Exponential Synchronization for Stochastic Neural Networks with Mixed Time Delays and Markovian Jump Parameters via Sampled Data

Abstract

The exponential synchronization issue for stochastic neural networks (SNNs) with mixed time delays and Markovian jump parameters using sampled-data controller is investigated. Based on a novel Lyapunov-Krasovskii functional, stochastic analysis theory, and linear matrix inequality (LMI) approach, we derived some novel sufficient conditions that guarantee that the master systems exponentially synchronize with the slave systems. The design method of the desired sampled-data controller is also proposed. To reflect the most dynamical behaviors of the system, both Markovian jump parameters and stochastic disturbance are considered, where stochastic disturbances are given in the form of a Brownian motion. The results obtained in this paper are a little conservative comparing the previous results in the literature. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.

1. Introduction

Neural networks, such as Hopfield neural networks, cellular neural networks, the Cohen-Grossberg neural networks, and bidirectional associative neural networks, are very important nonlinear circuit networks and, in the past few decades, have been extensively studied due to their potential applications in classification, signal and image processing, parallel computing, associate memories, optimization, cryptography, and so forth; see [1–7]. Many results, which deal with the dynamics of various neural networks such as stability, periodic oscillation, bifurcation, and chaos, have been obtained by applying the Lyapunov stability theory; see, for example, [8–10] and the references therein. As a special case, synchronization issues of the neural network systems have been extensively investigated too, and a lot of criteria have been developed to guarantee the global synchronization of the network systems in [11–17].

It has been widely reported that a neural network sometimes has finite modes that switch from one mode to another at different times; such a switching (jumping) signal between different neural network models can be governed by a Markovian chain; see [18–25] and the references therein. This class of systems has the advantage of modeling the dynamic systems subject to abrupt variation in their structures and has many applications such as target tracking problems, manufactory processes, and fault-tolerant systems. In [24], delay-interval dependent stability criteria are obtained for neural networks with Markovian jump parameters and time-varying delays, which are based on free-weighing matrix method and LMIs technique. In [25], by introducing some free-weighting matrices, delay-dependent stochastic exponential synchronization conditions are derived for chaotic neural networks with Markovian jump parameters and mixed time delays in terms of the Jensen inequality and linear matrix inequalities.

It is well known that noise disturbance widely exists in biological networks due to environmental uncertainties, which is a major source of instability and can lead to poor performances in neural networks. Such systems are described by stochastic differential systems which have been used efficiently in modeling many practical problems that arise in the fields of engineering, physics, and science as well. Therefore, the theory of stochastic differential equation is also attracting much attention in recent years and many results have been reported in the literature [26–30]. In addition to the noise disturbance, time delay is also a major source for causing instability and poor performances in neural networks; see, for example, [31–35]. It is known that time delays are often encountered in real neural networks, and the existence of time delays may cause oscillation or instability in neural networks, which are harmful to the applications of neural networks. Therefore, the stability analysis for neural networks with time delays has been widely studied in the literature.

On the other hand, as the rapid development of computer hardware, the sampled-data control technology has shown superiority over other control approaches because it is difficult to guarantee that the state variables transmitted to controllers are continuous in many real-world applications. In [36], Wu et al. investigated the synchronization problem of neural networks with time-varying delay under sampled-data control in the presence of a constant input delay. In [37], by using sampled-data controller, the global synchronization of the chaotic Lur'e systems is discussed and sufficient conditions are obtained in terms of effective synchronization linear matrix inequality by constructing the new discontinuous Lyapunov functionals. Wu et al. studied the sampled-data synchronization for Markovian jump neural networks with time-varying delay; some new and useful synchronization conditions in the framework of the input delay approach and the linear matrix inequality technique are derived in [38].

Motivated by the above discussion, in this paper we study the delay-dependent exponential synchronization of neural networks with stochastic perturbation, discrete and distributed time-varying delays, and Markovian jump parameters. Here, it should be mentioned that our results are delay dependent, which depend on not only the upper bounds of time delays but also their lower bounds. Moreover, the derivatives of time delays are not necessarily zero or smaller than one since several free matrices are introduced in our results. By constructing an appropriate Lyapunov-Krasovskii functional based on delay partitioning, several improved delay-dependent criteria are developed to achieve the exponential synchronization in mean square in terms of linear matrix inequalities. Two numerical examples are also provided to demonstrate the advantage of the theoretical results.

The rest of this paper is organized as follows. In Section 2, the model of stochastic neural network with both mixed time delays and Markovian jump parameters under sampled-data control is introduced, together with some definitions and lemmas. Exponential synchronization is proposed for neural networks with both Markovian jump parameters and mixed time delays via sampled data in Section 3. In Section 4, exponential synchronization is proved for stochastic neural networks with both Markovian jump parameters and mixed time delays under sampled-data control. In Section 5, two illustrative examples are given to demonstrate the validity of the proposed results. Finally, some conclusions are drawn in Section 6.

Notations. Throughout this paper, 𝑅 denotes the set of real numbers, 𝑅^𝑛 denotes the n-dimensional Euclidean space, and 𝑅^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. For any matrix 𝐴, 𝐴^𝑇 denotes the transpose of 𝐴. If 𝐴 is a real symmetric matrix, 𝐴>0(𝐴<0) means that 𝐴 is positive definite (negative definite). 𝜆_min(⋅) and 𝜆_max(⋅) represent minimum and maximum eigenvalues of a real symmetric matrix, respectively. (Ωℱ𝒫) is a complete probability space, where Ω is the sample space, ℱ is the 𝜎-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. diag{⋯} denotes a block-diagonal matrix and col{⋯} stands for a matrix column with blocks given by the matrices in {⋯}. 𝔼{⋅} denotes the expectation operator with respect to some probability measure 𝒫. Given the column vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇, 𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈𝑅^𝑛, 𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖, |𝑥|=(|𝑥₁|,…,|𝑥_𝑛|)^𝑇, and ‖𝑥‖=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2. 𝑥̇(𝑡) denotes the derivative of 𝑥(𝑡) and ∗ represents the symmetric form of matrix. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Model Description and Preliminaries

Let {𝑟(𝑡),𝑡≥0} be a right-continuous Markovian chain on the probability space (Ω  ℱ  𝒫) taking values in a finite state space 𝒮={1,2,…,𝑠} with generator Υ=(𝜋_𝑖𝑗)_𝑠×𝑠 given by 𝑃𝑟(𝑡+Δ𝑡)=𝑗∣𝑟(𝑡)=𝑖=𝑃_𝑖𝑗()=𝜋Δ𝑡^𝑖𝑗Δ𝑡+𝑜(Δ𝑡),if𝑖≠𝑗,1+𝜋_𝑖𝑖Δ𝑡+𝑜(Δ𝑡),if𝑖=𝑗, (1) where Δ𝑡>0 and lim_Δ𝑡→0(𝑜(Δ𝑡)/Δ𝑡)=0. Here, 𝜋_𝑖𝑗≥0(𝑗≠𝑖) is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 at time 𝑡+Δ𝑡, and 𝜋_𝑖𝑖 is the transition rate from 𝑖 to 𝑖 at time 𝑡+Δ𝑡.

Remark 1. The probability defined in (1) is called time-homogeneous transition probability, which is only relevant to the time internal Δ𝑡; that is, 𝑃_𝑖𝑗(Δ𝑡) is not relevant to the starting point 𝑡. Moreover, for the time-homogeneous transition probability defined in (1), the following two properties should be satisfied: 𝑃_𝑖𝑗(Δ𝑡)≥0,^𝑠_𝑗=1𝑃_𝑖𝑗(Δ𝑡)=1. (2) Accordingly, for any Δ𝑡 satisfying the conditions in (2), the matrix 𝑃=(𝑃_𝑖𝑗(Δ𝑡))_𝑠×𝑠 is called the probability transition matrix for the right-continuous Markovian chain {𝑟(𝑡),𝑡≥0}.

Fix a probability space (Ωℱ𝒫) and consider the neural networks with mixed time delays and Markovian jump described by the following differential equation system: 𝑥̇(𝑡)=−𝐶(𝑟(𝑡))𝑥(𝑡)+𝑊₀(𝑟(𝑡))𝑔(𝑥(𝑡))+𝑊₁(𝑟(𝑡))𝑔(𝑥(𝑡−𝑑(𝑡)))+𝑊₂(𝑟(𝑡))^𝑡_{𝑡−𝜏(𝑡)}𝑔(𝑥(𝑠))𝑑𝑠+𝐼(𝑡), (3) where 𝑥(𝑡)=[𝑥₁(𝑡)𝑥₂(𝑡)⋯𝑥_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the neuron state vector and 𝑥_𝑖(𝑡) is the state of the 𝑖th neuron at time 𝑡; 𝑔(𝑥(𝑡))=[𝑔₁(𝑥₁(𝑡))𝑔₂(𝑥₂(𝑡))⋯𝑔_𝑛(𝑥_𝑛(𝑡))]^𝑇 denotes the neuron activation function; 𝐶(𝑟(𝑡))=diag{𝑐₁,𝑐₂,…,𝑐_𝑛} is a diagonal matrix with positive entries; 𝑊₀(𝑟(𝑡))=(𝑤_0𝑖𝑗)_𝑛×𝑛, 𝑊₁(𝑟(𝑡))=(𝑤_1𝑖𝑗)_𝑛×𝑛, and 𝑊₂(𝑟(𝑡))=(𝑤_2𝑖𝑗)_𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weight matrix; 𝐼(𝑡)=[𝐼₁(𝑡)𝐼₂(𝑡)⋯𝐼_𝑛(𝑡)]^𝑇 is an external input vector; 𝑑(𝑡) and 𝜏(𝑡) denote the discrete delay and the distributed delay.

Throughout this paper, we make the following assumptions.(ℋ₁)There exist positive constants 𝑑, 𝜇, and 𝜏 such that 𝑑₁≤𝑑(𝑡)≤𝑑₂,𝑑̇(𝑡)≤𝜇,0≤𝜏(𝑡)≤𝜏. (4)(ℋ₂)Each activation function 𝑔_𝑖 in (3) is continuous and bounded, and there exist constants 𝐹⁻_𝑖 and 𝐹⁺_𝑖 such that 𝐹⁻_𝑖≤𝑔_𝑖𝛼₁−𝑔_𝑖𝛼₂𝛼₁−𝛼₂≤𝐹⁺_𝑖,𝑖=1,2,…,𝑛, (5) where 𝛼₁,𝛼₂∈𝑅 and 𝛼₁≠𝛼₂.

