Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 490515, 20 pages
doi:10.1155/2009/490515
Research Article

Stochastic Stability of Neural Networks with Both Markovian Jump Parameters and Continuously Distributed Delays

Quanxin Zhu1,2 and Jinde Cao1

1Department of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China
2Department of Mathematics, Ningbo University, Ningbo 315211, Zhejiang, China

Received 4 March 2009; Accepted 29 June 2009

Academic Editor: Manuel De La Sen

Copyright © 2009 Quanxin Zhu and Jinde Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problem of stochastic stability is investigated for a class of neural networks with both Markovian jump parameters and continuously distributed delays. The jumping parameters are modeled as a continuous-time, finite-state Markov chain. By constructing appropriate Lyapunov-Krasovskii functionals, some novel stability conditions are obtained in terms of linear matrix inequalities (LMIs). The proposed LMI-based criteria are computationally efficient as they can be easily checked by using recently developed algorithms in solving LMIs. A numerical example is provided to show the effectiveness of the theoretical results and demonstrate the LMI criteria existed in the earlier literature fail. The results obtained in this paper improve and generalize those given in the previous literature.

1. Introduction

In recent years, neural networks (especially recurrent neural networks, Hopfield neural networks, and cellular neural networks) have been successfully applied in many areas such as signal processing, image processing, pattern recognition, fault diagnosis, associative memory, and combinatorial optimization; see, for example, [1–5]. One of the best important works in these applications is to study the stability of the equilibrium point of neural networks. A major purpose that is concerned with is to find stability conditions (i.e., the conditions for the stability of the equilibrium point of neural networks). To do this, existensive literature has been presented; see, for example, [6–22] and references therein. It should be noted that the methods in the literature have seldom considered the case that the systems have Markovian jump parameters due to the difficulty of mathematics. However, neural networks in real life often have a phenomenon of information latching. It is recognized that a way for dealing with this information latching problem is to extract finite state representations (also called modes or clusters). In fact, such a neural network with information latching may have finite modes, and the modes may switch (or jump) from one to another at different times, and the switching (or jumping) between two arbitrarily different modes can be governed by a Markov chain. Hence, the neural networks with Markovian jump parameters are of great significance in modeling a class of neural networks with finite modes.

On the other hand, the time delay is frequently a major source of instability and poor performance in neural networks (e.g., see [6, 23, 24]), and so the stability analysis for neural networks with time delays is an important research topic. The existing works on neural networks with time delays can be classified into three categories: constant delays, time-varying delays, and distributed delays. It is noticed that most works in the literature have focused on the former two simple cases: constant delays or time-varying delays (e.g., see [6, 8–10, 12–16, 19–22]). However, as pointed out in [18], neural networks usually have a spatial nature due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and so it is desired to model them by introducing continuously distributed delays on a certain duration of time such that the distant past has less influence than the recent behavior of the state. But discussions about the neural networks with continuously distributed delays are only a few researchers [18, 25]. Therefore, there is enough room to develop novel stability conditions for improvement.

Motivated by the above discussion, the objective of this paper is to study the stability for a class of neural networks with both Markovian jump parameters and continuously distributed delays. Moreover, to make the model more general and practical, the factor of noise disturbance is considered in this paper since noise disturbance is also a major source leading to instability [7]. To the best of the authors' knowledge, up to now, the stability analysis problem for a class of stochastic neural networks with both Markovian jump parameters and continuously distributed delays is still an open problem that has not been properly studied. Therefore, this paper is the first attempt to introduce and investigate the problem of stochastic stability for a class of neural networks with both Markovian jump parameters and continuously distributed delays. By utilizing the Lyapunov stability theory and linear matrix inequality (LMI) technique, some novel delay-dependent conditions are obtained to guarantee the stochastically asymptotic stability of the equilibrium point. The proposed LMI-based criteria are computationally efficient as they can be easily checked by using recently developed standard algorithms such as interior point methods [24] in solving LMIs. Finally, a numerical example is provided to illustrate the effectiveness of the theoretical results and demonstrate the LMI criteria existed in the earlier literature fail. The results obtained in this paper improve and generalize those given in the previous literature.

The remainder of this paper is organized as follows. In Section 2, the model of a class of stochastic neural networks with both Markovian jump parameters and continuously distributed delays is introduced, and some assumptions needed in this paper are presented. By means of Lyapunov-Krasovskii functional approach, our main results are established in Section 3. In Section 4, a numerical example is given to show the effectiveness of the obtained results. Finally, in Section 5, the paper is concluded with some general conclusions.

