜕𝑡,𝑒))²𝑉(𝑡,𝑒(𝑡))𝜕𝑒_𝑖𝜕𝑒_{𝑗𝑛×𝑛}. (10)

Definition 4. Systems (1) and (2) are said to be exponentially synchronized in mean square if there exist constants 𝜆>0 and ℳ>0, such that 𝐸‖𝑒(𝑡)‖²≤ℳexp(−𝜆𝑡) and 𝑡>0, where 𝜆 is called the decay rate of exponential synchronization.

Lemma 5 (Gronwally’s inequality (see [35])). Let 𝑇>0 and 𝑢(⋅) be a Borel measurable bounded nonnegative function on [0,𝑇]. If 𝑢(𝑡)≤𝑐+𝑣^𝑡₀𝑢(𝑠)𝑑𝑠,∀0≤𝑡≤𝑇, (11) for some constants 𝑐, 𝑣, then 𝑢(𝑡)≤𝑐exp(𝑣𝑡),∀0≤𝑡≤𝑇. (12)

Lemma 6 (see [36]). If 𝑋 and 𝑌 are real matrices with appropriate dimensions, then there exists a number 𝜀>0, such that 𝑋^𝑇𝑌+𝑌^𝑇𝑋≤𝜀𝑋^𝑇1𝑋+𝜀𝑌^𝑇𝑌. (13)

3. Main Results

In this section, the robust adaptive exponential synchronization in mean square for the drive system (1) and response system (2) is studied under Hypotheses (H1)–(H4). The robust adaptive feedback controller is designed as 𝑢(𝑡)=−𝐾𝑒(𝑡)−𝑙𝝌𝑒^𝑇𝑃⁻¹(𝑡)𝛼𝑒𝑃𝑒−𝑙𝝌^𝑇𝑃⁻¹𝑃𝑒(𝑡)𝛽, (14) where 𝐾 is the feedback gain matrix to be scheduled; 𝛼 and 𝛽 are the estimation of the bounds 𝛼 and 𝛽, respectively. The update laws 𝛼 and 𝛽 are designed as 𝛼̇=𝑘_𝛼𝑙𝝌𝑒^𝑇𝑃,𝛽̇=𝑘_𝛽𝑙𝝌𝑒^𝑇𝑃, (15) with 𝑘_𝛼, 𝑘_𝛽>0.

Theorem 7. Assume that Hypotheses (H1)–(H4) hold. If there exist an 𝑛×𝑛 feedback gain matrix 𝐾, two 𝑛×𝑛 symmetric matrices 𝑃>0 and 𝑅>0, and four positive constants 𝜌, 𝜃₁, 𝜃₂, and 𝜅, such that the following conditions hold: (C1)𝑃≤𝜌𝐼;(C2)Ω₁₁𝑃𝐴𝑃𝐸₂𝑃𝐵𝑃𝐸^′₂1⋆−𝜃₁1𝐼000⋆⋆−𝜃₁𝐻₂²100⋆⋆⋆−𝜃₂1𝐼0⋆⋆⋆⋆−𝜃₂𝐻^′₂²2≤0,𝜃₂𝐿²+𝜌𝑄₂−1−𝜇𝑅≤0, (16) where Ω₁₁=𝑃(−𝐾−𝐶)+(−𝐾−𝐶)^𝑇2𝑃+𝜃₁𝐿²+𝑅+𝜌𝑄₁+𝜅𝐼, (17) then the drive system (1) and response system (2) can be exponentially synchronized in mean square.

