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Volume 2018, Article ID 8094292, 16 pages
Research Article

Synchronization of Semi-Markovian Jump Neural Networks with Randomly Occurring Time-Varying Delays

1School of Electrical and Information Engineering, Anhui University of Technology, Ma'anshan 243002, China
2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Zhen Wang; moc.621@ds_gnawnehz

Received 23 January 2018; Accepted 14 June 2018; Published 12 September 2018

Academic Editor: Bernhard C. Geiger

Copyright © 2018 Mengping Xing et al. 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.


Based on the Lyapunov stability theory, this paper mainly investigates the synchronization problem for semi-Markovian jump neural networks (semi-MJNNs) with randomly occurring time-varying delays (TVDs). The continuous-time semi-MJNNs, where the transition rates are dependent on sojourn time, are introduced to make the issue under our consideration more general. One of the main characteristics of our work is the handling of TVDs. In addition to using the improved Jensen inequality and the reciprocal convexity lemma to deal with the integral inequality, we also employ Schur complement and the projection lemma to achieve the decoupling between the square term of TVDs. Finally, we verify the validity and feasibility of our method by a couple of simulation examples.

1. Introduction

In recent years, with the unceasingly thorough research on large data and artificial intelligence, the theory and application of neural networks have been greatly developed. It has tremendous application prospect, especially in robotics [1], pattern recognition [2, 3], associative memory [46], identification [7, 8], and combinatorial optimization [912]. Neural networks can be simply divided into the deterministic neural networks and stochastic neural networks based on whether they are disturbed by outside noise [13]. When the system is undisturbed, the deterministic neural network can describe the actual system accurately [13]. Nevertheless, as far as we know, the actual system is generally uncertain and most of the physical system will be affected by random parameter variation and structure change [1416]. These changes may be caused by some sudden phenomena, such as components or connection failure and the deviation of parameter. In this circumstance, the stochastic neural networks can be described by a hybrid model, where a discrete stochastic variable called mode or pattern is attached to continuous state variables to describe the random jump of system parameters as well as the appearance of discontinuous points. It allows policymakers to respond to discrete events, which significantly perturb or alter the normal working condition of the system, by combining the empirical knowledge of events and the statistical information of their rates, adequately [17, 18].

Markovian jump neural networks (MJNNs), as we all know, as a kind of typical hybrid dynamic systems are widely used in the field of aerospace, industrial production, and biological, medical, and social construction in the past few decades due to its strong modeling ability and therefore draw great attention from researchers. For instance, the stability analysis, state estimation, filter design, passivity analysis, and stochastic synchronization for MJNNs were discussed in [19, 20], respectively. But one obvious drawback of MJNNs is that its jump time obeys the exponential distribution, which is a memoryless distribution and makes the transition probability of jump system an invariant function matrix; that is, the transition probability of the system obeys a stochastic process which is not relevant with the mode of the past [21]. Because of this limitation, it brings great restriction to the application of MJNNs. Therefore, semi-MJNNs were put forward later, where a transition probability matrix and a fixed dwell time probability density function matrix are used to represent the stochastic neural networks [22]. It has a wider range of application background due to the relaxation to the constraint condition where the probability density distribution function obeys exponential distribution. Compared with the abundant research achievements on MJNNs, the research efforts devoted to semi-MJNNs are relatively scarce. The robust stochastic stability condition for semi-MJNNs was derived in [23] and the relevant controller was also designed there. The synchronization controller for the semi-MJNNs was designed in [24] where the semi-MJNNs were transformed into associated MJNNs. An exponential passive filter was designed, and a cone complementarity linearization method was applied to manage the nonconvex feasibility issue in [25]. As mentioned above, the semi-MJNNs have more extensive application, such as in complex medical procedures [26].

Due to the finite signal transition speed as well as the limited switching speed of hardware facilities, the time-delay phenomenon exists in various practical industrial control systems widely, such as chemical system, process control system, and network control system [2736]. It is known that delay argument existing in the system is often unknown or time-varying and the occurrence of delay tends to be random, which makes the analysis and control of the system more difficult. Also, the existence of time-delay tends to result in the degradation of the performance index and can even make the system unstable [37]. Therefore, it has important theoretical significance and practical applying value to study the system with time-varying delays (TVDs). As the system that considers that time-delay is more in line with the actual situation, an increasing number of researches have been made on the time-delay systems in recent years, and considerable results have been presented. To mention a few, Gun and Niculescu discussed the problem of stability analysis for the systems with time-delay and gave a summary on literature about the stability analysis and controller design of systems with time-delay in [38]. Park and Wan Ko studied the stability and robust stability criteria for TVD systems in [39], and then the reciprocally convex approach and the second-order reciprocally convex approach were proposed for stability analysis of TVD systems in [40].

Synchronization refers to two or more dynamic systems whose properties are identical or close to each other. Through the interaction between the systems, the state of the dynamic system that evolves under different initial conditions is gradually close to each other and finally reaches the same. Synchronization analysis is particularly important in many dynamic behaviors of neural networks and therefore has been widely studied. Exponential synchronization, adaptive synchronization, finite-time synchronization, mixed /passive synchronization, and new delay-dependent exponential synchronization were considered in [41, 42], respectively.

