Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay

We regard the stochastic functional differential equation with infinite delay as the result of the effects of stochastic perturbation to the deterministic functional differential equation , where is defined by x_t(θ)=x(t+θ),θ(−∞,0]. We assume that the deterministic system with infinite delay is exponentially stable. In this paper, we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable.     

Attraction, stability and robustness for stochastic functional differential equations with infinite delay

This paper establishes the stochastic LaSalle theorem to locate limit sets for stochastic functional differential equations with infinite delay, from which some criteria on attraction, boundedness, stability and robustness are obtained. To illustrate the applications of our results clearly, this paper considers a scalar stochastic integro-differential equation with infinite delay as an example.     

Stability analysis of stochastic recurrent neural networks with unbounded time-varying delays

In this paper, the stability analysis issue of stochastic recurrent neural networks with unbounded time-varying delays is investigated. By the idea of Lyapunov function and the semi-martingale convergence theorem, both pth moment exponential stability and almost sure exponential stability are obtained. Moreover, the M-matrix technique is borrowed to make the results more applicable. Our criteria can be used not only in the case of bounded delay but also in the case of unbounded delay. Some earlier results are improved and generalized. An example is also given to demonstrate our results.     

Stability analysis for recurrent neural networks with time-varying delay

This paper is concerned with the stability analysis for static recurrent neural networks （RNNs） with time-varying delay. By Lyapunov functional method and linear matrix inequality technique, some new delay-dependent conditions are established to ensure the asymptotic stability of the neural network. Expressed in linear matrix inequalities （LMIs）, the proposed delay-dependent stability conditions can be checked using the recently developed algorithms. A numerical example is given to show that the obtained conditions can provide less conservative results than some existing ones.     

Stability of linear stochastic delay differential equations with infinite Markovian switchings

This paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.     

Passivity analysis for neural networks with a time-varying delay

This paper deals with the problem of passivity analysis for neural networks with both time-varying delay and norm-bounded parameter uncertainties by employing an improved free-weighting matrix approach. Some useful terms have been retained, which were used to be ignored in the derivative of Lyapunov-Krasovskii functional. Furthermore, the relationship among the time-varying delay, its upper bound and their difference is taken into account. As a result, for two types of time-varying delays, less conservative delay-dependent passivity conditions are obtained in terms of linear matrix inequalities (LMIs), respectively. Finally, a numerical example is given to demonstrate the effectiveness of the proposed techniques.     

pth moment stability analysis of stochastic recurrent neural networks with time-varying delays

This paper addresses, in great detail, the issue of pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. With the help of the Dini-derivative of the expectation of V(t,X(t)) “along” the solution X(t) of the model and the technique of Halanay-type inequality, some novel sufficient conditions on pth moment exponential stability of the trivial solution has been established. Results of the development as presented in this paper are more general than those reported in some previously published papers. An example is also given to illustrate that our results are correct and effectiveness.     

Stability analysis for stochastic BAM neural networks with Markovian jumping parameters

This paper is concerned with the stability analysis issue for stochastic delayed bidirectional associative memory (BAM) neural network with Markovian jumping parameters. Assume that the jumping parameters are generated from continue-time discrete-state homogeneous Markov process and the delays are time-invariant. By employing the Lyapunov stability theory, some inequality techniques and the stochastic analysis, sufficient conditions are derived to achieve the global exponential stability in the mean square of the stochastic BAM neural network. One example is also provided in the end of this paper to illustrate the effectiveness of our results.     

Training neural networks by stochastic optimisation

We present a stochastic learning algorithm for neural networks. The algorithm does not make any assumptions about transfer functions of individual neurons and does not depend on a functional form of a performance measure. The algorithm uses a random step of varying size to adapt weights. The average size of the step decreases during learning. The large steps enable the algorithm to jump over local maxima/minima, while the small ones ensure convergence in a local area. We investigate convergence properties of the proposed algorithm as well as test the algorithm on four supervised and unsupervised learning problems. We have found a superiority of this algorithm compared to several known algorithms when testing them on generated as well as real data.     

Exponential stability of hybrid stochastic neural networks with mixed time delays and nonlinearity

This paper is concerned with the problem of robust exponential stability for a class of hybrid stochastic neural networks with mixed time-delays and Markovian jumping parameters. In this paper, free-weighting matrices are employed to express the relationship between the terms in the Leibniz–Newton formula. Based on the relationship, a linear matrix inequality (LMI) approach is developed to establish the desired sufficient conditions for the mixed time-delays neural networks with Markovian jumping parameters. Finally, two simulation examples are provided to demonstrate the effectiveness of the results developed.     

Stability of stochastic interval systems with time delays

Consider a given exponentially stable system undergoing a random perturbation which is dependent on a past state of the solution of the system. Suppose this stochastically perturbed system is described by a stochastic differential-functional equation. In this paper, we establish a sufficient condition that the perturbed system remains exponentially stable. Using a specific example, we show how this condition may be used, and we extend it to deal with multiple time delays.     