Remark 2. In the earlier literature, the activation functions 𝑔_𝑖(𝑖=1,2,…,𝑛) are supposed to be continuous, differentiable, monotonically increasing, and bounded. Moreover, the constants 𝐹⁻_𝑖≡0 for 𝑖=1,2,…,𝑛 or the constants 𝐹⁻_𝑖=−𝐹⁺_𝑖 for 𝑖=1,2,…,𝑛. However, in this paper, the resulting activation functions may be not monotonically increasing and more general than the usual Lipschitz-type conditions. Moreover, the constants 𝐹⁻_𝑖 and 𝐹⁺_𝑖(𝑖=1,2,…,𝑛) are allowed to be positive, negative, or zero. Hence, Assumption 2 of this paper is weaker than those given in the earlier literature (see, e.g., [39, 40]).

In this paper, we consider system (3) as the master system and a slave system for (3) can be described by the following equation: 𝑦̇(𝑡)=−𝐶(𝑟(𝑡))𝑦(𝑡)+𝑊₀(𝑟(𝑡))𝑔𝑦(𝑡)+𝑊₁(𝑟(𝑡))𝑔𝑦(𝑡−𝑑(𝑡))+𝑊₂(𝑟(𝑡))^𝑡_{𝑡−𝜏(𝑡)}𝑔𝑦(𝑠)𝑑𝑠+𝐼(𝑡)+𝑢(𝑡), (6) where 𝐶(𝑟(𝑡)) and 𝑊_𝑖(𝑟(𝑡)) for 𝑖=0,1,2,…, are matrices given in (3) and 𝑢(𝑡)∈𝑅^𝑛 is the appropriate control input.

In order to investigate the problem of exponential synchronization between systems (3) and (6), we define the error signal 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡). Therefore, the error dynamical system between (3) and (6) is given as follows: 𝑒̇(𝑡)=−𝐶(𝑟(𝑡))𝑒(𝑡)+𝑊₀(𝑟(𝑡))𝑓(𝑒(𝑡))+𝑊₁(𝑟(𝑡))𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊₂(𝑟(𝑡))^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝑢(𝑡), (7) where 𝑓(𝑒(𝑡))=𝑔(𝑦(𝑡))−𝑔(𝑥(𝑡)). It can be found that the functions 𝑓_𝑖(⋅) satisfy the following condition: 𝐹⁻_𝑖≤𝑓_𝑖(𝛼)𝛼≤𝐹⁺_𝑖,𝑖=1,2,…,𝑛, (8) where 𝛼∈ℝ and 𝛼≠0.

The control signal is assumed to be generated by using a zero-order-hold function with a sequence of hold times 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘<⋯. Therefore, the mode-independent state feedback controller takes the following form: 𝑢(𝑡)=𝐾𝑒𝑡_𝑘,𝑡_𝑘≤𝑡<𝑡_𝑘+1, (9) where 𝐾 is a sampled-data feedback controller gain matrix to be determined, 𝑒(𝑡_𝑘) is a discrete measurement of 𝑒(𝑡) at the sampling instant 𝑡_𝑘, and lim_𝑘→∞𝑡_𝑘=+∞. It is assumed that 𝑡_𝑘+1−𝑡_𝑘=ℎ_𝑘≤ℎ for any integer 𝑘≥0, where ℎ is a positive scalar and represents the largest sampling interval.

By substituting (9) into (7), we obtain 𝑒̇(𝑡)=−𝐶(𝑟(𝑡))𝑒(𝑡)+𝑊₀(𝑟(𝑡))𝑓(𝑒(𝑡))+𝑊₁(𝑟(𝑡))𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊₂(𝑟(𝑡))^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘. (10)

For convenience, in the following, each possible value of 𝑒(𝑡) is denoted by 𝑖, 𝑖∈𝒮. Then we have 𝐶_𝑖=𝐶(𝑟(𝑡)), 𝑊_0𝑖=𝑊₀(𝑟(𝑡)), 𝑊_1𝑖=𝑊₁(𝑟(𝑡)), and 𝑊_2𝑖=𝑊₂(𝑟(𝑡)), where 𝐶_𝑖, 𝑊_0𝑖, and 𝑊_1𝑖𝑊_2𝑖, for any 𝑖∈𝒮, are known constant matrices of appropriate dimensions. The system (10) can be written as 𝑒̇(𝑡)=−𝐶_𝑖𝑒(𝑡)+𝑊_0𝑖𝑓(𝑒(𝑡))+𝑊_1𝑖𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊_2𝑖^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘. (11)

The first purpose of this paper is to design a controller with the form (9) to achieve the exponential synchronization of the master system (3) and slave system (6). In other words, we are interested in finding a feedback gain matric 𝐾 such that the error system (11) is exponentially stable.

As mentioned earlier, it is often the case in practice that the neural network is disturbed by environmental noises that affect the stability of the equilibrium. Motivated by this we express a stochastic system whose consequent parts are a set of stochastic uncertain recurrent neural networks with mixed time delays: (𝑡)=(𝑟(𝑡𝑥(𝑡)𝑑𝑥−𝐶))+𝑊₀(𝑟(𝑡))𝑔(𝑥(𝑡))+𝑊₁(𝑟(𝑡))𝑔(𝑥(𝑡−𝑑(𝑡)))+𝑊₂(𝑟(𝑡))^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠(𝑡((𝑡)((𝑡((𝑡(𝑡(𝑡),))𝑑𝑠+𝐼)𝑑𝑡+𝜌𝑡,𝑥,𝑥𝑡−𝑑)),𝑥𝑡−𝜏)),𝑟))𝑑𝜔 (12) where 𝜔(𝑡)=(𝜔₁(𝑡),𝜔₂(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an n-dimensional Brownian motion defined on a complete probability space (Ωℱ𝒫) satisfying 𝔼{𝑑𝜔(𝑡)}=0 and 𝔼{𝑑𝜔²(𝑡)}=𝑑𝑡 and 𝜌(𝑡,𝑥(𝑡),𝑥(𝑡−𝑑(𝑡)), 𝑥(𝑡−𝜏(𝑡)),𝑟(𝑡))∶𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑛×𝒮→𝑅^𝑛×𝑚 is the noise intensity function matrix.

And a slave system for (12) can be described by the following equation: (𝑡)=(𝑟(𝑡𝑦(𝑡)𝑑𝑦−𝐶))+𝑊₀(𝑟(𝑡))𝑔𝑦(𝑡)+𝑊₁(𝑟(𝑡))𝑔𝑦(𝑡−𝑑(𝑡))+𝑊₂(𝑟(𝑡))^𝑡_{(𝑡)𝑡−𝜏}𝑔𝑦(𝑠)(𝑡)(𝑡(𝑡)((𝑡((𝑡(𝑡(𝑡).𝑑𝑠+𝐼+𝑢)𝑑𝑡+𝜌𝑡,𝑦,𝑦𝑡−𝑑)),𝑦𝑡−𝜏)),𝑟)𝑑𝜔 (13)

The mode-independent state feedback controller is made as the form of (9) and each possible value of 𝑒(𝑡) is denoted by 𝑖, 𝑖∈𝒮; then we have the final stochastic error system: (𝑡)=𝑑𝑒−𝐶_𝑖𝑒(𝑡)+𝑊_0𝑖𝑓(𝑒(𝑡))+𝑊_1𝑖𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊_2𝑖^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘(𝑡)((𝑡((𝑡𝑡𝑑𝑡+𝜌𝑡,𝑒,𝑒𝑡−𝑑)),𝑒𝑡−𝜏)),𝑒_𝑘,𝑖𝑑𝜔(𝑡). (14)

We impose the following assumption:(ℋ₃)𝜌∶𝑅⁺×𝑅^𝑛×𝑅^𝑛×𝑅^𝑛×𝑅^𝑛×𝒮→𝑅^𝑛×𝑚 is locally Lipschitz continuous and satisfies trace𝜌^𝑇𝑡,𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑖𝜌𝑡,𝑒₁,𝑒₂,𝑒₃,𝑒₄,𝑖≤𝑒^𝑇₁𝑌_1𝑖𝑒₁+𝑒^𝑇₂𝑌_2𝑖𝑒₂+𝑒^𝑇₃𝑌_3𝑖𝑒₃+𝑒^𝑇₄𝑌_4𝑖𝑒₄. (15)

Our second purpose of this paper is to find a feedback gain matric 𝐾 in the controller with the form (9) to ensure that the error system (14) is exponentially stable, so that the master system (12) and slave system (13) are exponentially synchronous.

To state our main results, the following definition and lemmas are first introduced, which are essential for the proof in the sequel.

Definition 3. Master system and slave system are said to be exponentially synchronous if error system is exponentially stable; that is, for any initial condition 𝑒(𝑠)=𝜑(𝑠) defined on the interval −𝜛0, 𝜛=max{𝑑₂,𝜏,ℎ}, the following condition is satisfied: 𝔼‖𝑒(𝑡)‖²≤𝜁𝑒^−𝜖𝑡𝔼sup_{−𝜛≤𝑠≤0}𝜑(𝑠)². (16)

Lemma 4 (the Jensen inequality, see [41]). For any constant matrix 𝑃∈𝑅^𝑚×𝑚, 𝑃>0, scalar 0<𝑧(𝑡)<𝑧, vector function 𝑉∶[𝑡−𝑧,𝑡]→𝑅^𝑚, 𝑡≥0, such that the integrations concerned are well defined; then ₀^𝑧(𝑡)𝑉(𝑠)𝑑𝑠^𝑇𝑃₀^𝑧(𝑡)𝑉(𝑠)(𝑡)∫𝑑𝑠≤𝑧₀^𝑧(𝑡)𝑉^𝑇(𝑠)𝑃𝑉(𝑠)𝑑𝑠. (17)

Lemma 5 (the Schur complement). Given one positive definite matrix 𝐺₂>0 and constant matrices 𝐺₁ and 𝐺₃, where 𝐺₁=𝐺^𝑇₁, 𝐺₁+𝐺^𝑇₃𝐺₂⁻¹𝐺₃<0 if and only if 𝐺₁𝐺^𝑇₃𝐺₃−𝐺₂<0𝑜𝑟−𝐺₂𝐺^𝑇₃𝐺₃𝐺₁<0. (18)