Proof. Construct a Lyapunov functional in the form of 𝑉(𝑡,𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃𝑒(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇(𝑠)Re(𝑠)𝑑𝑠+𝑘_𝛼⁻¹𝛼²+𝑘_𝛽⁻¹𝛽², (18) where 𝛼=𝛼−𝛼,𝛽=𝛽−𝛽.
From Definition 3, we obtain the operator 𝔏𝑉(𝑡,𝑒(𝑡))𝔏𝑉(𝑡,𝑒(𝑡))=2𝑒^𝑇(𝑡)𝑃−𝐶𝑒(𝑡)+𝐴+Δ𝐴₂(𝑡)ℎ(𝑒(𝑡+))𝐵+Δ𝐵₂(𝑡)ℎ(𝑒((𝑡+𝑡−𝜏)))Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)𝑓(𝑥(𝑡+))Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)𝑓(𝑥((𝑡(𝑡)𝜎𝑡−𝜏)))+𝑢+trace^𝑇((𝑡)((𝑡((𝑡)((𝑡𝑡,𝑒,𝑒𝑡−𝜏)))×𝑃𝜎𝑡,𝑒,𝑒𝑡−𝜏)))+𝑒^𝑇(𝑡)Re(𝑡)−(1−𝜏̇(𝑡))×𝑒^𝑇((𝑡((𝑡−𝑡−𝜏))Re𝑡−𝜏))2𝑘_𝛼⁻¹𝛼𝛼̇+2𝑘_𝛽⁻¹𝛽̇𝛽=𝑒^𝑇(𝑡)−𝑃𝐶−𝐶^𝑇𝑃+𝑅𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑃𝐴+Δ𝐴₂(𝑡)ℎ(𝑒(𝑡))+2𝑒^𝑇(𝑡)𝑃𝐵+Δ𝐵₂(𝑡)ℎ(𝑒(𝑡−𝜏(𝑡)))+2𝑒^𝑇(𝑡)𝑃Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)𝑓(𝑥(𝑡))+2𝑒^𝑇(𝑡)𝑃Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)×𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑒^𝑇(𝑡)(𝑡)𝜎𝑃𝑢+trace^𝑇((𝑡)((𝑡((𝑡)((𝑡−(𝜏̇(𝑡𝑒𝑡,𝑒,𝑒𝑡−𝜏)))×𝑃𝜎𝑡,𝑒,𝑒𝑡−𝜏)))1−))^𝑇((𝑡((𝑡−𝑡−𝜏))Re𝑡−𝜏))2𝑘_𝛼⁻¹𝛼𝛼̇+2𝑘_𝛽⁻¹𝛽̇𝛽. (19)
It is clear from Lemma 6 that 2𝑒^𝑇(𝑡)𝑃𝐴+Δ𝐴₂(𝑡)ℎ(𝑒(𝑡))≤𝜃₁𝑒^𝑇(𝑡)𝑃𝐴𝐴^𝑇(𝑡)+1𝑃𝑒𝜃₁ℎ^𝑇(𝑒(𝑡))ℎ(𝑒(𝑡))+𝜃₁𝑒^𝑇(𝑡)𝑃𝐸₂𝐹₂(𝑡)𝐻₂𝐻^𝑇₂𝐹^𝑇₂(𝑡)𝐸^𝑇₂(𝑡)+1𝑃𝑒𝜃₁ℎ^𝑇(𝑒(𝑡))ℎ(𝑒(𝑡))≤𝜃₁𝑒^𝑇(𝑡)𝑃𝐴𝐴^𝑇(𝑡)+2𝑃𝑒𝜃₁𝑒^𝑇(𝑡)𝐿²𝑒(𝑡)+𝜃₁𝐻₂²𝑒^𝑇(𝑡)𝑃𝐸₂𝐸^𝑇₂𝑃𝑒(𝑡)=𝑒^𝑇(𝑡)𝜃₁𝑃𝐴𝐴^𝑇𝑃+𝐻₂²𝑃𝐸₂𝐸^𝑇₂𝑃+2𝜃₁𝐿²𝑒(𝑡),2𝑒^𝑇(𝑡)𝑃𝐵+Δ𝐵₂(𝑡)ℎ(𝑒(𝑡−𝜏(𝑡)))≤𝜃₂𝑒^𝑇(𝑡)𝑃𝐵𝐵^𝑇(𝑡)+1𝑃𝑒𝜃₂ℎ^𝑇(𝑒(𝑡−𝜏(𝑡)))ℎ(𝑒(𝑡−𝜏(𝑡)))+𝜃₂𝑒^𝑇(𝑡)𝑃𝐸^′₂𝐹^′₂(𝑡)𝐻^′₂𝐻₂^′𝑇𝐹₂^′𝑇(𝑡)𝐸₂^′𝑇(𝑡)+1𝑃𝑒𝜃₂ℎ^𝑇(𝑒(𝑡−𝜏(𝑡)))ℎ(𝑒(𝑡−𝜏(𝑡)))≤𝜃₂𝑒^𝑇(𝑡)𝑃𝐵𝐵^𝑇(𝑡)+2𝑃𝑒𝜃₂𝑒^𝑇(𝑡−𝜏(𝑡))𝐿²𝑒(𝑡−𝜏(𝑡))+𝜃₂𝐻^′₂²𝑒^𝑇(𝑡)𝑃𝐸^′₂𝐸₂^′𝑇𝑃𝑒(𝑡)=𝑒^𝑇(𝑡)𝜃₂𝑃𝐵𝐵^𝑇𝑃+𝐻^′₂²𝑃𝐸^′₂𝐸₂^′𝑇𝑃𝑒(𝑡)+2𝜃₂𝑒^𝑇(𝑡−𝜏(𝑡))𝐿²𝑒(𝑡−𝜏(𝑡)),2𝑒^𝑇(𝑡)𝑃Δ𝐴₂(𝑡)−Δ𝐴₁(𝑡)𝑓(𝑥(𝑡))+2𝑒^𝑇(𝑡)𝑃Δ𝐵₂(𝑡)−Δ𝐵₁(𝑡)𝑓(𝑥((𝑡𝑒𝑡−𝜏)))≤2^𝑇𝑃𝛼𝑙𝝌+2𝑒^𝑇𝑃𝑒𝛽𝑙𝝌=2^𝑇𝑃𝑙𝝌𝛼+𝛽. (20) From Hypothesis (H3), we have