Notation. Throughout this paper, the following notations will be used. ℝ 𝑛 and ℝ 𝑛 × 𝑛 denote the 𝑛 -dimensional Euclidean space and the set of all 𝑛 × 𝑛 real matrices, respectively. The superscript “ 𝑇 " denotes the transpose of a matrix or vector. Trace ( ⋅ ) denotes the trace of the corresponding matrix, and 𝐼 denotes the identity matrix with compatible dimensions. For square matrices 𝑀 1 and 𝑀 2 , the notation 𝑀 1 > ( ≥ , < , ≤ ) 𝑀 2 denotes that 𝑀 1 − 𝑀 2 is positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. Let 𝑤 ( 𝑡 ) = ( 𝑤 1 , … , 𝑤 𝑛 ) 𝑇 be an 𝑛 -dimensional Brownian motion defined on a complete probability space ( Ω , ℱ , 𝑃 ) with a natural filtration { ℱ 𝑡 } 𝑡 ≥ 0 . Also, let 𝜏 > 0 and 𝐶 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) denote the family of continuous function 𝜙 from [ − 𝜏 , 0 ] to ℝ 𝑛 with the uniform norm ‖ 𝜙 ‖ = s u p − 𝜏 ≤ 𝜃 ≤ 0 | 𝜙 ( 𝜃 ) | . Denote by 𝐿 2 ℱ 𝑡 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) the family of all ℱ 𝑡 measurable, 𝐶 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) -valued stochastic variables 𝜉 = { 𝜉 ( 𝜃 ) ∶ − 𝜏 ≤ 𝜃 ≤ 0 } such that ∫ 0 − 𝜏 𝐸 | 𝜉 ( 𝑠 ) | 2 𝑑 𝑠 < ∞ , where 𝐸 [ ⋅ ] stands for the correspondent expectation operator with respect to the given probability measure 𝑃 .

2. Model Description and Problem Formulation

Let { 𝑟 ( 𝑡 ) , 𝑡 ≥ 0 } be a right-continuous Markov chain on a complete probability space ( Ω , ℱ , 𝑃 ) taking values in a finite state space 𝑆 = { 1 , 2 , … , 𝑁 } with generator 𝑄 = ( 𝑞 𝑖 𝑗 ) 𝑁 × 𝑁 given by 𝑞 𝑃 { 𝑟 ( 𝑡 + Δ 𝑡 ) = 𝑗 ∣ 𝑟 ( 𝑡 ) = 𝑖 } = 𝑖 𝑗 Δ 𝑡 + 𝑜 ( Δ 𝑡 ) , i f 𝑖 ≠ 𝑗 , 1 + 𝑞 𝑖 𝑖 Δ 𝑡 + 𝑜 ( Δ 𝑡 ) , i f 𝑖 = 𝑗 , ( 2 . 1 ) where Δ 𝑡 > 0 and l i m Δ 𝑡 → 0 ( 𝑜 ( Δ 𝑡 ) / Δ 𝑡 ) = 0 . Here, 𝑞 𝑖 𝑗 ≥ 0 is the transition rate from 𝑖 to 𝑗 if 𝑖 ≠ 𝑗 while 𝑞 𝑖 𝑖 ∑ = − 𝑗 ≠ 𝑖 𝑞 𝑖 𝑗 .

In this paper we consider a class of neural networks with both Markovian jump parameters and continuously distributed delays, which is described by the following integro-differential equation: ̇ 𝑥 ( 𝑡 ) = − 𝐷 ( 𝑟 ( 𝑡 ) ) 𝑥 ( 𝑡 ) + 𝐴 ( 𝑟 ( 𝑡 ) ) 𝑓 ( 𝑥 ( 𝑡 ) ) + 𝐵 ( 𝑟 ( 𝑡 ) ) 𝑔 ( 𝑥 ( 𝑡 − 𝜏 ) ) + 𝐶 ( 𝑟 ( 𝑡 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑥 ( 𝑠 ) ) 𝑑 𝑠 + 𝑉 , ( 2 . 2 ) where 𝑥 ( 𝑡 ) = [ 𝑥 1 ( 𝑡 ) , 𝑥 2 ( 𝑡 ) , … , 𝑥 𝑛 ( 𝑡 ) ] 𝑇 is the state vector associated with the 𝑛 neurons, and the diagonal matrix 𝐷 ( 𝑟 ( 𝑡 ) ) = d i a g ( 𝑑 1 ( 𝑟 ( 𝑡 ) ) , 𝑑 2 ( 𝑟 ( 𝑡 ) ) , … , 𝑑 𝑛 ( 𝑟 ( 𝑡 ) ) ) has positive entries 𝑑 𝑖 ( 𝑟 ( 𝑡 ) ) > 0 ( 𝑖 = 1 , 2 , … , 𝑛 ) . The matrices 𝐴 ( 𝑟 ( 𝑡 ) ) = ( 𝑎 𝑖 𝑗 ( 𝑟 ( 𝑡 ) ) ) 𝑛 × 𝑛 , 𝐵 ( 𝑟 ( 𝑡 ) ) = ( 𝑏 𝑖 𝑗 ( 𝑟 ( 𝑡 ) ) ) 𝑛 × 𝑛 , and 𝐶 ( 𝑟 ( 𝑡 ) ) = ( 𝑐 𝑖 𝑗 ( 𝑟 ( 𝑡 ) ) ) 𝑛 × 𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weight matrix. 𝑓 ( 𝑥 ( 𝑡 ) ) = [ 𝑓 1 ( 𝑥 1 ( 𝑡 ) ) , 𝑓 2 ( 𝑥 2 ( 𝑡 ) ) , … , 𝑓 𝑛 ( 𝑥 𝑛 ( 𝑡 ) ) ] 𝑇 , 𝑔 ( 𝑥 ( 𝑡 ) ) = [ 𝑔 1 ( 𝑥 1 ( 𝑡 ) ) , 𝑔 2 ( 𝑥 2 ( 𝑡 ) ) , … , 𝑔 𝑛 ( 𝑥 𝑛 ( 𝑡 ) ) ] 𝑇 , and ℎ ( 𝑥 ( 𝑡 ) ) = [ ℎ 1 ( 𝑥 1 ( 𝑡 ) ) , ℎ 2 ( 𝑥 2 ( 𝑡 ) ) , … , ℎ 𝑛 ( 𝑥 𝑛 ( 𝑡 ) ) ] 𝑇 denote the neuron activation functions, and 𝑉 = [ 𝑉 1 , 𝑉 2 , … , 𝑉 𝑛 ] 𝑇 denotes a constant external input vector. The constant 𝜏 > 0 denotes the time delay, and 𝑅 = [ 𝑅 1 , 𝑅 2 , … , 𝑅 𝑛 ] 𝑇 denotes the delay kernel vector, where 𝑅 𝑖 is a real value nonnegative continuous function defined on [ 0 , ∞ ) and such that ∫ ∞ 0 𝑅 𝑖 ( 𝑠 ) 𝑑 𝑠 = 1 for 𝑖 = 1 , 2 , … , 𝑛 .