Lemma 6 (see [42]). For any constant matric 𝑀∈𝑅^𝑘×𝑛, symmetric positive definite matrix 𝑅∈𝑅^𝑛×𝑛, two functions 𝜈₁(𝑡) and 𝜈₂(𝑡) satisfying 0<𝜈_𝑚≤𝜈₁(𝑡)<𝜈₂(𝑡)≤𝜈_𝑀(𝑡≥0), and vector function 𝑉∶[𝜈_𝑚,𝜈_𝑀]→𝑅^𝑛 such that the integrations concerned are well defined, let ^𝜈2_𝜈^(𝑡)_1(𝑡)𝑉(𝑠)𝑑𝑠=𝜉̃^𝑇𝜙(𝑡), (19) where 𝜉̃∈𝑅^𝑘×𝑛 and 𝜙(𝑡)∈𝑅^𝑘. Then the following inequality holds: 𝜙(𝑡)𝑀𝜉̃^𝑇+𝜉̃𝑀^𝑇−𝜈₂(𝑡)−𝜈₁(𝑡)𝑀𝑅⁻¹𝑀𝜙(𝑡)≤^𝜈2_𝜈^(𝑡)_1(𝑡)𝑉^𝑇(𝑠)𝑅𝑉(𝑠)𝑑𝑠. (20)

3. Exponential Synchronization for Markovian Jump Neural Networks with Mixed Time Delays via Sampled Data

To present the main results of this section, we denote 𝐹₁=diag{𝐹⁻₁𝐹⁺₁,𝐹⁻₂𝐹⁺₂,…,𝐹⁻_𝑛𝐹⁺_𝑛}, 𝐹₂=diag{(𝐹⁻₁+𝐹⁺₁)/2,(𝐹⁻₂+𝐹⁺₂)/2,…,(𝐹⁻_𝑛+𝐹⁺_𝑛)/2}, 𝑔₁=𝐼0, and 𝑔₂=0𝐼.

Here, some LMI conditions will be developed to ensure that master system (3) and slave system (6) are exponential synchronous by employing the Lyapunov functionals.

Theorem 7. Under Assumptions ℋ₁ and ℋ₂, for a given scalar 𝛾, if there exist matrices 𝑃_𝑖>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0, 𝑍₁>0, 𝑍₂>0, 𝑍₃>0, 𝑈>0, 𝑆, 𝑋, 𝑋₁, 𝐺, 𝐿, and 𝐻_𝑖=𝐻_1𝑖𝐻_2𝑖𝐻_3𝑖 and diagonal matrices 𝑉_1𝑖>0, 𝑉_2𝑖>0, 𝑉_3𝑖>0, and 𝑉_4𝑖>0 such that, for any 𝑖∈𝒮, inequalities (21)–(24) hold, 𝑍₂𝑆∗𝑍₂Σ>0,_𝑖(ℎ)=𝑃_𝑖+ℎ𝑋+𝑋^𝑇2−ℎ𝑋+ℎ𝑋₁∗−ℎ𝑋₁−ℎ𝑋^𝑇₁+ℎ𝑋+𝑋^𝑇2Σ>0,_1𝑖(ℎ)=Σ_11𝑖Σ₁₂Σ_13𝑖0Σ_15𝑖Σ_16𝑖Σ_17𝑖+Λ₁(ℎ)∗Σ_22𝑖Σ₂₃Σ₂₄000∗∗Σ_33𝑖Σ₃₄00Σ_37𝑖∗∗∗Σ₄₄000∗∗∗∗Σ₅₅0Σ_57𝑖∗∗∗∗∗Σ_66𝑖Σ_67𝑖+Λ₂(ℎ)∗∗∗∗∗∗Σ₇₇Σ+ℎ𝑈<0,_2𝑖=Σ_11𝑖Σ₁₂Σ_13𝑖0Σ_15𝑖Σ_16𝑖Σ_17𝑖ℎ𝑔^𝑇₁𝐻^𝑇_1𝑖∗Σ_22𝑖Σ₂₃Σ₂₄0000∗∗Σ_33𝑖Σ₃₄00Σ_37𝑖0∗∗∗Σ₄₄0000∗∗∗∗Σ₅₅0Σ_57𝑖0∗∗∗∗∗Σ_66𝑖Σ_67𝑖ℎ𝐻^𝑇_2𝑖∗∗∗∗∗∗Σ₇₇ℎ𝐻^𝑇_3𝑖∗∗∗∗∗∗∗−ℎ𝑈<0 (21)(22)(23)(24)

are satisfied, where Σ_11𝑖=𝑔^𝑇₁^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑔₁+𝑄₁+𝑄₂+𝑄₃−𝑔^𝑇₁𝑍₁𝑔₁+𝜏²𝑔^𝑇₁𝑍₃𝑔₁−𝑔^𝑇₁𝑋+𝑋^𝑇2𝑔₁+𝑔^𝑇₁𝐻_1𝑖𝑔₁+𝑔^𝑇₁𝐻^𝑇_1𝑖𝑔₁−𝑔^𝑇₁𝐹₁𝑉_1𝑖𝑔₁+𝑔^𝑇₂𝑉_1𝑖𝐹₂𝑔₁+𝑔^𝑇₁𝐹₂𝑉_1𝑖𝑔₂−𝑔^𝑇₂𝑉_1𝑖𝑔₂−𝑔^𝑇₁𝐺𝐶_𝑖𝑔₁−𝑔^𝑇₁𝐶^𝑇_𝑖𝐺^𝑇𝑔₁+𝑔^𝑇₁𝐺𝑊_0𝑖𝑔₁+𝑔^𝑇₁𝑊^𝑇_0𝑖𝐺^𝑇𝑔₁,Σ₁₂=𝑔^𝑇₁𝑍₁𝑔₁,Σ_13𝑖=𝑔^𝑇₁𝐺𝑊_1𝑖𝑔₂,Σ_15𝑖=𝑔^𝑇₁𝐺𝑊_2𝑖,Σ_16𝑖=𝑔^𝑇₁𝑋+𝑋₁−𝑔^𝑇₁𝐻^𝑇_1𝑖+𝑔^𝑇₁𝐻^𝑇_2𝑖+𝑔^𝑇₁Σ𝐿,_17𝑖=𝑔^𝑇₁𝑃_𝑖+𝑔^𝑇₁𝐻_3𝑖−𝑔^𝑇₁𝐺−𝛾𝑔^𝑇₁𝐶^𝑇_𝑖𝐺^𝑇+𝛾𝑔^𝑇₂𝑊^𝑇_0𝑖𝐺^𝑇,Σ_22𝑖=−𝑄₁−𝑔^𝑇₁𝑍₁𝑔₁−𝑔^𝑇₁𝑍₂𝑔₁−𝑔^𝑇₁𝐹₁𝑉_2𝑖𝑔₁+𝑔^𝑇₂𝑉_2𝑖𝐹₂𝑔₁+𝑔^𝑇₁𝐹₂𝑉_2𝑖𝑔₂−𝑔^𝑇₂𝑉_2𝑖𝑔₂,Σ₂₃=𝑔^𝑇₁𝑍₂−𝑆𝑔₁,Σ₂₄=𝑔^𝑇₁𝑆𝑔₁,Σ_33𝑖=−1−𝜇𝑄₂+𝑔^𝑇₁−2𝑍₂+𝑆+𝑆^𝑇𝑔₁−𝑔^𝑇₁𝐹₁𝑉_3𝑖𝑔₁+𝑔^𝑇₂𝑉_3𝑖𝐹₂𝑔₁+𝑔^𝑇₁𝐹₂𝑉_3𝑖𝑔₂−𝑔^𝑇₂𝑉_3𝑖𝑔₂,Σ₃₄=𝑔^𝑇₁−𝑆+𝑍₂𝑔₁,Σ_37𝑖=𝛾𝑔^𝑇₂𝑊^𝑇_1𝑖𝐺^𝑇,Σ₄₄=−𝑄₃−𝑔^𝑇₁𝑍₃𝑔₁−𝑔^𝑇₁𝐹₁𝑉_4𝑖𝑔₁+𝑔^𝑇₂𝑉_4𝑖𝐹₂𝑔₁+𝑔^𝑇₁𝐹₂𝑉_4𝑖𝑔₂−𝑔^𝑇₂𝑉_4𝑖𝑔₂,Σ₅₅=−𝑍₃,Σ_57𝑖=𝛾𝑊^𝑇_2𝑖𝐺^𝑇,Σ_66𝑖=𝑋₁+𝑋^𝑇₁−𝑋+𝑋^𝑇2−𝐻_2𝑖−𝐻^𝑇_2𝑖,Σ_67𝑖=−𝐻_3𝑖+𝛾𝐿^𝑇,Σ₇₇=𝑑²₁𝑍₁+𝑑²₁₂𝑍₂−𝛾𝐺−𝛾𝐺^𝑇,Λ₁(ℎ)=ℎ𝑔^𝑇₁𝑋+𝑋^𝑇2,Λ₂(ℎ)=ℎ−𝑋^𝑇+𝑋^𝑇₁, (25) and then the master system (3) and the slave system (6) are exponentially synchronous. Moreover, the desired controller gain matrix in (9) can be given by 𝐾=𝐺⁻¹𝐿.