trace𝜎^𝑇𝑃𝜎≤𝜆_max(𝑃)trace𝜎^𝑇𝜎≤𝜌𝑒^𝑇(𝑡)𝑄₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏(𝑡)). (21) Substituting (14) and (15), (20) and (21) to (19), we obtain 𝔏𝑉(𝑡,𝑒(𝑡))≤𝑒^𝑇(𝑡)Ω₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))Ω₂𝑒((𝑡𝑒𝑡−𝜏))+2^𝑇𝑃𝑙𝝌𝛼+𝛽−2𝑒^𝑇(𝑡)𝑃𝑙𝝌𝑒^𝑇𝑃⁻¹(𝑡)𝛼𝑒𝑃𝑒+𝑙𝝌^𝑇𝑃⁻¹𝑃𝑒(𝑡)𝛽−2𝑘_𝛼⁻¹𝛼𝑘_𝛼𝑙𝝌𝑒^𝑇𝑃−2𝑘_𝛽⁻¹𝛽𝑘_𝛽𝑙𝝌𝑒^𝑇𝑃=𝑒^𝑇(𝑡)Ω₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))Ω₂𝑒((𝑡𝑒𝑡−𝜏))+2^𝑇𝑃𝑙𝝌𝛼+𝛽−𝛼−𝛽−2𝑒^𝑇𝑃𝑙𝝌𝛼+𝛽=𝑒^𝑇(𝑡)Ω₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))Ω₂𝑒(𝑡−𝜏(𝑡))=𝑒^𝑇(𝑡)Ω₁+𝜅𝐼𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))Ω₂𝑒(𝑡−𝜏(𝑡))−𝜅𝑒^𝑇(𝑡)𝑒(𝑡)=𝑒^𝑇(𝑡)Ω^′₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))Ω₂𝑒(𝑡−𝜏(𝑡))−𝜅𝑒^𝑇(𝑡)𝑒(𝑡), (22) where Ω₁=𝑃(−𝐾−𝐶)+(−𝐾−𝐶)^𝑇𝑃+𝑅+𝜌𝑄₁+2𝜃₁𝐿²+𝜃₁𝑃𝐴𝐴^𝑇𝑃+𝐻₂²𝑃𝐸₂𝐸^𝑇₂𝑃+𝜃₂𝑃𝐵𝐵^𝑇𝑃+𝐻^′₂²𝑃𝐸^′₂𝐸₂^′𝑇𝑃,Ω^′₁=Ω₁Ω+𝜅𝐼,₂=2𝜃₂𝐿²+𝜌𝑄₂−1−𝜇𝑅. (23)
Considering condition (𝐶2) of Theorem 7 and Schur complement [37], we get 𝔏𝑉(𝑡,𝑒(𝑡))≤−𝜅𝑒^𝑇(𝑡)𝑒(𝑡). (24)
It follows from Itô’s formula that ((𝑡((0=𝐸𝑉𝑡,𝑒))−𝐸𝑉0,𝑒))_𝑡0𝐸𝔏𝑉≤−𝜅^𝑡₀𝐸𝑒(𝑠)^𝑇𝑒(𝑠)𝑑𝑠,𝑡>0, (25) which means that 𝜆_min(𝑃)𝐸‖𝑒(𝑡)‖²((𝑡((0≤𝐸𝑉𝑡,𝑒))≤𝐸𝑉0,𝑒))−𝜅^𝑡₀𝐸‖𝑒(𝑠)‖²𝑑𝑠. (26) By using Lemma 5, we have 𝐸‖𝑒(𝑡)‖²≤𝑐exp(𝑣𝑡), (27) where 𝑐=𝐸𝑉(0,𝑒(0))/𝜆_min(𝑃), 𝑣=−𝜅/𝜆_min(𝑃).
On the other hand, we denote 𝐸𝑉(0,𝑒(0))=𝐸𝑒^𝑇(0)(0)+𝑃𝑒_0(0)−𝜏𝐸𝑒^𝑇(𝑠)Re(𝑠)+𝑘_𝛼⁻¹𝛼²(0)+𝑘_𝛽⁻¹𝛽²(0)≤𝜆_max(𝑃)𝐸sup_{−𝜏≤𝜃≤0}𝜉(𝜃)²+𝜏𝜆_max(𝑅)𝐸sup_{−𝜏≤𝜃≤0}𝜉(𝜃)²+𝑘_𝛼⁻¹𝛼²(0)+𝑘_𝛽⁻¹𝛽²(0)=𝜆_max(𝑃)+𝜏𝜆_max(𝑅)𝐸sup_{−𝜏≤𝜃≤0}𝜉(𝜃)²+𝑘_𝛼⁻¹𝛼²(0)+𝑘_𝛽⁻¹𝛽²(0). (28) Combining (27) and (28), we finally obtain 𝐸‖𝑒(𝑡)‖²≤ℳexp−𝜅𝜆_min(𝑃)𝑡, (29) where ℳ=𝑐=𝜆_max(𝑃)+𝜏𝜆_max(𝑅)𝐸sup_{−𝜏≤𝜃≤0}𝜉(𝜃)²+𝑘_𝛼⁻¹𝛼²(0)+𝑘_𝛽⁻¹𝛽²(0)×𝜆_min(𝑃)⁻¹. (30) Therefore, by Definition 4 we see that the drive system (1) and response system (2) can be exponentially synchronized in mean square with a decay rate 𝜆=𝜅/𝜆_min(𝑃). The proof of Theorem 7 is completed.