In this paper we will investigate a more general model in which the environmental noise is considered on system (2.2), and so this model can be written as the following integrodifferential equation: 𝑑 𝑥 ( 𝑡 ) = − 𝐷 ( 𝑟 ( 𝑡 ) ) 𝑥 ( 𝑡 ) + 𝐴 ( 𝑟 ( 𝑡 ) ) 𝑓 ( 𝑥 ( 𝑡 ) ) + 𝐵 ( 𝑟 ( 𝑡 ) ) 𝑔 ( 𝑥 ( 𝑡 − 𝜏 ) ) + 𝐶 ( 𝑟 ( 𝑡 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑥 ( 𝑠 ) ) 𝑑 𝑠 + 𝑉 𝑑 𝑡 + 𝜎 ( 𝑥 ( 𝑡 ) , 𝑥 ( 𝑡 − 𝜏 ) , 𝑡 , 𝑟 ( 𝑡 ) ) 𝑑 𝑤 ( 𝑡 ) , ( 2 . 3 ) where 𝜎 ∶ ℝ 𝑛 × ℝ 𝑛 × ℝ + × 𝑆 → ℝ 𝑛 × 𝑛 is the noise perturbation.

Throughout this paper, the following conditions are supposed to hold.

Assumption 2.1. There exist six diagonal matrices 𝑈 − = d i a g ( 𝑢 − 1 , 𝑢 − 2 , … , 𝑢 − 𝑛 ) , 𝑈 + = d i a g ( 𝑢 + 1 , 𝑢 + 2 , … , 𝑢 + 𝑛 ) , 𝑉 − = d i a g ( 𝑣 − 1 , 𝑣 − 2 , … , 𝑣 − 𝑛 ) , 𝑉 + = d i a g ( 𝑣 + 1 , 𝑣 + 2 , … , 𝑣 + 𝑛 ) , 𝑊 − = d i a g ( 𝑤 − 1 , 𝑤 − 2 , … , 𝑤 − 𝑛 ) , and 𝑊 + = d i a g ( 𝑤 + 1 , 𝑤 + 2 , … , 𝑤 + 𝑛 ) satisfying 𝑢 − 𝑖 ≤ 𝑓 𝑖 ( 𝛼 ) − 𝑓 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑢 + 𝑖 , 𝑣 − 𝑖 ≤ 𝑔 𝑖 ( 𝛼 ) − 𝑔 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑣 + 𝑖 , 𝑤 − 𝑖 ≤ ℎ 𝑖 ( 𝛼 ) − ℎ 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑤 + 𝑖 ( 2 . 4 ) for all 𝛼 , 𝛽 ∈ ℝ , 𝛼 ≠ 𝛽 , 𝑖 = 1 , 2 , … , 𝑛 .

Assumption 2.2. There exist two positive definite matrices Σ 1 𝑖 and Σ 2 𝑖 such that 𝜎 t r a c e 𝑇 ( 𝑥 , 𝑦 , 𝑡 , 𝑟 ( 𝑡 ) ) 𝜎 ( 𝑥 , 𝑦 , 𝑡 , 𝑟 ( 𝑡 ) ) ≤ 𝑥 𝑇 Σ 1 𝑖 𝑥 + 𝑦 𝑇 Σ 2 𝑖 𝑦 ( 2 . 5 ) for all 𝑥 , 𝑦 ∈ ℝ 𝑛 and 𝑟 ( 𝑡 ) = 𝑖 , 𝑖 ∈ 𝑆 .