Proof. Denote 𝜂(𝑡)=[𝑒^𝑇(𝑡)𝑓^𝑇(𝑒(𝑡))]^𝑇 and ℋ=𝑋+𝑋^𝑇2−𝑋+𝑋₁∗−𝑋₁−𝑋^𝑇₁+𝑋+𝑋^𝑇2, (26) and then consider the following Lyapunov functional for error system (11): 𝑉(𝑒(𝑡),𝑡,𝑟(𝑡))=⁷_𝑙=1𝑉_𝑙(𝑒(𝑡)(𝑡,𝑡,𝑡,𝑟))𝑡∈_𝑘,𝑡_𝑘+1, (27) where 𝑉₁(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑒^𝑇(𝑡)𝑃(𝑟(𝑡𝑒(𝑡),𝑉))₂(𝑒(𝑡)(𝑡=,𝑡,𝑟))_{𝑡𝑡−𝑑1}𝜂^𝑇(𝑠)𝑄₁𝜂(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝑑}𝜂^𝑇(𝑠)𝑄₂𝜂(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑2}𝜂^𝑇(𝑠)𝑄₃𝜂(𝑠)𝑉𝑑𝑠,₃(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑑₁⁰_−𝑑1^𝑡_𝑡+𝜃𝑒^𝑇̇(𝑠)𝑍₁𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₄(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑑₁₂^−𝑑1_−𝑑2^𝑡_𝑡+𝜃𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₅(𝑒(𝑡),𝑡,𝑟(𝑡))=𝜏⁰_−𝜏^𝑡_𝑡+𝜃𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠𝑉))𝑑𝑠𝑑𝜃,₆(𝑒(𝑡)(𝑡=𝑡,𝑡,𝑟))_𝑘+1−𝑡^𝑡_𝑡𝑘𝑒^𝑇̇(𝑠)𝑈𝑒̇(𝑠)𝑉𝑑𝑠,₇(𝑒(𝑡)(𝑡=𝑡,𝑡,𝑟))_𝑘+1−𝑡𝑒(𝑡)𝑒(𝑡_𝑘)^𝑇ℋ𝑒(𝑡)𝑒𝑡_𝑘. (28)
Let ℒ be the weak infinitesimal generator of the random process (𝑒(𝑡),𝑡≥0,𝑟(𝑡)) along system (11). Next, we will compute ℒ𝑉₁(𝑒(𝑡),𝑡,𝑖), ℒ𝑉₂(𝑒(𝑡),𝑡,𝑖), ℒ𝑉₃(𝑒(𝑡),𝑡,𝑖), ℒ𝑉₄(𝑒(𝑡),𝑡,𝑖), ℒ𝑉₅(𝑒(𝑡),𝑡,𝑖), ℒ𝑉₆(𝑒(𝑡),𝑡,𝑖), and ℒ𝑉₇(𝑒(𝑡),𝑡,𝑖) along the trajectories of the error system (11), respectively, for 𝑟(𝑡)=𝑖∈𝒮: ℒ𝑉₁(𝑒(𝑡),𝑡,𝑖)=2𝑒^𝑇(𝑡)𝑃_𝑖𝑒̇(𝑡)+𝑒^𝑇(𝑡)^𝑠_𝑗=1𝜋_𝑖𝑗𝑝_𝑗𝑒(𝑡)=2𝜂^𝑇(𝑡)𝑔^𝑇₁𝑃_𝑖𝑒̇(𝑡)+𝜂^𝑇(𝑡)𝑔^𝑇₁^𝑠_𝑗=1𝜋_𝑖𝑗𝑝_𝑗𝑔₁𝜂(𝑡),ℒ𝑉₂(𝑒(𝑡),𝑡,𝑖)=𝜂^𝑇(𝑡)𝑄₁+𝑄₂+𝑄₃𝜂(𝑡)−𝜂^𝑇𝑡−𝑑₁𝑄₁𝜂𝑡−𝑑₁−1−𝑑̇(𝑡)𝜂^𝑇(𝑡−𝑑(𝑡))𝑄₁𝜂(𝑡−𝑑(𝑡))−𝜂^𝑇𝑡−𝑑₂𝑄₃𝜂𝑡−𝑑₂≤𝜂^𝑇(𝑡)𝑄₁+𝑄₂+𝑄₃𝜂(𝑡)−𝜂^𝑇𝑡−𝑑₁𝑄₁𝜂𝑡−𝑑₁−1−𝜇𝜂^𝑇(𝑡−𝑑(𝑡))𝑄₁𝜂(𝑡−𝑑(𝑡))−𝜂^𝑇𝑡−𝑑₂𝑄₃𝜂𝑡−𝑑₂,ℒ𝑉₃(𝑒(𝑡),𝑡,𝑖)=𝑑²₁𝑒^𝑇̇(𝑡)𝑍₁𝑒̇(𝑡)−𝑑₁^𝑡_𝑡−𝑑1𝑒^𝑇̇(𝑠)𝑍₁𝑒̇(𝑠)𝑑𝑠,ℒ𝑉₄(𝑒(𝑡),𝑡,𝑖)=𝑑²₁₂𝑒^𝑇̇(𝑡)𝑍₂𝑒̇(𝑡)−𝑑₁₂^𝑡−𝑑1_𝑡−𝑑2𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)𝑑𝑠,ℒ𝑉₅(𝑒(𝑡),𝑡,𝑖)=𝜏²𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠))−𝜏^𝑡_𝑡−𝜏𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠))𝑑𝑠≤𝜏²𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠))−𝜏^𝑡_{(𝑡)𝑡−𝜏}𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠))𝑑𝑠≤𝜏²𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠−))_{𝑡(𝑡)𝑡−𝜏}𝑓(𝑒(𝑠))𝑑𝑠^𝑇𝑍₃^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑒(𝑠))𝑑𝑠,ℒ𝑉₆(𝑒(𝑡),𝑡,𝑖)=−^𝑡_𝑡𝑘𝑒(̇𝑠)^𝑇𝑈𝑒̇(𝑠)𝑑𝑠+𝑡_𝑘+1−𝑡𝑒(̇𝑡)^𝑇𝑈𝑒̇(𝑡),ℒ𝑉₇(𝑒(𝑡))𝑒(𝑡)𝑒𝑡,𝑡,𝑖=−_𝑘^𝑇ℋ𝑒(𝑡)𝑒𝑡_𝑘+2𝑡_𝑘+1−𝑡𝑒(𝑡)𝑒(𝑡_𝑘)^𝑇ℋ𝑒̇(𝑡)0. (29)(30)(31)(32)(33)(34)(35) According to Lemma 4, it follows that −𝑑₁^𝑡_𝑡−𝑑1𝑒^𝑇̇(𝑡)𝑍₁𝑒̇(𝑡)𝑑𝑠≤−^𝑡_𝑡−𝑑1𝑒^𝑇̇(𝑡)𝑑𝑠𝑍₁^𝑡_𝑡−𝑑1𝑒̇(𝑡)𝑑𝑠=−𝜂(𝑡)𝜂(𝑡−𝑑₁)^𝑇𝑔^𝑇₁𝑍₁𝑔₁−𝑔^𝑇₁𝑍₁𝑔₁∗𝑔^𝑇₁𝑍₁𝑔₁𝜂(𝑡)𝜂𝑡−𝑑₁. (36) On the other hand, denote 𝒥₁(𝑡)=^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑒̇(𝑠)𝒥𝑑𝑠,₂(𝑡)=^{𝑡−𝑑(𝑡)}_𝑡−𝑑2𝑒̇(𝑠)𝑑𝑠. (37) When 𝑑₁<𝑑(𝑡)<𝑑₂, by Lemma 4, we have that 𝑑₁₂^𝑡−𝑑1_𝑡−𝑑2𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)𝑑𝑠=𝑑₁₂^𝑡−𝑑1_{𝑡−𝑑(𝑡)}𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)𝑑𝑠+𝑑₁₂^{𝑡−𝑑(𝑡)}_{𝑡−𝑑12}𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)≥𝑑𝑑𝑠₁₂𝑑(𝑡)−𝑑₁𝒥^𝑇₁(𝑡)𝑍₂𝒥₁(𝑡)+𝑑₁₂𝑑₂(𝑡)𝒥−𝑑^𝑇2(𝑡)𝑍₂𝒥₂(𝑡). (38) It is clear from (21) that 𝒥₁𝒥(𝑡)₂(𝑡)^𝑇𝑍₂𝑆∗𝑍₂𝒥₁(𝑡)𝒥₂(𝑡)=𝒥^𝑇₁(𝑡)𝑍₂𝒥₁(𝑡)𝒥^𝑇₁(𝑡)𝑆𝒥₂(𝑡)∗𝒥^𝑇₂(𝑡)𝑍₂𝒥₂(𝑡)≥0. (39) Moreover, 1𝑑₁₂/𝑑(𝑡)−𝑑₁+1𝑑₁₂/𝑑₁−𝑑(𝑡)=1. (40) Based on the lower bounds lemma of [43], we have from (38)–(40) that 𝑑₁₂^𝑡−𝑑1_𝑡−𝑑2𝑒^𝑇̇(𝑠)𝑍₂𝑒̇(𝑠)𝑑𝑠≥𝒥^𝑇₁(𝑡)𝑍₂𝒥₁(𝑡)+𝒥^𝑇₂(𝑡)𝑍₂𝒥₂(𝑡)+𝒥^𝑇₁(𝑡)𝑆𝒥₂(𝑡)+𝒥^𝑇₂(𝑡)𝑆𝒥₁(𝑡)=𝒥₁𝒥(𝑡)₂(𝑡)^𝑇𝑍₂𝑆∗𝑍₂𝒥₁(𝑡)𝒥₂(𝑡). (41)
Note that, when 𝑑(𝑡)=𝑑₁ or 𝑑(𝑡)=𝑑₂, we have 𝒥₁(𝑡)=0 or 𝒥₂(𝑡)=0, respectively. Thus, (41) still holds.
Therefore, −𝑑²₁₂^𝑡−𝑑1_𝑡−𝑑2𝑒^𝑇̇(𝑡)𝑍₂𝑒̇(𝑡)≤𝒩^𝑇(𝑡)𝒲𝒩(𝑡), (42) where 𝒩(𝑡)=𝜂^𝑇𝑡−𝑑₁𝜂^𝑇(𝑡−𝑑(𝑡))𝜂^𝑇𝑡−𝑑₂^𝑇,𝒲=−𝑔^𝑇₁𝑍₂𝑔₁𝑔^𝑇₁𝑍₂−𝑆𝑔₁𝑔^𝑇₁𝑆𝑔₁∗𝑔^𝑇₁−2𝑍₂+𝑆+𝑆^𝑇𝑔₁𝑔^𝑇₁−𝑆+𝑍₂𝑔₁∗∗−𝑔^𝑇₁𝑍₂𝑔₁. (43)
Inspired by the free-weighting matric approach [44], we can find that, for any appropriately dimensioned matrix 𝐻_𝑖, ^{𝐻𝑇𝑖𝑈−1𝐻𝑖𝐻𝑇𝑖}_∗𝑈≥0. Hence, the following inequality holds: ^𝑡𝑡𝑘̇𝜓(𝑡)𝑒(𝑠)^𝑇𝐻^𝑇_𝑖𝑈⁻¹𝐻_𝑖𝐻^𝑇_𝑖𝜓(𝑡)∗𝑈𝑒̇(𝑠)𝑑𝑠≥0, (44) where 𝜓(𝑡)=[𝑒^𝑇(𝑡)𝑒^𝑇(𝑡_𝑘)𝑒^𝑇̇(𝑡)]^𝑇.
From (44), we can immediately get that −^𝑡_𝑡𝑘𝑒^𝑇̇(𝑠)𝑈𝑒̇(𝑠)≤𝑑𝑠𝑡−𝑡_𝑘𝜓^𝑇(𝑡)𝐻^𝑇_𝑖𝑈⁻¹𝐻_𝑖𝜓(𝑡)+2𝜓^𝑇(𝑡)𝐻^𝑇_𝑖𝑒(𝑡)−𝑒𝑡_𝑘. (45)
Furthermore, according to error system (11), for any appropriately dimensioned matrix 𝐺 and scalar 𝛾, the following equality is satisfied: 2𝜂^𝑇(𝑡)𝑔^𝑇₁𝐺+𝛾𝑒^𝑇̇(𝑡)𝐺×−𝑒̇(𝑡)−𝐶_𝑖𝑒(𝑡)+𝑊_0𝑖𝑓(𝑒(𝑡))+𝑊_1𝑖𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊_2𝑖^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘=0. (46)