Let 𝑌=𝑃𝐾; the feedback gain matrix 𝐾 can be calculated by 𝐾=𝑃⁻¹𝑌; then, the following Theorem holds.

Theorem 8. Assume that Hypotheses (H1)–(H4) hold. If there exist an 𝑛×𝑛 matrix 𝑌, two 𝑛×𝑛 symmetric matrices 𝑃>0 and 𝑅>0, and four positive constants 𝜌, 𝜃₁, 𝜃₂, and 𝜅, such that the following conditions hold (𝐶1)𝑃≤𝜌𝐼;(𝐶2^′)Ω^′₁₁𝑃𝐴𝑃𝐸₂𝑃𝐵𝑃𝐸^′₂1⋆−𝜃₁1𝐼000⋆⋆−𝜃₁𝐻₂²100⋆⋆⋆−𝜃₂1𝐼0⋆⋆⋆⋆−𝜃₂𝐻^′₂²2≤0,𝜃₂𝐿²+𝜌𝑄₂−1−𝜇𝑅≤0, (31) where Ω^′₁₁=−𝑃𝐶−𝑌−𝐶^𝑇𝑃−𝑌^𝑇+2𝜃₁𝐿²+𝑅+𝜌𝑄₁+𝜅𝐼, (32) then the drive system (1) and response system (2) can be exponentially synchronized in mean square.