Assumption 2.3. 𝜎 ( 0 , 0 , 𝑡 , 𝑟 ( 𝑡 ) ) ≡ 0 .

Under Assumptions 2.1 and 2.2, it is well known (see, e.g., Mao [16]) that for any initial data 𝑥 ( 𝜃 ) = 𝜉 ( 𝜃 ) on − 𝜏 ≤ 𝜃 ≤ 0 in 𝐿 2 ℱ 𝑡 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) , (2.3) has a unique equilibrium point. Now, let 𝑥 ∗ = ( 𝑥 ∗ 1 , 𝑥 ∗ 2 , … , 𝑥 ∗ 𝑛 ) be the unique equilibrium point of (2.3), and set 𝑦 ( 𝑡 ) = 𝑥 ( 𝑡 ) − 𝑥 ∗ . Then we can rewrite system (2.3) as 𝑑 𝑦 ( 𝑡 ) = − 𝐷 ( 𝑟 ( 𝑡 ) ) 𝑦 ( 𝑡 ) + 𝐴 ( 𝑟 ( 𝑡 ) ) 𝑓 ( 𝑦 ( 𝑡 ) ) + 𝐵 ( 𝑟 ( 𝑡 ) ) 𝑔 ( 𝑦 ( 𝑡 − 𝜏 ) ) + 𝐶 ( 𝑟 ( 𝑡 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 𝑑 𝑡 + 𝜎 ( 𝑦 ( 𝑡 ) , 𝑦 ( 𝑡 − 𝜏 ) , 𝑡 , 𝑟 ( 𝑡 ) ) 𝑑 𝑤 ( 𝑡 ) , ( 2 . 6 ) where 𝑓 ( 𝑦 ) = [ 𝑓 1 ( 𝑦 1 ) , 𝑓 2 ( 𝑦 2 ) , … , 𝑓 𝑛 ( 𝑦 𝑛 ) ] 𝑇 , 𝑔 ( 𝑦 ) = [ 𝑔 1 ( 𝑦 1 ) , 𝑔 2 ( 𝑦 2 ) , … , 𝑔 𝑛 ( 𝑦 𝑛 ) ] 𝑇 , ℎ ( 𝑦 ) = [ ℎ 1 ( 𝑦 1 ) , ℎ 2 ( 𝑦 2 ) , … , ℎ 𝑛 ( 𝑦 𝑛 ) ] 𝑇 , and 𝑓 𝑖 ( 𝑦 𝑖 ) = 𝑓 𝑖 ( 𝑦 𝑖 + 𝑥 ∗ 𝑖 ) − 𝑓 𝑖 ( 𝑥 ∗ 𝑖 ) , 𝑔 𝑖 ( 𝑦 𝑖 ) = 𝑔 𝑖 ( 𝑦 𝑖 + 𝑥 ∗ 𝑖 ) − 𝑔 𝑖 ( 𝑥 ∗ 𝑖 ) , ℎ 𝑖 ( 𝑦 𝑖 ) = ℎ 𝑖 ( 𝑦 𝑖 + 𝑥 ∗ 𝑖 ) − ℎ 𝑖 ( 𝑥 ∗ 𝑖 ) ( 𝑖 = 1 , 2 , … , 𝑛 ) .

Noting the facts that 𝑓 ( 0 ) = 𝑔 ( 0 ) = ℎ ( 0 ) = 0 and 𝜎 ( 0 , 0 , 𝑡 , 𝑟 ( 𝑡 ) ) = 0 , the trivial solution of system (2.6) exists. Hence, to prove the stability of 𝑥 ∗ of (2.3), it is sufficient to prove the stability of the trivial solution of system (2.6). On the other hand, by Assumption 2.1 we have 𝑢 − 𝑖 ≤ 𝑓 𝑖 ( 𝛼 ) − 𝑓 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑢 + 𝑖 , 𝑣 ( 2 . 7 ) − 𝑖 ≤ 𝑔 𝑖 ( 𝛼 ) − 𝑔 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑣 + 𝑖 , 𝑤 ( 2 . 8 ) − 𝑖 ≤ ℎ 𝑖 ( 𝛼 ) − ℎ 𝑖 ( 𝛽 ) 𝛼 − 𝛽 ≤ 𝑤 + 𝑖 ( 2 . 9 ) for all 𝛼 , 𝛽 ∈ ℝ , 𝛼 ≠ 𝛽 , 𝑖 = 1 , 2 , … , 𝑛 .