具有不对称结构的广义时滞神经网络的动态分析

研究一类具有不对称互连结构的广义时滞神经网络的动态行为．通过构造适当的Lyapunov泛函及扇区条件，给出了平衡点渐近稳定的充分条件，并对由推论给出的一种小增益条件进行了分析．仿真结果进一步证明了结论的有效性．     

Delay-dependent robust stability of stochastic delay systems with Markovian switching

In recent years, the stability of hybrid stochastic delay systems, one of the important issues in the study of stochastic systems, has received considerable attention. However, the existing results do not deal with the structure of the diffusion but estimate its upper bound, which induces conservatism. This paper studies delay-dependent robust stability of hybrid stochastic delay systems. A delay-dependent criterion for robust exponential stability of hybrid stochastic delay systems is presented in terms of linear matrix inequalities （LMIs）, which exploits the structure of the diffusion. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method.     

Adaptive finite-time neural network control for nonlinear stochastic systems with state constraints

This paper focuses on the adaptive finite-time neural network control problem for nonlinear stochastic systems with full state constraints. Adaptive controller and adaptive law are designed by backstepping design with log-type barrier Lyapunov function. Radial basis function neural networks are employed to approximate unknown system parameters. It is proved that the tracking error can achieve finite-time convergence to a small region of the origin in probability and the state constraints are confirmed in probability. Different from deterministic nonlinear systems, here the stochastic system is affected by two random terms including continuous Brownian motion and discontinuous Poisson jump process. Therefore, it will bring difficulties to the controller design and the estimations of unknown parameters. A simulation example is given to illustrate the effectiveness of the designed control method.     

时变时滞随机非线性系统的自适应神经网络跟踪控制

针对一类具有时变时滞的不确定随机非线性严格反馈系统的自适应跟踪问题,利用Razumikhin引理和backstepping方法,提出一种新的自适应神经网络跟踪控制器.该控制器可保证闭环系统的所有误差变量皆四阶矩半全局一致最终有界,并且跟踪误差可以稳定在原点附近的邻域内.仿真例子表明所提出控制方案的有效性.     

Stability of stochastic systems with uncertain time delays

For linear delay systems with bilinear noise sufficient conditions are given for the global asymptotic stochastic stability independent of the length of the delay(s). For linear stochastic noise terms, sufficient conditions for the existence of an invariant distribution, for all values of the delay are given. It is shown that the gaussian distribution is the unique invariant distribution. The covariance and correlation matrix function of the resulting stationary process are completely characterized by a Lyapunov-type equation. All these sufficient conditions are obtained in the form of the existence of some positive definite matrices satisfying certain Riccati-type equations.     

Exponential stability analysis of stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed delays

In this paper, we study a new class of stochastic Cohen-Grossberg neural networks with reaction-diffusion and mixed delays. Without the aid of nonnegative semimartingale convergence theorem, the method of variation parameter and linear matrix inequalities technique, a set of novel sufficient conditions on the exponential stability for the considered system is obtained by utilizing a new Lyapunov-Krasovskii functional, the Poincaré inequality and stochastic analysis theory. The obtained results show that the reaction-diffusion term does contribute to the exponentially stabilization of the considered system. Therefore, our results generalize and improve some earlier publications. Moreover, two numerical examples are given to show the effectiveness of the theoretical results and demonstrate that the stability criteria existed in the earlier literature fail.     

Delay-dependent exponential stability for uncertain neutral stochastic neural networks with interval time-varying delay

This paper is mainly concerned with the problem for the robustly exponential stability in mean square moment of uncertain neutral stochastic neural networks with interval time-varying delay. With an appropriate augmented Lyapunov–Krasovskii functional (LKF) formulated, the convex combination method is utilised to estimate the derivative of the LKF. Some new delay-dependent exponential stability criteria for such systems are obtained in terms of linear matrix inequalities, which involve fewer matrix variables and have less conservatism. Finally, two illustrative numerical examples are given to show the effectiveness of our obtained results.     

Stability analysis of hybrid stochastic delay differential equations with asynchronous switching and discrete observations

This paper studies the hybrid stochastic delay differential equations (SDDEs) with asynchronous switching and discrete observations. For SDDEs based on discrete observations, there are two methods: The discrete-time approach and the input time delay method. For linear solvable equations, the discrete-time approach is feasible but for unsolvable nonlinear hybrid SDSs, the results of the discrete-time approach have not been discussed. So, it is natural to ask: Is the discrete-time approach still workable for nonlinear hybrid SDSs? This paper focuses on this problem. By using tools of stochastic analysis, constructing Lyapunov functional and using the discrete-time approach, the stability of hybrid SDSs by discrete-time feedback control is obtained. Finally, a numerical example is presented to verify the theoretical result.     

New robust exponential stability analysis for uncertain neural networks with time-varying delay

In this paper, the global robust exponential stability is considered for a class of neural networks with parametric uncer-tainties and time-varying delay. By using Lyapunov functional method, and by resorting to the new technique for estimating the upper bound of the derivative of the Lyapunov functional, some less conservative exponential stability criteria are derived in terms of linear matrix inequalities (LMIs). Numerical examples are presented to show the effectiveness of the proposed method.