This paper mainly studies the synchronization of semi-MJNNs with randomly occurring TVDs. First of all, by Lyapunov stability theory, we can get that the key point to establish a Lyapunov functional is to contain more useful information about the delays, which is useful to obtain the results with less conservatism [43]. As a result of the existence of TVDs, some novel inequality techniques derived from the Park inequality and the improved Jensen inequality [44] are employed to handle the time-varying items. At the same time, considering the existence of the square term of TVDs in the formulas, we use the projection lemma to achieve the decoupling between time-varying items. By using convex optimization techniques, the synchronization control of semi-MJNNs is investigated in this paper. The corresponding main results are presented by three theorems: Theorem 1 provides sufficient conditions for the stochastic stability and synchronization of the closed-loop dynamic error system; Theorem 2 conducts the decoupling arithmetic; and Theorem 3 is then presented to get strict LMI-based conditions, and a numerical method to calculate controller gains is presented, which is simple and easily conducted.

Compared with the existing literature, this article has the following characteristics: (1) Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account simultaneously; (2) with the introduction of some advanced inequalities, combining with Schur complement lemma and projection lemma, synchronization conditions with less conservatism are derived; (3) we use the LMI control toolbox to carry out the relevant simulation, and the corresponding controller can be obtained which can verify the correctness and feasibility of the proposed method. Throughout this work, the notations used are standard.

2. Problem Formulation

Firstly, given the following semi-MJNNs with randomly occurring TVDs , where is the system state vector which is associated with the neurons; denotes the neuron activation functions of the system, which is assumed to be bounded and satisfies where , and are real known scalars, and they could be zero, positive, or negative. For the purpose of simplifying the symbols, we set

stands for external input; denotes the TVDs satisfying and where the nonnegative scalars and refer to the minimum and maximum time-delay, respectively. ( is a positive integer) represents a continuous-time and discrete-state homogeneous semi-Markovian process whose trajectories are right continuous. Assuming takes value in a finite state space , the transition rate matrix can be given by where and ; stands for the transition rates from to , and is a Bernoulli-distributed white sequence that takes values of and and obeys the following probability distribution laws where is a known constant.

Remark 1. Different from the previous literature, a more general system model is introduced in this paper, in which both the semi-MJNNs and the random TVDs are taken into account. The time-delay phenomenon, which occurs randomly and tends to be time-varying, exists in various practical neural networks. Therefore, the stochastic variable is introduced to express the randomly occurring TVDs in this paper to make the issue under consideration more practical and more reasonable.

In this paper, the slave system could be represented as the following forms where and are the response state vector and the response output, respectively; is the advisable control input.

For presenting a better explanation to the addressed problem, we introduce as the synchronization error vector, as the output error, and as the nonlinear error. Then, the following error system is obtained:

The controller input for the error system is established as where the controller gain matrix will be designed in the sequel. In order to simplify the notation, and are denoted by , and , for each , respectively. Then, the closed-loop dynamic error system can be obtained:

Before making further derivation, we need the definition and lemmas as shown in the following.

Lemma 1. [24] The following inequalities hold for any diagonal matrices

Lemma 2. [45] (projection lemma) For a symmetric matrix , two matrices with the same column dimension of , the problem can be solved with respect to matrix if and only if where are any basis of the nullspace of .

Lemma 3. [30] (Schur complement lemma). Given constant matrices , , and , where and , then, if and only if

Lemma 4. [40] Given a matrix , a differentiable function , it is easy to obtain the following inequalities where

Lemma 5. [40] (reciprocal convexity lemma) For any vector ; matrices , , ; and nonnegative real scalars and meeting , the following inequality holds subject to .

Definition 1. ( synchronization) The system is said to be synchronization with the disturbance attenuation under the condition that the following requirements are met: (1)The system is stochastically stable when the disturbance input is always equal to .(2)For a positive scalar , the following inequality is satisfied under zero initial conditions

3. Main Results

Theorem 1. Given scalars , , , , , and the considered system is stochastically stable and synchronized; if there exist symmetric matrices , positive definite matrices , positive definite diagonal matrices , matrices , and such that for each the following matrix inequalities hold where with

Proof 1. We firstly consider a stochastic semi-Markovian Lyapunov-Krasovskii functional as follows: where with

First of all, we define where is the probability density function of sojourn time resting on mode .

Before further analysis, we consider the weak infinitesimal operator as the following forms and we have

Considering the integral items in (35) and (36), we can get that

For the integral items , and in (37), by using Lemma 4, we can obtain that

Similarly, it is easy for us to get the following conditions:

Synthesizing the above results, we can get that

Considering the single integral items in (41) and (42), we can get that

For the integral items , , and in (43), by using Lemma 4, we can obtain that

By using a similar method, it is not difficult to get that

Considering the double integral items in (41) and (42), we can get that

For in (47), by using Lemma 4, we can obtain that