Substituting 𝐾=𝑃⁻¹𝑌 to Theorem 7, the above result is easy to be obtained. So the proof of Theorem 8 is omitted.

Remark 9. Our method does not need to construct a complex Lyapunov-Krasovskii function which contains quadratic integral terms and triple integral terms. Thus, the amount of calculation is greatly reduced. Furthermore, we do not require that the matrix 𝑃>0 is diagonal (see [8, 9, 30]) just only being symmetric. This partly shows the less conservativeness of our control strategy. If the drive system (1) and response system (2) have no nonidentical parametric uncertainties, the corresponding synchronization problems have been addressed in [8, 9, 29, 30]. Therefore, our results are more general than those given in [8, 9, 29, 30].

By constructing another Lyapunov functional 𝑉^′(𝑡,𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃𝑒(𝑡)+∫^𝑡_{𝑡−𝜏(𝑡)}𝑒^𝑇(𝑠)Re(𝑠)𝑑𝑠, we can derive the following corollaries by Theorem 7 and Theorem 8, respectively.

Corollary 10. Assume that Hypotheses (H1)–(H4) hold, and the feedback controller is designed as 𝑢(𝑡)=−𝐾(𝑦(𝑡)−𝑥(𝑡)). If there exist an 𝑛×𝑛 feedback gain matrix 𝐾, two 𝑛×𝑛 symmetric matrices 𝑃>0 and 𝑅>0, and four positive constants 𝜌, 𝜃₁, 𝜃₂, and 𝜅, such that the following conditions hold (𝑖)𝑃≤𝜌𝐼;(𝑖𝑖)Ξ₁₁1𝑃𝐴𝑃𝐵⋆−𝜃₁1𝐼0⋆⋆−𝜃₂𝐼2≤0,𝜃₂𝐿²+𝜌𝑄₂−1−𝜇𝑅≤0, (33) where Ξ₁₁=𝑃(−𝐾−𝐶)+(−𝐾−𝐶)^𝑇𝑃+2𝜃₁𝐿²+𝑅+𝜌𝑄₁+𝜅𝐼, (34) then the drive system (1) and response system (2) can be exponentially synchronized in mean square.

Corollary 11. Assume that Hypotheses (H1)–(H4) hold, and the feedback controller is designed as 𝑢(𝑡)=−𝐾(𝑦(𝑡)−𝑥(𝑡)). If there exist an 𝑛×𝑛 matrix 𝑌, two 𝑛×𝑛 symmetric matrices 𝑃>0 and 𝑅>0, and four positive constants 𝜌, 𝜃₁, 𝜃₂, and 𝜅>0, such that the following conditions hold(𝑖)𝑃≤𝜌𝐼;(𝑖𝑖^′)Ξ^′₁₁1𝑃𝐴𝑃𝐵⋆−𝜃₁1𝐼0⋆⋆−𝜃₂𝐼2≤0,𝜃₂𝐿²+𝜌𝑄₂−1−𝜇𝑅≤0, (35) where Ξ^′₁₁=−𝑃𝐶−𝑌−𝐶^𝑇𝑃−𝑌^𝑇+2𝜃₁𝐿²+𝑅+𝜌𝑄₁+𝜅𝐼,𝑌=𝑃𝐾, (36) then the drive system (1) and response system (2) can be exponentially synchronized in mean square.