Let 𝑦 ( 𝑡 ; 𝜉 ) denote the state trajectory from the initial data 𝑦 ( 𝜃 ) = 𝜉 ( 𝜃 ) on − 𝜏 ≤ 𝜃 ≤ 0 in 𝐿 2 ℱ 𝑡 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) . Clearly, system (2.6) admits a trivial solution 𝑦 ( 𝑡 ; 0 ) ≡ 0 corresponding to the initial data 𝜉 = 0 . For simplicity, we write 𝑦 ( 𝑡 ; 𝜉 ) = 𝑦 ( 𝑡 ) . Let 𝐶 2 1 ( ℝ + × ℝ 𝑛 × 𝑆 ; ℝ + ) denote the family of all nonnegative functions 𝑉 ( 𝑡 , 𝑦 , 𝑖 ) on ℝ + × ℝ 𝑛 × 𝑆 which are continuously twice differentiable in 𝑦 and differentiable in 𝑡 . If 𝑉 ∈ 𝐶 2 1 ( ℝ + × ℝ 𝑛 × 𝑆 ; ℝ 𝑛 ) , then along the trajectory of system (2.6) we define an operator ℒ 𝑉 from ℝ + × ℝ 𝑛 × 𝑆 to ℝ by ℒ 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑉 𝑡 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) + 𝑉 𝑦 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) − 𝐷 ( 𝑟 ( 𝑡 ) ) 𝑦 ( 𝑡 ) + 𝐴 ( 𝑟 ( 𝑡 ) ) 𝑓 ( 𝑦 ( 𝑡 ) ) + 𝐵 ( 𝑟 ( 𝑡 ) ) 𝑔 ( 𝑦 ( 𝑡 − 𝜏 ) ) + 𝐶 ( 𝑟 ( 𝑡 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ + 1 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 2 𝜎 t r a c e 𝑇 ( 𝑦 ( 𝑡 ) , 𝑦 ( 𝑡 − 𝜏 ) , 𝑡 , 𝑟 ( 𝑡 ) ) 𝑉 𝑦 𝑦 + ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) 𝜎 ( 𝑦 ( 𝑡 ) , 𝑦 ( 𝑡 − 𝜏 ) , 𝑡 , 𝑟 ( 𝑡 ) ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑗 ) , ( 2 . 1 0 ) where 𝑉 𝑡 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝜕 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) 𝜕 𝑡 , 𝑉 𝑦 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝜕 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) 𝜕 𝑦 1 , … , 𝜕 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) 𝜕 𝑦 𝑛 , 𝑉 𝑦 𝑦 𝜕 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 2 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) 𝜕 𝑦 𝑖 𝜕 𝑦 𝑗 𝑛 × 𝑛 . ( 2 . 1 1 ) Now we give the definition of stochastic asymptotically stability for system (2.6).

Definition 2.4. The equilibrium point of (2.6) (or (2.3) equivalently) is said to be stochastic asymptotically stable in the mean square if, for every 𝜉 ∈ 𝐿 2 ℱ 0 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) , the following equality holds: l i m 𝑡 → ∞ 𝐸 | 𝑦 ( 𝑡 ; 𝜉 ) | 2 = 0 . ( 2 . 1 2 )

In the sequel, for simplicity, when 𝑟 ( 𝑡 ) = 𝑖 , the matrices 𝐶 ( 𝑟 ( 𝑡 ) ) , 𝐴 ( 𝑟 ( 𝑡 ) ) , and 𝐵 ( 𝑟 ( 𝑡 ) ) will be written as 𝐶 𝑖 , 𝐴 𝑖 and 𝐵 𝑖 , respectively.

3. Main Results and Proofs

In this section, the stochastic asymptotically stability in the mean square of the equilibrium point for system (2.6) is investigated under Assumptions 2.1–2.3.

