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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 stability of switched systems with impulsive effect

The exponential stability of a class of switched systems containing stable and unstable subsystems with impulsive effect is analyzed by using the matrix measure concept and the average dwell-time approach. It is shown that if appropriately a large amount of the average dwell-time and the ratio of the total activation time of the subsystems with negative matrix measure to the total activation time of the subsystems with nonnegative matrix measure is chosen, the exponential stability of a desired degree is guaranteed. Using the proposed switching scheme, we studied the robust exponential stability for a class of switched systems with impulsive effect and structure perturbations. Simulations validate the main 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.     

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.     

Pinning exponential synchronisation and passivity of coupled delayed reaction–diffusion neural networks with and without parametric uncertainties

In this paper, we focus on the pinning exponential synchronisation and passivity of coupled reaction–diffusion neural networks (CRDNNs) with and without parametric uncertainties, respectively. On the one hand, with the help of designed nonlinear pinning controllers and Lyapunov functional method, sufficient conditions are established to let the CRDNNs with hybrid coupling and mixed time-varying delays realise exponential synchronisation and passivity. On the other hand, considering that the external perturbations may lead the reaction–diffusion neural networks (RDNNs) parameters to containing uncertainties, the robust pinning exponential synchronisation and robust pinning passivity for coupled delayed RDNNs with parametric uncertainties are investigated by designing appropriate pinning control strategies. Finally, the effectiveness of the theoretical results are substantiated by the two given numerical examples.     

Finite-time stability analysis for fractional-order Cohen–Grossberg BAM neural networks with time delays

In this paper, the problem of finite-time stability for a class of fractional-order Cohen–Grossberg BAM neural networks with time delays is investigated. Using some inequality techniques, differential mean value theorem and contraction mapping principle, sufficient conditions are presented to ensure the finite-time stability of such fractional-order neural models. Finally, a numerical example and simulations are provided to demonstrate the effectiveness of the derived theoretical 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.     

Synchronisation in an array of spatial diffusion coupled reaction–diffusion neural networks via pinning control

The authors investigate pinning synchronisation for two spatial diffusion coupled reaction–diffusion neural networks under undirected and directed topologies. Combined with stability theory, some inequalities and Lyapunov functional method, several sufficient conditions are derived to assure the synchronisation of the considered networks by designing appropriate pinning controllers. It should be pointed out that a new type of spatial diffusion feedback pinning controller is designed in this paper. The correctness of the obtained results is confirmed by two simulation examples.     

Implementation and solution of the diffusion–reaction equation using high-order methods

The orthogonal collocation, Galerkin, tau and least-squares methods are applied to solve a diffusion–reaction problem. In general, the least-squares method suffers from lower accuracy than the other weighted residual methods. The least-squares method holds the most complex linear algebra theory and is thus associated with the most complex implementation. On the other hand, an advantage of the least-squares method is that it always produces a symmetric and positive definite system matrix which can be solved with an efficient iterative technique such as the conjugate gradient method or its preconditioned version. For the present problem, neither the Galerkin, tau and orthogonal collocation techniques produce symmetric and positive definite system matrices, hence the conjugate gradient method is not applicable for these numerical techniques.