pth moment asymptotic stability of stochastic delayed hybrid systems with Lévy noise

The problem of pth moment asymptotic stability analysis is considered for stochastic delayed hybrid systems with Lévy noise. By virtue of Itô’s formula and M-matrix theories, we propose some sufficient conditions to guarantee the asymptotic stability and exponential stability of the system. The criterion of mean square asymptotic stability is derived as well for delayed neural networks with Lévy noise. A numerical example is provided to show the usefulness of the proposed asymptotic stability criterion.     

Delay-independent stability of stochastic reaction–diffusion neural networks with Dirichlet boundary conditions

This paper deals with the problem of global stability of stochastic reaction–diffusion recurrent neural networks with continuously distributed delays and Dirichlet boundary conditions. The influence of diffusion, noise and continuously distributed delays upon the stability of the concerned system is discussed. New stability conditions are presented by using of Lyapunov method, inequality techniques and stochastic analysis. Under these sufficient conditions, globally exponential stability in the mean square holds, regardless of system delays. The proposed results extend those in the earlier literature and are easier to verify.     

Exponential stability of impulsive stochastic fuzzy cellular neural networks with mixed delays and reaction–diffusion terms

The problem of mean square exponential stability for a class of impulsive stochastic fuzzy cellular neural networks with distributed delays and reaction–diffusion terms is investigated in this paper. By using the properties of M-cone, eigenspace of the spectral radius of nonnegative matrices, Lyapunov functional, Itô’s formula and inequality techniques, several new sufficient conditions guaranteeing the mean square exponential stability of its equilibrium solution are obtained. The derived results are less conservative than the results recently presented in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008). In fact, the systems discussed in Wang and Xu (Chaos Solitons Fractals 42:2713–2721, 2009), Zhang and Li (Stability analysis of impulsive stochastic fuzzy cellular neural networks with time varying delays and reaction–diffusion terms. World Academy of Science, Engineering and Technology 2010), Huang (Chaos Solitons Fractals 31:658–664, 2007), and Wang (Chaos Solitons Fractals 38:878–885, 2008) are special cases of ours. Two examples are presented to illustrate the effectiveness and efficiency of the results.     

Input-to-state stability of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with the event-triggered control

ABSTRACT

In this paper, we investigate a class of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with feedback control and an unknown exogenous disturbance. By using the Lyapunov function, Itô's formula, Dynkin's formula, Comparison principle and stochastic analysis theory, we show that the considered system is input-to-state stable with the help of the designed event-triggered mechanism. Moreover, we also guarantee that the internal execution time intervals of control task will not be arbitrarily small. Finally, some remarks and discussions have been provided to show that our results are meaningful.     

Mean square exponential stability of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays

In this paper, we establish a method to study the mean square exponential stability of the zero solution of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays. By using the properties of M-cone and inequality technique, we obtain some sufficient conditions ensuring mean square exponential stability of the zero solution of impulsive stochastic reaction-diffusion Cohen–Grossberg neural networks with delays. The sufficient conditions are easily checked in practice by simple algebra methods and have a wider adaptive range. Two examples are also discussed to illustrate the efficiency of the obtained results.     

Almost sure stability of stochastic neutral Cohen–Grossberg neural networks with Lévy noise and time-varying delays

The almost sure stability for the stochastic neutral Cohen–Grossberg neural networks (SNCGNNs) with Lévy noise, time-varying delays, and Markovian switching would be deliberated in this article. By means of the nonnegative semimartingale convergence theorem (NSCT), the neutral Itô formula, M-matrix method, and selecting appropriate Lyapunov function, several almost sure stability criterions for the SNCGNNs could be derived. Moreover, according to the M-matrix theory, the upper bounds of the coefficients at any mode are given. Finally, two examples and numerical simulations verify the correctness of theoretical analysis for the stability criterions proposed in the article.     

Further stability analysis on Cohen–Grossberg neural networks with time-varying and distributed delays

The article is concerned with asymptotical stability for Cohen–Grossberg neural networks with both interval time-varying (0?≤?τ₀?≤?τ(t)?≤?τ _m ) and distributed delays, in which two types of distributed delays are treated: one is bounded while the other is unbounded. Through partitioning the delay intervals [0,?τ₀] and [τ₀,?τ _m ], and choosing two augmented Lyapunov–Krasovskii functionals, some sufficient conditions are obtained to guarantee the global stability by employing the simplified free-weighting matrix method and convex combination. These stability criteria are presented in terms of linear matrix inequalities (LMIs) and can be easily checked by resorting to LMI in Matlab toolbox. Finally, three numerical examples are given to illustrate the effectiveness and reduced conservatism of the theoretical results.     

Exponential stability of impulsive high-order Hopfield-type neural networks with delays and reaction–diffusion

The problem of global exponential stability analysis of Impulsive high-order Hopfield-type neural networks with time-varying delays and reaction–diffusion terms has been investigated in this paper. Using the Lyapunov function method and M-matrix theory, we establish the global exponential stability of the neural networks with its estimated exponential convergence rate. As an illustration, a numerical example is given using the results.     

Estimates of equilibrium points and global asymptotic stability of Cohen–Grossberg neural networks with delays

This article studies a class of Cohen–Grossberg neural networks (CGNNs) with variable and distributed delays. Some novel conditions guaranteeing the existence, uniqueness and the estimated location of the equilibrium points are obtained. Using these results, the global asymptotic stability of the CGNNs can be derived without demanding the boundedness and the globally Lipschitz condition of the activation functions. Two numerical examples are demonstrated to verify the theoretical results.     