Theorem 3.1. Under Assumptions 2.1–2.3, the equilibrium point of (2.6) (or (2.3) equivalently) is stochastic asymptotically stable, if there exist positive scalars 𝜆 𝑖 , positive definite matrices 𝐺 , 𝐻 , 𝑀 , 𝑃 𝑖 , 𝐾 𝑖 , 𝐿 𝑖 ( 𝑖 ∈ 𝑆 ) , and four positive diagonal matrices 𝑄 1 , 𝑄 2 , 𝑄 3 , and 𝐹 = d i a g ( 𝑓 1 , 𝑓 2 , … , 𝑓 𝑛 ) such that the following LMIs hold: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Γ 1 1 0 𝑃 𝑖 𝐴 𝑖 0 0 0 𝑃 𝑖 𝐵 𝑖 𝑃 𝑖 𝐶 𝑖 ⋆ Γ 2 2 0 0 0 0 0 0 ⋆ ⋆ Γ 3 3 0 0 0 0 0 ⋆ ⋆ ⋆ Γ 4 4 0 0 0 0 ⋆ ⋆ ⋆ ⋆ − 𝑄 3 + 𝐹 0 0 0 ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐾 𝑖 0 0 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐿 𝑖 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 𝑃 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐹 < 0 , ( 3 . 1 ) 𝑖 ≤ 𝜆 𝑖 𝐼 , ( 3 . 2 ) 𝐻 ≥ 𝑁 ∑ 𝑗 = 1 𝑞 𝑖 𝑗 𝐾 𝑗 , ( 3 . 3 ) 𝑀 ≥ 𝑁 ∑ 𝑗 = 1 𝑞 𝑖 𝑗 𝐿 𝑗 , ( 3 . 4 ) where the symbol “ ⋆ " denotes the symmetric term of the matrix: Γ 1 1 = − 2 𝑃 𝑖 𝐷 𝑖 + 𝐺 + 𝜆 𝑖 Σ 1 𝑖 + 𝑈 𝑄 1 𝑈 + 𝑉 𝑄 2 𝑉 + 𝑊 𝑄 3 𝑊 + 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑃 𝑗 , Γ 2 2 = − 𝐺 + 𝜆 𝑖 Σ 2 𝑖 , Γ 3 3 = − 𝑄 1 + 𝐾 𝑖 + 𝜏 𝐻 , Γ 4 4 = − 𝑄 2 + 𝐿 𝑖 𝑢 + 𝜏 𝑀 , 𝑈 = d i a g 1 , 𝑢 2 , … , 𝑢 𝑛 𝑣 , 𝑉 = d i a g 1 , 𝑣 2 , … , 𝑣 𝑛 𝑤 , 𝑊 = d i a g 1 , 𝑤 2 , … , 𝑤 𝑛 , 𝑢 𝑖 | | 𝑢 = m a x − 𝑖 | | , | | 𝑢 + 𝑖 | | , 𝑣 𝑖 | | 𝑣 = m a x − 𝑖 | | , | | 𝑣 + 𝑖 | | , 𝑤 𝑖 | | 𝑤 = m a x − 𝑖 | | , | | 𝑤 + 𝑖 | | ( 𝑖 = 1 , 2 , … , 𝑛 ) . ( 3 . 5 )