On the other hand, we have from (8) that, for any 𝜀=1,2,…,𝑛, 𝑓_𝜀𝑒_𝜀(𝑡)−𝐹⁻_𝜀𝑒_𝜀(𝑡)𝑓_𝜀𝑒_𝜀(𝑡)−𝐹⁺_𝜀𝑒_𝜀(𝑡)≤0, (47) which is equivalent to 𝜂^𝑇(𝑡)𝐹⁻_𝜀𝐹⁺_𝜀𝑒_𝜀̂𝑒^𝑇_𝜀̂−𝐹⁻_𝜀+𝐹⁺_𝜀2𝑒_𝜀̂𝑒^𝑇_𝜀̂−𝐹⁻_𝜀+𝐹⁺_𝜀2𝑒_𝜀̂𝑒^𝑇_𝜀̂𝑒_𝜀̂𝑒^𝑇_𝜀̂𝜂(𝑡)≤0, (48) where 𝑒_𝜀̂ denotes the unit column vector with one element on its 𝜀th row and zeros elsewhere. Thus, for any appropriately dimensioned diagonal matrices 𝑉_1𝑖>0, the following inequality holds [45]: 𝜂^𝑇(𝑡)−𝐹₁𝑉_1𝑖𝐹₂𝑉_1𝑖∗−𝑉_1𝑖𝜂(𝑡)≥0, (49) which implies 𝜂^𝑇(𝑡)×−𝑔^𝑇₁𝐹₁𝑉_1𝑖𝑔₁+𝑔^𝑇₂𝑉_1𝑖𝐹₂+𝑔^𝑇₁𝐹₂𝑉_1𝑖𝑔₂−𝑔^𝑇₂𝑉_1𝑖𝑔₂×𝜂(𝑡)≥0. (50)
Similarly, for any appropriately dimensioned diagonal matrices 𝑉_2𝑖>0, 𝑉_3𝑖>0, and 𝑉_4𝑖>0, the following inequalities also hold: 𝜂^𝑇𝑡−𝑑₁×−𝑔^𝑇₁𝐹₁𝑉_2𝑖𝑔₁+𝑔^𝑇₂𝑉_2𝑖𝐹₂+𝑔^𝑇₁𝐹₂𝑉_2𝑖𝑔₂−𝑔^𝑇₂𝑉_2𝑖𝑔₂×𝜂𝑡−𝑑₁𝜂≥0,_𝑇((𝑡×𝑡−𝑑))−𝑔^𝑇₁𝐹₁𝑉_3𝑖𝑔₁+𝑔^𝑇₂𝑉_3𝑖𝐹₂+𝑔^𝑇₁𝐹₂𝑉_3𝑖𝑔₂−𝑔^𝑇₂𝑉_3𝑖𝑔₂((𝑡𝜂×𝜂𝑡−𝑑))≥0,_𝑇𝑡−𝑑₂×−𝑔^𝑇₁𝐹₁𝑉_4𝑖𝑔₁+𝑔^𝑇₂𝑉_4𝑖𝐹₂+𝑔^𝑇₁𝐹₂𝑉_4𝑖𝑔₂−𝑔^𝑇₂𝑉_4𝑖𝑔₂×𝜂𝑡−𝑑₂≥0. (51)
Adding the left-hand sides of (46)–(51) to ℒ𝑉(𝑒(𝑡),𝑡,𝑖) and letting 𝐿=𝐺𝐾, we have from (29)–(36), (42), and (45) that, for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), ℒ𝑉(𝑒(𝑡),𝑡,𝑖)≤𝒳^𝑇(𝑡𝑡)^𝑘+1−𝑡ℎ_𝑘Σ_1𝑖ℎ_𝑘+𝑡−𝑡_𝑘ℎ_𝑘Γ_𝑖ℎ_𝑘𝒳(𝑡), (52)
whereΓ_𝑖ℎ_𝑘=Σ_1𝑖(0)+ℎ_𝑘𝑔^𝑇₁𝐻^𝑇_1𝑖000𝐻^𝑇_2𝑖𝐻^𝑇_3𝑖𝑈⁻¹𝑔^𝑇₁𝐻^𝑇_1𝑖000𝐻^𝑇_2𝑖𝐻^𝑇_3𝑖^𝑇,𝒳(𝑡)=𝜂^𝑇(𝑡)𝜂^𝑇𝑡−𝑑₁𝜂^𝑇(𝑡−𝑑(𝑡))𝜂^𝑇𝑡−𝑑₂∫^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑒(𝑠))𝑑𝑠^𝑇𝑒^𝑇𝑡_𝑘𝑒^𝑇̇(𝑡)^𝑇. (53)
According to Lemma 5, (24) is equivalent to Σ_1𝑖ℎ_𝑘𝑔+ℎ^𝑇1𝐻^𝑇_1𝑖000𝐻^𝑇_2𝑖𝐻^𝑇_3𝑖𝑈⁻¹𝑔^𝑇₁𝐻^𝑇_1𝑖000𝐻^𝑇_2𝑖𝐻^𝑇_3𝑖^𝑇<0, (54) which implies Γ_𝑖ℎ_𝑘<0,Σ_1𝑖(0)<0. (55)
From (23) and (55), it can be seen that Σ_1𝑖ℎ_𝑘=ℎ_𝑘ℎΣ_1𝑖(ℎ)+ℎ−ℎ_𝑘ℎΣ_1𝑖(0)<0. (56)
Thus, we can show from (52)–(56) that (𝑒(𝑡))‖‖ℒ𝑉,𝑡,𝑖≤−𝜉𝑒(𝑡)²+‖𝑒(𝑡−𝑑(𝑡))‖²+‖𝑒(𝑡−𝜏(𝑡))‖²+𝑒(𝑡_𝑘)². (57)
From the definition of 𝑉(𝑒(𝑡),𝑡,𝑟(𝑡)), 𝑒(̇𝑡), and 𝑓(𝑒(𝑡)), there exist positive scalars 𝛿₀, 𝛿₁, 𝛿₂, 𝛿₃, and 𝛿₄ such that the following inequality holds: 𝑉(𝑒(𝑡),𝑡,𝑖)≤𝛿₀𝑒(𝑡_𝑘)²+𝛿₁‖𝑒(𝑡)‖²+𝛿₂^𝑡_𝑡−𝜛‖𝑒(𝑠)‖²𝑑𝑠+𝛿₃^𝑡_𝑡−𝜛‖𝑒(𝑠−𝑑(𝑠))‖²𝑑𝑠+𝛿₄^𝑡_𝑡−𝜛‖𝑒(𝑠−𝜏(𝑠))‖²𝑑𝑠. (58)
Define a new function 𝑉(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑒^𝜖𝑡𝑉(𝑒(𝑡),𝑡,𝑟(𝑡)), where 𝜖>0 and 𝜖max{𝛿₀,𝛿₁+𝛿₂𝜛𝑒^𝜖𝜛,𝛿₃𝜛𝑒^𝜖𝜛,𝛿₄𝜛𝑒^𝜖𝜛}≤𝜉. It can be found that 𝑉(𝑒(𝑡)(𝑡≥𝑉,𝑡,𝑟))₁(𝑒(𝑡),𝑡,𝑟(𝑡))+𝑉₇(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑒^𝜖𝑡𝑒(𝑡)𝑒𝑡_𝑘^𝑇𝑡_𝑘+1−𝑡ℎ_𝑘Σ_𝑖ℎ_𝑘+𝑡−𝑡_𝑘ℎ_𝑘Σ_𝑖(0×𝑒(𝑡)𝑒𝑡_𝑘), (59) where Σ_𝑖ℎ_𝑘=ℎ_𝑘ℎΣ_𝑖(ℎ)+ℎ−ℎ_𝑘ℎΣ_𝑖(0). (60)
Due to the fact that 𝑃_𝑖>0 and Σ_𝑖(ℎ)>0, we can find a sufficiently small scalar 𝛿>0 such that 𝑃_𝑖>𝛿𝐼 and Σ_𝑖(ℎ)>𝛿𝐼, which implies 𝑉(𝑒(𝑡),𝑡,𝑟(𝑡))>𝛿𝑒^𝜖𝑡‖𝑒(𝑡)‖²>0.
On the other hand, from (57) and (58) we have ℒ𝑉(𝑒(𝑡),𝑡,𝑖)≤𝑒^𝜖𝑡𝜖𝛿₀−𝜉𝑒(𝑡_𝑘)²+𝜖𝛿₁−𝜉‖𝑒(𝑡)‖²−𝜉‖𝑒(𝑡−𝑑(𝑡))‖²−𝜉‖𝑒(𝑡−𝜏(𝑡))‖²+𝜖𝛿₂^𝑡_𝑡−𝜛‖𝑒(𝑠)‖²𝑑𝑠+𝜖𝛿₃^𝑡_𝑡−𝜛‖𝑒(𝑡−𝑑(𝑠))‖²𝑑𝑠+𝜖𝛿₄^𝑡_𝑡−𝜛‖𝑒(𝑡−𝜏(𝑠))‖²𝑑𝑠. (61)
By using Dynkin's formula, for 𝑇>0, we have 𝔼𝑉(𝑒(𝑡),𝑡,𝑖)≤𝐽₁+𝜖𝛿₀−𝜉𝔼^𝑇₀𝑒^𝜖𝑡𝑒𝑡_𝑘²+𝑑𝑡𝜖𝛿₁−𝜉𝔼^𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡)‖²𝑑𝑡−𝜉𝔼^𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡−𝑑(𝑡))‖²𝑑𝑡−𝜉𝔼^𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡−𝜏(𝑡))‖²𝑑𝑡+𝜖𝛿₂𝔼^𝑇₀^𝑡_𝑡−𝜛𝑒^𝜖𝑡‖𝑒(𝑠)‖²𝑑𝑠𝑑𝑡+𝜖𝛿₃𝔼^𝑇₀^𝑡_𝑡−𝜛𝑒^𝜖𝑡‖𝑒(𝑠−𝑑(𝑠))‖²𝑑𝑠𝑑𝑡+𝜖𝛿₄𝔼^𝑇₀^𝑡_𝑡−𝜛𝑒^𝜖𝑡‖𝑒(𝑠−𝜏(𝑠))‖²𝑑𝑠𝑑𝑡, (62) where 𝐽₁=[𝛿₀+𝛿₁+𝜛𝛿₂+𝜛𝛿₃+𝜛𝛿₄]sup_{−𝜛≤𝑠≤0}𝔼‖𝜑(𝑠)‖².
Consequently, by changing the integration sequence, the following inequalities hold: ^𝑇₀^𝑡_𝑡−𝜛𝑒^𝜖𝑡‖𝑒(𝑠)‖²≤𝑑𝑠𝑑𝑡_{𝑇−𝜛𝜃𝜃+𝜛}𝑒^𝜖𝑡𝑑𝑡‖𝑒(𝑠)‖²≤𝑑𝑠_𝑇−𝜛𝜛𝑒^{𝜖(𝜃+𝜛)}‖𝑒(𝑠)‖²𝑑𝑠≤𝜛𝑒^𝜖𝜛𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡)‖²𝑑𝑡+𝜛𝑒^𝜖𝜛0_−𝜛𝜑(𝑠)²𝑑𝑠≤𝜛𝑒^𝜖𝜛𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡)‖²𝑑𝑡+𝜛𝑒^𝜖𝜛sup_{−𝜛≤𝑠≤0}𝜑(𝑠)²,_{𝑇0𝑡𝑡−𝜛}𝑒^𝜖𝑡‖𝑒(𝑠−𝑑(𝑠))‖²𝑑𝑠𝑑𝑡≤𝜛𝑒^𝜖𝜛𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡−𝑑(𝑡))‖²𝑑𝑡+𝜛𝑒^𝜖𝜛sup_{−𝜛≤𝑠≤0}𝜑(𝑠)²,_{𝑇0𝑡𝑡−𝜛}𝑒^𝜖𝑡‖𝑒(𝑠−𝜏(𝑠))‖²𝑑𝑠𝑑𝑡≤𝜛𝑒^𝜖𝜛𝑇₀𝑒^𝜖𝑡‖𝑒(𝑡−𝜏(𝑡))‖²𝑑𝑡+𝜛𝑒^𝜖𝜛sup_{−𝜛≤𝑠≤0}𝜑(𝑠)². (63)
After substituting (63) into the right side of (62) and then using 𝜖max{𝛿₀,𝛿₁+𝛿₂𝜛𝑒^𝜖𝜛,𝛿₃𝜛𝑒^𝜖𝜛,𝛿₄𝜛𝑒^𝜖𝜛}≤𝜉, we can obtain 𝔼𝑉(𝑒(𝑡),𝑡,𝑖)≤𝐽₁+𝐽₂, (64) where 𝐽₂=(𝜖𝛿₂𝜛²𝑒^𝜖𝜛+𝜖𝛿₃𝜛²𝑒^𝜖𝜛+𝜖𝛿₄𝜛²𝑒^𝜖𝜛)sup_{−𝜛≤𝑠≤0}𝔼‖𝜑(𝑠)‖².
So, 𝔼‖𝑒(𝑇)‖²≤𝐽₁+𝐽₂𝛿𝑒^−𝜖𝑡. (65) Then it can be shown that, for any 𝑡>0, 𝔼‖𝑒(𝑡)‖²≤𝜁𝑒^−𝜖𝑡𝔼sup_{−𝜛≤𝑠≤0}𝜑(𝑠)², (66) where 𝜁=(1/𝛿)[𝛿₀+𝛿₁+𝜏𝛿₂+𝜏𝛿₃+𝜏𝛿₄+𝜖𝛿₂𝜛²𝑒^𝜖𝜛+𝜖𝛿₃𝜛²𝑒^𝜖𝜛+𝜖𝛿₄𝜛²𝑒^𝜖𝜛].
Consequently, according to the Lyapunov-Krasovskii stability theory and Definition 3, we know that the error system (11) is exponentially stable. This completes the proof.

