New stability and stabilization conditions for stochastic neural networks of neutral type with Markovian jumping parameters

Abstract

This paper discusses the stabilization criteria for stochastic neural networks of neutral type with both Markovian jump parameters. First, delay-dependent conditions to guarantee the globally exponential stability in mean square and almost surely exponential stability of such systems are obtained by combining an appropriate constructed Lyapunov–Krasovskii functional with the semi-martingale convergence theorem. These conditions are in terms of the linear matrix inequalities (LMIs), which can be some less conservative than some existing results. Second, based on the obtained stability conditions, the state feedback controller is designed. Finally, four numerical examples are provided to illustrate the effectiveness and significant improvement of the proposed method.

Introduction

It is well known that many physical systems are subject to frequent unpredictable structural changes, such as random failures [1], repairs of components [2], sudden environment disturbances [3], abrupt variation of the operating point on a nonlinear plant [4], etc. Markovian jump systems are often used to describe such systems. Generally speaking, a Markovian jump system is a hybrid system with state vector that has two components x(t) and r(t), where x(t) denotes the state and r(t) is a continuous-time Markov chain, which is usually regarded as the mode. In its operation, the jump system will switch from one mode to another in a random way, which is determined by a continuous-time Markov chain r(t), based on a Markov chain with finite state space $S=(1,2,\dots ,s)$. Therefore, it is important to research the dynamics behaviors of systems with Markovian jumping parameters.

Recently, a large number of stability and stabilization results on stochastic systems with Markovian jumping parameters have been reported in the literature, see [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] and the references therein. In [5], [6], Ji and Chizeck and Mariton studied the stability of linear jump equations; Boukas and Yang studied the exponential stability problem for stochastic systems with Markovian jumping parameters in [7]. Mao investigated the exponential stability for general nonlinear stochastic differential equations with Markovian jumping parameters [8]. The robustly exponential stability of stochastic neural networks with mixed time delays and Markovian switching was investigated in [9]. Balasubramaniam and Rakkiyappan [10] studied the globally asymptotic stability for a class of Markovian jumping stochastic Cohen–Grossberg neural networks with discrete interval and distributed delays. Very recently, the exponential stability for several new classes of Markovian jump stochastic neural networks with mixed time delays with/without impulse control were discussed in [11], [12], [13]. The globally asymptotic stability for a class of stochastic recurrent neural networks with Markovian jumping parameters and continuously distributed delays were investigated in [14]. In the area of control, the linear quadratic regulator problem were studied in [15], [16]. Mariton and Bertrand [17] extended those quadratic regulator to the case of output feedback. The robust stability and stabilization of linear jump systems were studied in [18]. Ghosh et al. [19] developed a dynamic programming approach to the optimal control of general stochastic differential equations with Markovian switching. Recently, Yuan and Lygeros [20] considered the problem of exponential stabilization by the state feedback for a class of stochastic differential equations with Markovian jumping parameters. In [27], the trajectory tracking control problem of a class of non-affine stochastic nonlinear switched systems with the non-lower triangular form under arbitrary switching was considered. Very recently, Li and Yang [30] studied the adaptive event-triggered control problem of nonlinear continuous-time systems in strict-feedback form.

On the other hand, neural networks of neutral type have been widely discussed since this model can be extensively used in the area of computer aided design, circuit analysis, real time simulation of mechanical systems, chemical process simulation, power systems, population dynamics and automatic control. For instance, Park and Kwon [31] studied the delay-dependent stability criterion for a class of neural networks of neutral-type with interval time-varying delays by the Lyapunov stability theory and linear matrix inequality approach. In [32], the authors obtained some delay-dependent conditions for a class of neutral type neural networks with both discrete and unbounded distributed delays. Park et al. [33] proved the globally asymptotic stability of equilibrium for a class of continuous bidirectional associative memory neural networks of neutral type. Mahmoud and Ismail [34] derived the robustly exponential stability condition for a class of neutral-type delayed neural networks via using the Lyapunov–Krasovskii functional and the integral inequality. In [35], the authors considered the adaptive pinning synchronization problem for neutral-type neural networks with stochastic perturbation. However, the results in [31], [32], [33], [34], [35] did not consider neural networks of neutral type with Markovian jumping parameters. The delay-dependent stability analysis for stochastic neural networks of neutral type with Markovian jumping parameters was considered in [36], [37]. The almost surely exponential stability for neutral stochastic neural networks with constant delay was discussed by using the semi-martingale convergence theorem in [38], [39]. However, the almost surely exponential stability analysis for stochastic systems with neural networks of neutral type with Markovian jumping parameters has not yet been investigated in [38], [39], which prompt our present research. In addition, to the best of our knowledge, a few little stabilization results on stochastic neural networks of neutral type with Markovian jumping parameters were reported [40], [41]. Therefore, it may exist large space to develop novel stabilization conditions for stochastic systems with neural networks of neutral type with Markovian jumping parameters. All of those motivate this research.

Motivated by the aforementioned discussions, the objective of this paper is to further study the stability and stabilization for a new class of stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. The main contributions are summarized as follows.

• By employing the semi-martingale convergence theorem, some delay-dependent sufficient criteria to guarantee the globally exponential stability in mean square and almost surely exponential stability conditions are both given.

• Based on the obtained less conservative stability conditions, the state feedback controller is designed by the linear matrix inequality technique.

• Compared with some existing stability criteria, our derived criteria are not only less conservative but also easily checked.

Notation: Throughout this paper, the following notations will be used. ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times n}$ denote, respectively, the n-dimensional Euclidean space and the set of all n × n real matrices. The superscript T denotes the transposition and the notation X ≥ Y (respectively, X > Y), where X and Y are symmetric matrices, means that $X-Y$ is positive semi-definite (respectively, positive definite). λ_max (X) and λ_min (X) denote the maximum and minimum eigenvalues of the corresponding matrix X, respectively. $sym\left(X\right)=X+X^T$. col{⋅⋅⋅} denotes a column vector. E{ · } stands for the mathematical expectation operator. I_n is the n × n identity matrix. | · | denotes the absolute value of the real number and the norm of a vector or a matrix.