Proof. Fixing 𝜉 ∈ 𝐿 2 ℱ 𝑡 ( [ − 𝜏 , 0 ] ; ℝ 𝑛 ) arbitrarily and writing 𝑦 ( 𝑡 ; 𝜉 ) = 𝑦 ( 𝑡 ) , consider the following Lyapulov-Krasovskii functional: 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 7 𝑘 = 1 𝑉 𝑘 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) , ( 3 . 6 ) where 𝑉 1 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝑉 𝑦 ( 𝑡 ) , 2 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑡 𝑡 − 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝐾 𝑖 𝑓 𝑉 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , 3 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 0 − 𝜏 𝑑 𝜃 𝑡 𝑡 + 𝜃 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝐻 𝑓 ( 𝑉 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , 4 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑛 𝑗 = 1 𝑓 𝑗 ∞ 0 𝑅 𝑗 ( 𝜃 ) 𝑡 𝑡 − 𝜃 ℎ 𝑗 2 𝑦 𝑗 𝑉 ( 𝑠 ) 𝑑 𝑠 𝑑 𝜃 , 5 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑡 𝑡 − 𝜏 𝑦 𝑇 ( 𝑉 𝑠 ) 𝐺 𝑦 ( 𝑠 ) 𝑑 𝑠 , 6 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑡 𝑡 − 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝐿 𝑖 𝑔 𝑉 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , 7 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 0 − 𝜏 𝑑 𝜃 𝑡 𝑡 + 𝜃 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑀 𝑔 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 . ( 3 . 7 ) For simplicity, denote 𝜎 ( 𝑦 ( 𝑡 ) , 𝑦 ( 𝑡 − 𝜏 ) , 𝑡 , 𝑟 ( 𝑡 ) ) by 𝜎 ( 𝑡 ) . Then it follows from (2.10) and (2.6) that ℒ 𝑉 1 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 − 𝐷 𝑖 𝑦 ( 𝑡 ) + 𝐴 𝑖 𝑓 ( 𝑦 ( 𝑡 ) ) + 𝐵 𝑖 𝑔 ( 𝑦 ( 𝑡 − 𝜏 ) ) + 𝐶 𝑖 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ + ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑗 𝜎 𝑦 ( 𝑡 ) + t r a c e 𝑇 ( 𝑡 ) 𝑃 𝑖 𝜎 ( 𝑡 ) = 𝑦 𝑇 ( 𝑡 ) − 2 𝑃 𝑖 𝐷 𝑖 𝑦 ( 𝑡 ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐴 𝑖 𝑓 ( 𝑦 ( 𝑡 ) ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐵 𝑖 𝑔 ( 𝑦 ( 𝑡 − 𝜏 ) ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐶 𝑖 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 + 𝑦 𝑇 ( 𝑡 ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑃 𝑗 𝜎 𝑦 ( 𝑡 ) + t r a c e 𝑇 ( 𝑡 ) 𝑃 𝑖 . 𝜎 ( 𝑡 ) ( 3 . 8 ) On the other hand, by Assumption 2.2 and condition (3.2) we obtain 𝜎 t r a c e 𝑇 ( 𝑡 ) 𝑃 𝑖 𝜎 ( 𝑡 ) ≤ 𝜆 𝑖 𝜎 t r a c e 𝑇 ( 𝑡 ) 𝜎 ( 𝑡 ) ≤ 𝜆 𝑖 𝑦 𝑇 ( 𝑡 ) Σ 1 𝑖 𝑦 ( 𝑡 ) + 𝜆 𝑖 𝑦 𝑇 ( 𝑡 − 𝜏 ) Σ 2 𝑖 𝑦 ( 𝑡 − 𝜏 ) , ( 3 . 9 ) which together with (3.8) gives ℒ 𝑉 1 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) ≤ 𝑦 𝑇 ( 𝑡 ) − 2 𝑃 𝑖 𝐷 𝑖 𝑦 ( 𝑡 ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐴 𝑖 𝑓 ( 𝑦 ( 𝑡 ) ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐵 𝑖 𝑔 ( 𝑦 ( 𝑡 − 𝜏 ) ) + 2 𝑦 𝑇 ( 𝑡 ) 𝑃 𝑖 𝐶 𝑖 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 + 𝑦 𝑇 ( 𝑡 ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑃 𝑗 𝑦 ( 𝑡 ) + 𝜆 𝑖 𝑦 𝑇 ( 𝑡 ) Σ 1 𝑖 𝑦 ( 𝑡 ) + 𝜆 𝑖 𝑦 𝑇 ( 𝑡 − 𝜏 ) Σ 2 𝑖 𝑦 ( 𝑡 − 𝜏 ) . ( 3 . 1 0 ) Also, from direct computations, it follows that ℒ 𝑉 2 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑓 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐾 𝑖 𝑓 ( 𝑦 ( 𝑡 ) ) − 𝑓 𝑇 ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝐾 𝑖 𝑓 + ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝑡 𝑡 − 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝐾 𝑗 𝑓 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , ℒ 𝑉 3 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐻 𝑓 ( 𝑦 ( 𝑡 ) ) − 𝑡 𝑡 − 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝐻 𝑓 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , ℒ 𝑉 4 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑛 𝑗 = 1 𝑓 𝑗 ∞ 0 𝑅 𝑗 ( 𝜃 ) ℎ 