Global exponential stability analysis of cellular neural networks with multiple time delays

Global exponential stability problems are investigated for cellular neural networks （CNN） with multiple time-varying delays. Several new criteria in linear matrix inequality form or in algebraic form are presented to ascertain the uniqueness and global exponential stability of the equilibrium point for CNN with multiple time-varying delays and with constant time delays. The proposed method has the advantage of considering the difference of neuronal excitatory and inhibitory effects, which is also computationally efficient as it can be solved numerically using the recently developed interior-point algorithm or be checked using simple algebraic calculation. In addition, the proposed results generalize and improve upon some previous works. Two numerical examples are used to show the effectiveness of the obtained results.     

Global exponential stability of delayed reaction–diffusion neural networks with time-varying coefficients

In the current paper, a class of general neural networks with time-varying coefficients, reaction–diffusion terms, and general time delays is studied. Several sufficient conditions guaranteeing its global exponential stability and the existence of periodic solutions are obtained through analytic methods such as Lyapunov functional and Poincaré mapping. The obtained results assume no boundedness, monotonicity or differentiability of activation functions and can be applied within a broader range of neural networks. Among the presented conditions, some are independent of time delay and expressed in terms of system parameters, so easy to verify and of leading significance in applications. For illustration, an example is given.     

Comments on ‘A multi-model adaptive predictor for stochastic processes with Markov switching parameters’

A prediction algorithm proposed by Qi Xiao-Jiang (1986) is shown to be non-optimal.     

Stability of general neural networks with reaction-diffusion

By constructing average Lyapunov functions, general neural networks (Cohen-Gross-berg's model) with reaction-diffusion are analyzed. A series of constructively algebraic criteria are presented. And, the existing results are included as the special cases of our results.     

Robust stability of uncertain fuzzy Cohen–Grossberg BAM neural networks with time-varying delays

In this paper, the Takagi–Sugeno (TS) fuzzy model representation is extended to the stability analysis for uncertain Cohen–Grossberg type bidirectional associative memory (BAM) neural networks with time-varying delays using linear matrix inequality (LMI) theory. A novel LMI-based stability criterion is obtained by using LMI optimization algorithms to guarantee the asymptotic stability of uncertain Cohen–Grossberg BAM neural networks with time varying delays which are represented by TS fuzzy models. Finally, the proposed stability conditions are demonstrated with numerical examples.     

Exponential p-stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays

In this paper, we study the impulsive stochastic Cohen–Grossberg neural networks with mixed delays. By establishing an L-operator differential inequality with mixed delays and using the properties of M-cone and stochastic analysis technique, we obtain some sufficient conditions ensuring the exponential p-stability of the impulsive stochastic Cohen–Grossberg neural networks with mixed delays. These results generalize a few previous known results and remove some restrictions on the neural networks. Two examples are also discussed to illustrate the efficiency of the obtained results.     

New stability criteria for neutral-type Cohen–Grossberg neural networks with discrete and distributed delays

This paper studies the existence, uniqueness and globally robust exponential stability for a class of uncertain neutral-type Cohen–Grossberg neural networks with time-varying and unbounded distributed delays. Based on Lyapunov–Krasovskii functional, by involving a free-weighting matrix, using the homeomorphism mapping principle, Cauchy–Schwarz inequality, Jensen integral inequality, linear matrix inequality techniques and matrix decomposition method, several delay-dependent and delay-independent sufficient conditions are obtained for the robust exponential stability of considered neural networks. Two numerical examples are given to show the effectiveness of our results.     

Delay-dependent H ∞ and generalized H 2 filtering for stochastic neural networks with time-varying delay and noise disturbance

This paper presents the delay-dependent \(H_\infty\) and generalized H ₂ filters design for stochastic neural networks with time-varying delay and noise disturbance. The stochastic neural networks under consideration are subject to time-varying delay in both the state and measurement equations. The aim is to design a stable full-order linear filter assuring asymptotical mean-square stability and a prescribed \(H_\infty\) or generalized H ₂ performance indexes for the filtering error systems. Delay-dependent sufficient conditions for the existence of \(H_\infty\) and generalized H ₂ filters are both proposed in terms of linear matrix inequalities. Finally, numerical example demonstrates that the proposed approaches are effective.     

Global exponential robust stability of reaction–diffusion interval neural networks with continuously distributed delays

This paper considers the existence of the equilibrium point and its global exponential robust stability for reaction-diffusion interval neural networks with variable coefficients and distributed delays by means of the topological degree theory and Lyapunov-functional method. The sufficient conditions on global exponential robust stability established in this paper are easily verifiable. An example is presented to demonstrate the effectiveness and efficiency of our results.     

Stabilization of stochastic Hopfield neural network with distributed parameters

In this paper, the stability of stochastic Hopfield neural network with distributed parameters is studied. To discuss the stability of systems, the main idea is to integrate the solution to systems in the space variable. Then, the integration is considered as the solution process of corresponding neural networks described by stochastic ordinary differential equations. A Lyapunov function is constructed and Ito formula is employed to compute the derivative of the mean Lyapunov function along the systems, with respect to the space variable. It is difficult to treat stochastic systems with distributed parameters since there is no corresponding Ito formula for this kind of system. Our method can overcome this difficulty. Till now, the research of stability and stabilization of stochastic neural networks with distributed parameters has not been considered.