4. Exponential Synchronization for Stochastic Neural Networks with Mixed Time Delays and Markovian Jump via Sampled Data

In this section, some sufficient conditions of exponential synchronization for stochastic error system (14) are obtained by employing the Lyapunov-Krasovskii functionals.

Theorem 8. Under Assumptions ℋ₁, ℋ₂, and ℋ₃, for given scalar 𝛾, the error system (14) is globally exponentially stable, which ensures that the master system (12) and slave system (13) are stochastically synchronized, if there exist positive scalars 𝜆_𝑖, symmetric positive definite matrices 𝑃_𝑖, 𝑄₁, 𝑄₂, 𝑄₃, 𝑍₁, 𝑍₂, 𝑍₃, and 𝑍₄, positive definite matrices 𝑌_1𝑖, 𝑌_2𝑖, 𝑌_3𝑖, and 𝑌_4𝑖, and matrices 𝑅_𝑘, 𝑄_𝑘, 𝑇_𝑘, (𝑘=1,2,…,8), 𝐺, and 𝐿, such that, for any 𝑖∈𝒮, the following matrix inequalities hold: 𝑃_𝑖≤𝜆_𝑖Π𝐼,Ω=𝑑₁𝑅𝑑₁₂𝑄√ℎ𝑇∗−𝑍₁00∗∗−𝑍₂0∗∗∗−𝑍₄<0, (67)(68) where 𝑅=𝑐𝑜𝑙{𝑅₁𝑅₂⋯𝑅₈}, 𝑄=𝑐𝑜𝑙{𝑄₁𝑄₂⋯𝑄₈}, 𝑇=𝑐𝑜𝑙{𝑇₁𝑇₂⋯𝑇₈}, and Π is given as follows: ΠΠ=^11𝑖Π₁₂Π_13𝑖Π₁₄Π₁₅Π₁₆Π_17𝑖Π_18𝑖∗Π₂₂Π₂₃Π₂₄Π₂₅Π₂₆Π₂₇Π₂₈∗∗Π₃₃−𝑄₃0−𝑇₃𝛾𝐹^𝑇𝑊^𝑇_1𝑖𝐺0∗∗∗Π₄₄−𝑄₅^𝑇Π₄₆−𝑄₇^𝑇−𝑄₈^𝑇∗∗∗∗𝜆_𝑖𝑌_3𝑖−𝑇₅00∗∗∗∗∗Π₆₆Π_67𝑖−𝑇₈^𝑇∗∗∗∗∗∗Π₇₇𝛾𝐺𝑊_2𝑖∗∗∗∗∗∗∗−𝑍₃, (69)where Π_11𝑖=^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝜆_𝑖𝑌_1𝑖+𝑄₁+𝑄₂+𝑄₃+𝜏²𝐹^𝑇𝑍₃𝐹−𝐺𝐶_𝑖−𝐶^𝑇_𝑖𝐺+𝐺𝑊_0𝑖𝐹+𝐹^𝑇𝑊^𝑇_0𝑖𝐺+𝑅₁+𝑅₁^𝑇+𝑇₁+𝑇₁^𝑇,Π₁₂=−𝑅₁+𝑅₂^𝑇+𝑄₁+𝑇₂^𝑇,Π_13𝑖=𝐺𝑊_1𝑖𝐹+𝑅₃^𝑇+𝑇₃^𝑇,Π₁₄=𝑅₄^𝑇−𝑄₁+𝑇₄^𝑇,Π₁₅=𝑅₅^𝑇+𝑇₅^𝑇,Π₁₆=𝐿+𝑅₆^𝑇+𝑇₆^𝑇−𝑇₁,Π_17𝑖=𝑃_𝑖−𝐺−𝛾𝐶^𝑇_𝑖𝐺+𝛾𝐹^𝑇𝑊^𝑇_0𝑖𝐺+𝑅₇^𝑇+𝑇₇^𝑇,Π_18𝑖=𝐺𝑊_2𝑖+𝑅₈^𝑇+𝑇₈^𝑇,Π₂₂=−𝑄₁−𝑅₂−𝑅₁^𝑇+𝑄₂+𝑄₂^𝑇,Π₂₃=−𝑅₃^𝑇+𝑄₃^𝑇,Π₂₄=−𝑅₄^𝑇−𝑄₂+𝑄₄^𝑇,Π₂₅=−𝑅₅^𝑇+𝑄₅^𝑇,Π₂₆=−𝑅₆^𝑇+𝑄₆^𝑇−𝑇₂,Π₂₇=−𝑅₇^𝑇+𝑄₇^𝑇,Π₂₈=−𝑅₈^𝑇+𝑄₈^𝑇,Π₃₃=𝜆_𝑖𝑌_2𝑖−1−𝜇𝑄₂,Π₄₄=−𝑄₃−𝑄₄−𝑄₄^𝑇,Π₄₆=−𝑄₆^𝑇−𝑇₄,Π₆₆=𝜆_𝑖𝑌_3𝑖−𝑇₆−𝑇₆^𝑇,Π_67𝑖=𝛾𝐿^𝑇−𝑇₇^𝑇+𝜆_𝑖𝑌_4𝑖,Π₇₇=𝑑₁𝑍₁+𝑑₁₂𝑍₂+ℎ𝑍₄−𝛾𝐺−𝛾𝐺^𝑇,𝐹=diag𝐹₁𝐹₂⋯𝐹_𝑛,𝐹_𝜀=max𝐹⁻_𝜀,𝐹⁺_𝜀,𝜀=1,2,…,𝑛. (70)