The proof of Corollaries 10 and 11 is direct, so it is omitted.

Remark 12. Suppose response system (2) without noise perturbation; then, the synchronization issue of the drive system (1) and response system (2) converted to the synchronization of chaotic delayed neural networks with parametric uncertainties, which has been investigated in [11, 23]. In addition, the authors in [33] only studied the time invariance parametric uncertainties. Subsequently, our method has a wider application range than those results in [11, 23, 33].

4. Numerical Results

In this section, a numerical example and its simulations are presented to demonstrate the effectiveness of the proposed scheme in the previous sections. Throughout the simulations, we use the Euler-Maruyama method [38] with step 10⁻³.

Example 13. Consider two-dimensional chaotic delayed neural networks [39] as the drive system: 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴+Δ𝐴₁(𝑡)𝑓(𝑥(𝑡+))𝐵+Δ𝐵₁(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐽𝑑𝑡, (37) where the initial condition 𝑥(𝑠)=(0.1,0.1)^𝑇,𝑠∈[−𝜏,0], 𝑓(𝑥)=0.5(|𝑥+1|−|𝑥−1|), and 𝜏(𝑡)=1,𝐽=(0,0)^𝑇; then, from Hypothesis (H1) and Remark 1, it is easy to verify that 𝐿₁=(1,1)^𝑇, 𝐿₂=(0,0)^𝑇, and 𝑙=1. The parameter matrices 𝐶, 𝐴, and 𝐵 and the parametric uncertainties are given as follows: 𝜋𝐶=1001,𝐴=1+4𝜋200.11+4,−√𝐵=2×1.3𝜋4√0.10.1−2×1.3𝜋4,Δ𝐴₁(𝑡)(𝑡)0(𝑡)=0.04∗,sin0sinΔ𝐵₁(𝑡)(𝑡)0(𝑡)=0.04∗;cos0cos (38) that is, 𝐹₁(𝑡)=(𝑡)0(𝑡),𝐹sin0sin_′1(𝑡)=(𝑡)0(𝑡),𝐸cos0cos₁=𝐻₁=𝐸^′₁=𝐻^′₁=0.2𝐼. (39)
The corresponding response system is taken as 𝑑𝑦(𝑡)=−𝐶𝑦(𝑡)+𝐴+Δ𝐴₂(𝑡)𝑓𝑦(𝑡)+𝐵+Δ𝐵₂(𝑡)𝑓𝑦((𝑡(𝑡)(𝑡)(𝑡)((𝑡((𝑡(𝑡),𝑡−𝜏))+𝑢+𝐽𝑑𝑡+𝜎𝑡,𝑦−𝑥,𝑦𝑡−𝜏))−𝑥𝑡−𝜏))𝑑𝜔 (40) where the initial condition 𝑦(𝑠)=(0.2,−0.2)^𝑇, 𝑠∈[−𝜏,0], the noise perturbation 𝜎((𝑡)((𝑡√𝑡,𝑒,𝑒𝑡−𝜏)))=diag0.005𝑒₁(𝑡),√0.025𝑒₂(𝑡)+diag√0.25𝑒₁(𝑡−𝜏(𝑡)),√0.005𝑒₂(𝑡−𝜏(𝑡)), (41) and 𝜔(𝑡) is a 2-dimensional Brownian motion satisfying 𝐸{𝑑𝜔(𝑡)}=0 and 𝐸{[𝑑𝜔(𝑡)]²}=𝑑𝑡; then, from Hypothesis (H2), it is easy to get 𝑄₁=diag(0.01,0.05) and 𝑄₂=diag(0.5,0.01). And the parametric uncertainties are given as Δ𝐴₂(𝑡)(𝑡)0(𝑡)=−0.04∗,sin0sinΔ𝐵₂(𝑡)(𝑡)0(𝑡)=−0.04∗;cos0cos (42) that is, 𝐹₂(𝑡)(𝑡)0(𝑡),𝐹=−sin0sin_′2(𝑡)(𝑡)0(𝑡),𝐸=−cos0cos₂=𝐻₂=𝐸^′₂=𝐻^′₂=0.2𝐼. (43) The robust adaptive feedback controller 𝑢(𝑡) satisfies (14). The update laws 𝛼 and 𝛽 are designed as (15).
By using MATLAB LMI toolbox, we can obtain the following feasible solutions to LMIs in Theorem 8: ,,𝑃=0.0040−0.0065−0.00650.0772,𝑌=0.52060.00190.00190.4607𝑅=0.4307−0.0002−0.00020.3292𝜌=0.2734,𝜃₁𝜃=6.1797,₂=9.4490,𝜅=0.0842. (44) Thus, we can calculate the feedback gain matrix 𝐾 and the decay rate, which are given as follows: 𝐾=𝑃⁻¹,𝜅𝑌=149.124411.601112.51586.9422𝜆=𝜆_min(𝑃)=24.2757. (45) We choose 𝝌=16, 𝑘_𝛼=𝑘_𝛽=0.1, 𝛼(0)=0.079, and 𝛽(0)=0.079. Figures 1(a)–1(d) show the time response of state variables and the phase plots of systems (37) and (40) without the robust adaptive feedback controller 𝑢(𝑡) designed in (14). Figures 2(a)–2(d) depict the time response of state variables, the synchronization errors, and the phase plots of systems (37) and (40) with the robust adaptive feedback controller 𝑢(𝑡) designed in (14), from which we can see that the drive system (37) and response system (40) can be exponentially synchronized in mean square. Figures 3(a) and 3(b) display the estimated parameters 𝛼 and 𝛽 converge asymptotically to some constants, which show the effectiveness of the proposed robust adaptive synchronization scheme.