𝑗 2 𝑦 𝑗 ( 𝑡 ) 𝑑 𝜃 − 𝑛 𝑗 = 1 𝑓 𝑗 ∞ 0 𝑅 𝑗 ( 𝜃 ) ℎ 𝑗 2 𝑦 𝑗 ( 𝑡 − 𝜃 ) 𝑑 𝜃 = ℎ 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐹 ℎ ( 𝑦 ( 𝑡 ) ) − 𝑛 𝑗 = 1 𝑓 𝑗 ∞ 0 𝑅 𝑗 ( 𝜃 ) 𝑑 𝜃 ∞ 0 𝑅 𝑗 ( 𝜃 ) ℎ 𝑗 2 𝑦 𝑗 ( 𝑡 − 𝜃 ) 𝑑 𝜃 ≤ ℎ 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐹 ℎ ( 𝑦 ( 𝑡 ) ) − 𝑛 𝑗 = 1 𝑓 𝑗 ∞ 0 𝑅 𝑗 ( 𝜃 ) ℎ 𝑗 ( 𝑦 𝑗 ( 𝑡 − 𝜃 ) ) 𝑑 𝜃 2 = ℎ 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐹 ℎ − ( 𝑦 ( 𝑡 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 𝑇 𝐹 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , ℒ 𝑉 5 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑦 𝑇 ( 𝑡 ) 𝐺 𝑦 ( 𝑡 ) − 𝑦 𝑇 ( 𝑡 − 𝜏 ) 𝐺 𝑦 ( 𝑡 − 𝜏 ) , ℒ 𝑉 6 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝑔 𝑇 ( 𝑦 ( 𝑡 ) ) 𝐿 𝑖 𝑔 ( 𝑦 ( 𝑡 ) ) − 𝑔 𝑇 ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝐿 𝑖 𝑔 + ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝑡 𝑡 − 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝐿 𝑗 𝑔 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 , ℒ 𝑉 7 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) = 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑡 ) ) 𝑀 𝑔 ( 𝑦 ( 𝑡 ) ) − 𝑡 𝑡 − 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑀 𝑔 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 . ( 3 . 1 1 ) It should be mentioned that the calculation of ℒ 𝑉 4 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) has applied the following inequality: ∞ 0 𝑙 1 ( 𝜃 ) 𝑙 2 ( 𝜃 ) 𝑑 𝜃 2 ≤ ∞ 0 𝑙 2 1 ( 𝜃 ) 𝑑 𝜃 ∞ 0 𝑙 2 2 ( 𝜃 ) 𝑑 𝜃 ( 3 . 1 2 ) with 𝑙 1 ( 𝜃 ) ∶ = [ 𝑅 ( 𝜃 ) ] 1 / 2 and 𝑙 2 ( 𝜃 ) ∶ = [ 𝑅 ( 𝜃 ) ] 1 / 2 ℎ ( 𝑦 ( 𝑡 − 𝜃 ) ) .
Furthermore, it follows from the conditions (3.3) and (3.4) that 𝑡 𝑡 − 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝐾 𝑗 𝑓 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 − 𝑡 𝑡 − 𝜏 𝑓 𝑇 ( 𝑦 ( 𝑠 ) ) 𝐻 𝑓 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 ≤ 0 , 𝑡 𝑡 − 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝐿 𝑗 𝑔 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 − 𝑡 𝑡 − 𝜏 𝑔 𝑇 ( 𝑦 ( 𝑠 ) ) 𝑀 𝑔 ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 ≤ 0 . ( 3 . 1 3 )
On the other hand, by Assumption 2.1 we have 𝑓 𝑇 ( 𝑦 ( 𝑡 ) ) 𝑄 1 𝑓 ( 𝑦 ( 𝑡 ) ) ≤ 𝑦 𝑇 ( 𝑡 ) 𝑈 𝑄 1 𝑔 𝑈 𝑦 ( 𝑡 ) , 𝑇 ( 𝑦 ( 𝑡 ) ) 𝑄 2 𝑔 ( 𝑦 ( 𝑡 ) ) ≤ 𝑦 𝑇 ( 𝑡 ) 𝑉 𝑄 2 ℎ 𝑉 𝑦 ( 𝑡 ) , 𝑇 ( 𝑦 ( 𝑡 ) ) 𝑄 3 ℎ ( 𝑦 ( 𝑡 ) ) ≤ 𝑦 𝑇 ( 𝑡 ) 𝑊 𝑄 3 𝑊 𝑦 ( 𝑡 ) . ( 3 . 1 4 )
Hence, by (3.8)–(3.14), we get ℒ 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) ≤ 𝜁 𝑇 ( 𝑡 ) Π 𝑖 𝜁 ( 𝑡 ) , ( 3 . 1 5 ) where 𝜁 𝑇 ( 𝑦 𝑡 ) = 𝑇 ( 𝑡 ) 𝑦 𝑇 ( 𝑡 − 𝜏 ) 𝑓 𝑇 ( 𝑦 ( 𝑡 ) ) 𝑔 𝑇 ( 𝑦 ( 𝑡 ) ) ℎ 𝑇 ( 𝑓 𝑦 ( 𝑡 ) ) 𝑇 ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝑔 𝑇 ( 𝑦 ( 𝑡 − 𝜏 ) ) 𝑡 − ∞ 𝑅 ( 𝑡 − 𝑠 ) ℎ ( 𝑦 ( 𝑠 ) ) 𝑑 𝑠 𝑇 , Π 𝑖 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ Γ 1 1 0 𝑃 𝑖 𝐴 𝑖 0 0 0 𝑃 𝑖 𝐵 𝑖 𝑃 𝑖 𝐶 𝑖 ⋆ Γ 2 2 0 0 0 0 0 0 ⋆ ⋆ Γ 3 3 0 0 0 0 0 ⋆ ⋆ ⋆ Γ 4 4 0 0 0 0 ⋆ ⋆ ⋆ ⋆ − 𝑄 3 + 𝐹 0 0 0 ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐾 𝑖 0 0 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐿 𝑖 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Γ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − 𝐹 1 1 = − 2 𝑃 𝑖 𝐷 𝑖 + 𝐺 + 𝜆 𝑖 Σ 1 𝑖 + 𝑈 𝑄 1 𝑈 + 𝑉 𝑄 2 𝑉 + 𝑊 𝑄 3 𝑊 + 𝑁 𝑗 = 1 𝑞 𝑖 𝑗 𝑃 𝑗 , Γ 2 2 = − 𝐺 + 𝜆 𝑖 Σ 2 𝑖 , Γ 3 3 = − 𝑄 1 + 𝐾 𝑖 + 𝜏 𝐻 , Γ 4 4 = − 𝑄 2 + 𝐿 𝑖 + 𝜏 𝑀 . ( 3 . 1 7 ) By condition (3.1), there must exist a scalar 𝛽 𝑖 > 0 ( 𝑖 ∈ 𝑆 ) such that Π 𝑖 + 𝛽 𝑖 𝐼 < 0 . Setting 𝛽 = m i n 𝑖 ∈ 𝑆 𝛽 𝑖 , it is clear that 𝛽 > 0 . Taking the mathematical expectation on both sides of (3.15), we obtain 𝐄 ℒ 𝑉 ( 𝑡 , 𝑦 ( 𝑡 ) , 𝑖 ) ≤ 𝐄 𝜁 𝑇 ( 𝑡 ) Π 𝑖 𝜁 ( 𝑡 ) ≤ − 𝛽 𝑖 𝐄 | | | | 𝑦 ( 𝑡 ; 𝜉 ) 2 |