Proof. Let 𝛽(𝑒(𝑡))=−𝐶(𝑟(𝑡))𝑒(𝑡)+𝑊₀(𝑟(𝑡))𝑓(𝑒(𝑡))+𝑊₁(𝑟(𝑡))𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊₂(𝑟(𝑡))∫^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒(𝑡_𝑘). Then, the system (14) can be written as (𝑡)(𝑒(𝑡((𝑡)((𝑡,𝑒((𝑡𝑡𝑑𝑒=𝛽))𝑑𝑡+𝜌𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘,𝑟(𝑡)𝑑𝜔(𝑡). (71)
To analyze the stability of error system (14), we construct the following stochastic Lyapunov functional candidate: 𝑉(𝑒(𝑡),𝑡,𝑟(𝑡))=⁶_𝑙=1𝑉_𝑙(𝑒(𝑡)(𝑡,𝑡,𝑡,𝑟))𝑡∈_𝑘,𝑡_𝑘+1, (72) where 𝑉₁(𝑒(𝑡),𝑡,𝑟(𝑡))=𝑒^𝑇(𝑡)𝑃(𝑟(𝑡𝑒(𝑡),𝑉))₂(𝑒(𝑡)(𝑡=,𝑡,𝑟))_{𝑡𝑡−𝑑1}𝑒^𝑇(𝑠)𝑄₁𝑒(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝑑}𝑒^𝑇(𝑠)𝑄₂𝑒(𝑠)+𝑑𝑠_{𝑡𝑡−𝑑2}𝑒^𝑇(𝑠)𝑄₃𝑒(𝑠)𝑉𝑑𝑠,₃(𝑒(𝑡),𝑡,𝑟(𝑡))=⁰_−𝑑1^𝑡_𝑡+𝜃𝛽^𝑇(𝑒(𝑠))𝑍₁𝛽(𝑒(𝑠𝑉))𝑑𝑠𝑑𝜃,₄(𝑒(𝑡),𝑡,𝑟(𝑡))=^−𝑑1_−𝑑2^𝑡_𝑡+𝜃𝛽^𝑇(𝑒(𝑠))𝑍₂𝛽(𝑒(𝑠𝑉))𝑑𝑠𝑑𝜃,₅(𝑒(𝑡),𝑡,𝑟(𝑡))=𝜏⁰_−𝜏^𝑡_𝑡+𝜃𝑓^𝑇(𝑒(𝑠))𝑍₃𝑓(𝑒(𝑠𝑉))𝑑𝑠𝑑𝜃,₆(𝑒(𝑡)(𝑡=,𝑡,𝑟))_{0−ℎ𝑡𝑡+𝜃}𝛽^𝑇(𝑒(𝑠))𝑍₄𝛽(𝑒(𝑠))𝑑𝑠𝑑𝜃. (73)
Let ℒ be the weak infinitesimal operator of stochastic process (𝑒(𝑡),𝑡≥0,𝑟(𝑡)) along the trajectories of error system (14). Then we obtain that (𝑒(𝑡))(𝑒(𝑡))𝑑𝑉,𝑡,𝑖=ℒ𝑉,𝑡,𝑖𝑑𝑡+2𝑒^𝑇(𝑡)𝑃_𝑖𝜌((𝑡)((𝑡,𝑒((𝑡𝑡𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘,𝑖𝑑𝜔(𝑡), (74) where ℒ𝑉(𝑒(𝑡),𝑡,𝑖)=𝑉_𝑡(𝑒(𝑡),𝑡,𝑖)+𝑉_𝑥(𝑒(𝑡))𝛽(𝑒(𝑡+1,𝑡,𝑖))2trace𝜌^𝑇((𝑡)((𝑡,𝑒((𝑡𝑡𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘,𝑖×𝑉_𝑒𝑒(𝑒(𝑡))((𝑡)((𝑡,𝑒((𝑡𝑡,𝑡,𝑖×𝜌𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘+,𝑖_𝒮𝑗=1𝜋_𝑖𝑗𝑉𝑒(𝑡),𝑉,𝑡,𝑗_𝑡(𝑒(𝑡))=(𝑒(𝑡)),𝑡,𝑖𝜕𝑉,𝑡,𝑖,𝑉𝜕𝑡_𝑒(𝑒(𝑡))=(𝑒(𝑡)),𝑡,𝑖𝜕𝑉,𝑡,𝑖𝜕𝑒₁,𝜕𝑉(𝑒(𝑡),𝑡,𝑖)𝜕𝑒₂(𝑒(𝑡)),…,𝜕𝑉,𝑡,𝑖𝜕𝑒_𝑛,𝑉_𝑒𝑒(𝑒(𝑡))=𝜕,𝑡,𝑖²𝑉(𝑒(𝑡),𝑡,𝑖)𝜕𝑒_𝑖𝜕𝑒_{𝑗𝑛×𝑛}. (75)
So, we have that ℒ𝑉₁(𝑒(𝑡),𝑡,𝑖)=2𝑒^𝑇(𝑡)𝑃_𝑖𝛽(𝑒(𝑡))+𝑒^𝑇(𝑡)^𝒮_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒(𝑡)+trace𝜌^𝑇𝑡,𝑒(𝑡),𝑒(𝑡−𝑑(𝑡)),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡_𝑘,𝑖×𝑃_𝑖𝜌((𝑡)((𝑡,𝑒((𝑡𝑡𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘,𝑖≤2𝑒^𝑇(𝑡)𝑃_𝑖𝛽(𝑒(𝑡))+𝑒^𝑇(𝑡)^𝒮_𝑗=1𝜋_𝑖𝑗𝑃_𝑗𝑒(𝑡)+𝜆_𝑖𝑒^𝑇(𝑡)𝑌_1𝑖𝑒(𝑡)+𝑒^𝑇(𝑡−𝑑(𝑡))×𝑌_2𝑖𝑒(𝑡−𝑑(𝑡))+𝑒^𝑇(𝑡−𝜏(𝑡))𝑌_3𝑖𝑒(𝑡−𝜏(𝑡))+𝑒^𝑇𝑡_𝑘𝑌_4𝑖𝑒𝑡_𝑘,ℒ𝑉₂(𝑒(𝑡),𝑡,𝑖)=𝑒^𝑇(𝑡)𝑄₁+𝑄₂+𝑄₃𝑒(𝑡)−𝑒^𝑇𝑡−𝑑₁𝑄₁𝑒𝑡−𝑑₁−1−𝜇𝑒^𝑇(𝑡−𝑑(𝑡))𝑄₂𝑒(𝑡−𝑑(𝑡))−𝑒^𝑇𝑡−𝑑₂𝑄₃𝑒𝑡−𝑑₂,ℒ𝑉₃(𝑒(𝑡),𝑡,𝑖)=𝑑₁𝛽^𝑇(𝑒(𝑡))𝑍₁𝛽(𝑒(𝑡−))_{𝑡𝑡−𝑑1}𝛽^𝑇(𝑒(𝑡))𝑍₁𝛽(𝑒(𝑡))𝑑𝑠,ℒ𝑉₄(𝑒(𝑡),𝑡,𝑖)=𝑑₁₂𝛽^𝑇(𝑒(𝑡))𝑍₂𝛽(𝑒(𝑡−))_{𝑡𝑡−𝑑1}𝛽^𝑇(𝑒(𝑡))𝑍₂𝛽(𝑒(𝑡))𝑑𝑠,ℒ𝑉₆(𝑒(𝑡),𝑡,𝑖)=ℎ𝛽^𝑇(𝑒(𝑡))𝑍₄𝛽(𝑒(𝑡−))_{𝑡𝑡−ℎ}𝛽^𝑇(𝑒(𝑡))𝑍₄𝛽(𝑒(𝑡))𝑑𝑠≤ℎ𝛽^𝑇(𝑒(𝑡))𝑍₄𝛽(𝑒(𝑡−))_𝑡𝑡𝑘𝛽^𝑇(𝑒(𝑡))𝑍₄𝛽(𝑒(𝑡))𝑑𝑠. (76)
It is easy to see from (71) that 𝑒(𝑡)−𝑒𝑡−𝑑₁=^𝑡_𝑡−𝑑1𝛽(𝑒(𝑠+))𝑑𝑠_{𝑡𝑡−𝑑1}𝜌((𝑡)((𝑡,𝑒((𝑡𝑡𝑡,𝑒,𝑒𝑡−𝑑))𝑡−𝜏)),𝑒_𝑘,𝑖𝑑𝜔(𝑡). (77)
Taking the mathematical expectation on both sides of (77) yields 𝔼^𝑡_𝑡−𝑑1𝛽(𝑒(𝑠𝑒(𝑡)))𝑑𝑠=𝔼−𝑒𝑡−𝑑₁=𝔼𝜉^𝑇₁𝒳(𝑡), (78) where𝒳(𝑡)=col𝑒(𝑡)𝑒𝑡−𝑑₁𝑒(𝑡−𝑑(𝑡))𝑒𝑡−𝑑₂𝑒(𝑡−𝜏(𝑡))𝑒𝑡_𝑘𝛽(𝑡)∫^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑒(𝑠,𝜉))𝑑𝑠₁=col𝐼−𝐼000000. (79)
Applying Lemma 6, we have that there exists a matrix 𝑅∈𝑅^8𝑛×𝑛 such that 𝔼−^𝑡_𝑡−𝑑1𝛽^𝑇(𝑒(𝑠))𝑍₁𝛽(𝑒(𝑠𝒳(𝑡)))𝑑𝑠≤𝔼^𝑇𝑅𝜉^𝑇₁+𝜉₁𝑅^𝑇+𝑑₁𝑅𝑍₁⁻¹𝑅^𝑇𝒳(𝑡). (80)
Similarly, it can be seen that there exist matrices 𝑄 and 𝑇∈𝑅^8𝑛×𝑛 such that 𝔼−^𝑡−𝑑1_𝑡−𝑑2𝛽^𝑇(𝑒(𝑠))𝑍₂𝛽(𝑒(𝑠𝒳(𝑡)))𝑑𝑠≤𝔼^𝑇𝑄𝜉^𝑇₂+𝜉₂𝑄^𝑇+𝑑₁₂𝑄𝑍₂⁻¹𝑄^𝑇𝒳(𝑡,𝔼−_𝑡𝑡𝑘)𝛽^𝑇(𝑒(𝑠))𝑍₄𝛽(𝑒(𝑠𝒳(𝑡)))𝑑𝑠≤𝔼^𝑇𝑇𝜉^𝑇₃+𝜉₃𝑇^𝑇+ℎ𝑇𝑍₄⁻¹𝑇^𝑇𝒳(𝑡), (81) where 𝜉₂,𝜉=col0𝐼0−10000₃=col𝐼0000−𝐼00. (82)
Furthermore, according to the definition of 𝛽(𝑒(𝑡)), for any appropriately dimensioned matrix 𝐺 and scalar 𝛾, the following equality is satisfied: 2𝑒^𝑇(𝑡)𝐺+𝛾𝛽^𝑇(𝑒(𝑡𝐺×(𝑒(𝑡))−𝛽))−𝐶_𝑖𝑒(𝑡)+𝑊_0𝑖𝑓(𝑒(𝑡))+𝑊_1𝑖𝑓(𝑒(𝑡−𝑑(𝑡)))+𝑊_2𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑓(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘=0. (83)
Taking the mathematical expectation on both sides of (74), letting 𝐿=𝐺𝐾, and considering (8), (33), (76), and (80)–(83), we obtain that 𝔼{(𝑒(𝑡)𝒳(𝑡)ℒ𝑉,𝑡,𝑖)}=𝔼^𝑇Π+𝑑₁𝑅𝑍₁⁻¹𝑅^𝑇+𝑑₁₂𝑄𝑍₂⁻¹𝑄^𝑇+ℎ𝑇𝑍₄⁻¹𝑇^𝑇𝒳(𝑡). (84)
Applying Lemma 5 and (68), we have that 𝔼{(𝑒(𝑡)‖‖ℒ𝑉,𝑡,𝑖)}≤𝔼−𝜉𝑒(𝑡)²+‖𝑒(𝑡−𝑑(𝑡))‖²+‖𝑒(𝑡−𝜏(𝑡))‖². (85)
Following the similar line of the proof of Theorem 7, we can get that, for any 𝑡>0, 𝔼‖𝑒(𝑡)‖²≤𝜁𝑒^−𝜖𝑡𝔼sup_{−𝜛≤𝑠≤0}𝜑(𝑠)², (86) where 𝜁=(1/𝛿)[𝛿₁+𝜏𝛿₂+𝜏𝛿₃+𝜏𝛿₄+𝜖𝛿₂𝜛²𝑒^𝜖𝜛+𝜖𝛿₃𝜛²𝑒^𝜖𝜛+𝜖𝛿₄𝜛²𝑒^𝜖𝜛].
Thus, the master system and the slave system are exponentially synchronized; the sampled-data feedback control gain is given by 𝐾=𝐺⁻¹𝐿. This completes the proof.