(a)

(b)

(c)

(d)

Figure 1 

Time response of state variables and the phase plots of systems (37) and (40) without the robust adaptive feedback controller 𝑢(𝑡).

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(b)

(c)

(d)

Figure 2 

Time response of state variables, the synchronization errors, and the phase plots of systems (37) and (40) with the robust adaptive feedback controller 𝑢(𝑡).

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Figure 3 

The curves of the estimated parameters 𝛼 and 𝛽.

Remark 14. The simulation results in Example 13 show that the effect of the nonidentical parametric uncertainties between the drive system (37) and response system (40) can be suppressed rapidly via the designed robust adaptive feedback controller, which means that the proposed strategy has strong robustness against the nonidentical parametric uncertainties. Meanwhile, we have a relatively large decay rate 𝜆.

Remark 15. Synchronization is encountered in various fields of science, in engineering and in social behavior [40, 41]. In fact, synchronization problem of networks system belongs to the category of control, which is an important branch of synthetic theory in model control field. The authors in [42] dealt with the synchronization problem of coupled switched neural networks with mode-dependent impulsive effects and time delays by using switching analysis techniques and a comparison principle. Zhang et al. in [43] have investigated the synchronization problem for a class of nonlinear delayed dynamical networks with heterogeneous impulsive effects. The distributed synchronization problem in networks of agent systems with controllers and nonlinearities subject to Bernoulli switchings has been studied in [44], in which the advantage of distributed adaptive controllers over conventional adaptive controllers has also been validated. The abovementioned literature is interesting synthesis problems of networks and will become our future investigative directions.

5. Conclusion

In this paper, we have studied robust exponential adaptive synchronization of stochastic perturbed chaotic delayed neural networks with nonidentical parametric uncertainties. A robust adaptive feedback controller has been designed to equalize the effect of the nonidentical parametric uncertainties. A numerical example has also been exploited to depict the usefulness of the obtained results. The nonidentical parametric uncertainties, which have been taken into account, exhibited the main advantage of the proposed scheme. The simulation results confirmed that our method has a high robustness property against parameter uncertainties mismatch.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their careful comments and suggestions to improve the quality of the paper. This work is jointly supported by Opening Fund of Geomathematics Key Laboratory of Sichuan Province under Grant sc-sxdz2011010 and the scientific research Fund of Sichuan University of Science and Engineering under Grant 2011PY08.