5. Illustrative Examples

In this section, two numerical examples are given to demonstrate the effectiveness of the theoretical results.

Example 1. Consider the second-order master system (3) and slave system (6) with the following parameters: 𝐶₁=0.8000.9,𝑊₀₁=,𝑊1.7−0.15−5.23.3₁₁=,𝐶−1.7−0.1−0.26−2.5₂=1001,𝑊₀₂=,𝑊1.5−0.16−5.13.4₁₂=,𝑊−1.9−0.1−0.25−2.6₂₁=0.70.152−0.12,𝑊₂₂=,0.80.151.5−0.12 (87) and the activation functions are taken as 𝑔(𝛼)=(|𝛼+1|−|𝛼−1|)/2.
It can be verified that 𝐹⁻₁=𝐹⁻₂=0 and 𝐹⁺₁=𝐹⁺₂=1. Thus, 𝐹₁=⁰⁰₀₀ and 𝐹₂=^0.50_00.5.
It is assumed that 𝐼(𝑡)=0, discrete delay 𝑑(𝑡)=𝑒^𝑡/𝑒^𝑡+1, and distributed delay 𝜏(𝑡)=0.5sin²(𝑡). Hence, a straightforward calculation gives 𝑑₁=0.5, 𝑑₂=1, 𝜇=0.25, and 𝜏=0.5. Moreover, the transition probability matrix is chosen as Υ=⁻⁵⁵₅₋₅.
The chaotic behaviors of the master system (3) and slave system (6) with 𝑢(𝑡)=0 are given in Figures 1 and 2, respectively, with the initial states chosen as 𝑥(𝑡)=−0.50.5^𝑇 and 𝑦(𝑡)=0.5−0.5^𝑇, 𝑡∈−10.
Choosing 𝛾=0.7 and applying Theorem 7, we can find that the upper bound on the samplings, which preserves that master system (3) and slave system (6) are exponentially synchronous, is 0.11. And using the Matlab LMI Control Toolbox to solve LMIs (21)–(24), we can also obtain the following matrices: ,.𝐺=1.78920.13260.13260.2696𝐿=−3.47410.05130.05130.1577 (88)
Thus, the corresponding gain matrix in (9) is given by 𝐾=−2.0298−0.01521.18820.5924. (89)
Under the obtained gain matrix in (89), the response curves of control input (9) and error system (11) are exhibited in Figures 3 and 4, respectively. It is obvious from Figure 4 that the slave system (3) exponentially synchronizes with the master system (6).

Figure 1 

Chaotic behavior of master system (3) with 𝑢(𝑡)=0.

Figure 2 

Chaotic behavior of slave system (6) with 𝑢(𝑡)=0.

Figure 3 

Control input 𝑢(𝑡).

Figure 4 

State responses of error system (10).

Example 2. In the following, we consider the second-order stochastic master system (12) and slave system (13) with 𝐼(𝑡)=(𝐼₁(𝑡),𝐼₂(𝑡))^𝑇; 𝜔(𝑡) is a second-order Brownian motion and 𝑟(𝑡) is a right-continuous Markovian chain taking values in 𝑆={1,2} with generator Υ=⁻³³₃₋₃.
For the two operating conditions (modes), the associated data are 𝐶₁=0.8000.9,𝑊₀₁=,𝑊0.15.13.20.1₁₁=,𝐶−0.13.25.1−0.1₂=1001,𝑊₀₂=,𝑊05.23.10₁₂=,𝑊−0.13.15.2−0.1₂₁=0.70.152−0.12,𝑊₂₂=,𝜌(𝑡)((𝑡((𝑡𝑡0.80.151.5−0.12𝑡,𝑒,𝑒𝑡−𝑑)),𝑒𝑡−𝜏)),𝑒_𝑘=𝑧,1₁(𝑡)00𝑧₂(𝑡),𝜌𝑡,𝑒(𝑡),𝑒(𝑡−𝑑(𝑡)),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡_𝑘=𝑧,2₃(𝑡)00𝑧₄(𝑡), (90) where 𝑧₁(𝑡)=0.2𝑒₁(𝑡)+0.3𝑒₁(𝑡−𝑑(𝑡))+0.1𝑒₁(𝑡−𝜏(𝑡))+0.4𝑒₁𝑡_𝑘,𝑧₂(𝑡)=0.1𝑒₂(𝑡)+0.2𝑒₂(𝑡−𝑑(𝑡))+0.2𝑒₂(𝑡−𝜏(𝑡))+0.3𝑒₂𝑡_𝑘,𝑧₃(𝑡)=0.2𝑒₁(𝑡)+0.3𝑒₁(𝑡−𝑑(𝑡))+0.1𝑒₁(𝑡−𝜏(𝑡))+0.2𝑒₁𝑡_𝑘,𝑧₄(𝑡)=0.2𝑒₂(𝑡)+0.1𝑒₂(𝑡−𝑑(𝑡))+0.1𝑒₂(𝑡−𝜏(𝑡))+0.3𝑒₂𝑡_𝑘 (91) and the activation functions are taken as 𝑔(𝛼)=(1/2)sin(𝛼).
It can be verified that 𝐹⁻₁=𝐹⁻₂=−0.5 and 𝐹⁺₁=𝐹⁺₂=0.5. Thus, 𝐹=^0.50_00.5.
In this example, 𝐼(𝑡)=0, discrete delay 𝑑(𝑡)=1+0.3sin(2𝑡), and distributed delay 𝜏(𝑡)=0.5sin²(𝑡). Then, a straightforward calculation gives 𝑑₁=0.7, 𝑑₂=1.3, 𝜏=0.5, and 𝑑(̇𝑡)=0.6cos(2𝑡)≤0.6, which implies 𝜇=0.6.
Figures 5 and 6 show the chaotic behavior of the master system (12) and slave system (13) with 𝑢(𝑡)=0, respectively, under the initial states 𝑥(𝑡)=−0.20.2^𝑇 and 𝑦(𝑡)=0.2−0.2^𝑇, 𝑡∈−1.30.
By using Matlab LMI Control Toolbox to solve the LMIs given in Theorem 8 with 𝛾=0.2, we can find that the upper bound on the samplings, which preserves that master system (12) and slave system (13) are exponentially synchronous, is 0.1. Moreover, we can also get the following matrices: ,.𝐺=0.1798−0.0507−0.05070.1495𝐿=−1.7755−0.0854−0.0854−1.6652 (92)
Thus, the corresponding gain matrix in (9) is given by 𝐾=𝐺⁻¹𝐿=−11.1000−3.9995−4.3360−12.4944. (93)
Under the above given gain matrix, Figures 7 and 8 show the response curves of control input (9) and error system (14), respectively. It is clear from Figures 7 and 8 that the obtained sampled-data controller achieves the exponential synchronization of master system (12) and slave system (13).

Figure 5 

Chaotic behavior of master system (12) with 𝑢(𝑡)=0.

Figure 6 

Chaotic behavior of slave system (13) with 𝑢(𝑡)=0.

Figure 7 

Control input 𝑢(𝑡).

Figure 8 

State responses of error system (14).

6. Conclusion

In this paper, the exponential synchronization issue for stochastic neural networks (SNNs) with mixed time delays and Markovian jump parameters under sampled-data control has been addressed. New delay-dependent conditions have been presented in terms of LMIs to ensure the exponential stability of the considered error systems, and, thus, the master systems exponentially synchronize with the slave systems. The results obtained in this paper are a little conservative comparing the previous results in the literature. The methods of this paper can be applied to other classes of neural networks such as complex neural networks and impulsive neural networks.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Science and Technology Major Project of China (2011ZX05020-006) and the Natural Science Foundation of Hebei Province of China (A2011203103).