Boundary Stabilization of Stochastic Delayed Cohen-Grossberg Neural Networks With Diffusion Terms

This study considers the boundary stabilization for stochastic delayed Cohen-Grossberg neural networks (SDCGNNs) with diffusion terms by the Lyapunov functional method. In the realization of NNs, sometimes time delays and diffusion phenomenon cannot be ignored, so Cohen-Grossberg NNs with time delays and diffusion terms are studied in this article. Moreover, different from the previously distributed control, the boundary control is used to stabilize the system, which can reduce the spatial cost of the controller and is easy to implement. Boundary controllers are presented for system with Neumann boundary and mixed boundary conditions, and criteria are derived such that the controlled system achieves mean-square exponential stabilization. Based on the criterion, the effects of diffusion matrix, coupling strength, coupling matrix, and time delays on exponentially stability are analyzed. In the process of analysis, two difficulties need to be addressed: 1) how to introduce boundary control into system analysis? and 2) how to analyze the influence of system parameters on stability? We deal with these problems by using Poincaré's inequality and Schur's complement lemma. Moreover, mean-square exponential synchronization of stochastic delayed Hopfield NNs with diffusion terms, as an application of the theoretical result, is considered under the boundary control. Examples are given to illustrate the effectiveness of the theoretical results.

Boundary Mittag-Leffler stabilization of fractional reaction-diffusion cellular neural networks

Mittag-Leffler stabilization is studied for fractional reaction-diffusion cellular neural networks (FRDCNNs) in this paper. Different from previous literature, the FRDCNNs in this paper are high-dimensional systems, and boundary control and observed-based boundary control are both used to make FRDCNNs achieve Mittag-Leffler stability. First, a state-dependent boundary controller is designed when system states are available. By employing the spatial integral functional method and some inequalities, a criterion ensuring Mittag-Leffler stability of FRDCNNs is presented. Then, when the information of system states is not fully accessible, an observer is presented to estimate the system states based on boundary output and an observer-based boundary controller is provided aiming to stabilize the considered FRDCNNs. Furthermore, a robust observer-based boundary controller is proposed to ensure the Mittag-Leffler stability for FRDCNNs with uncertainties. Examples are given to illustrate the effectiveness of obtained theoretical results.

Suppression of Coherence Collapse in Semiconductor Fano Lasers

We show that semiconductor Fano lasers strongly suppress dynamic instabilities induced by external optical feedback. A comparison with conventional Fabry-Perot lasers shows orders of magnitude improvement in feedback stability and in many cases even total suppression of coherence collapse, which is of major importance for applications in integrated photonics. The laser dynamics are analyzed using a generalization of the Lang-Kobayashi model for semiconductor lasers with external feedback, and an analytical expression for the critical feedback level is derived.

Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

Improved delay-dependent robust stability criteria for recurrent neural networks with time-varying delays

In this paper, the problem of improved delay-dependent robust stability criteria for recurrent neural networks (RNNs) with time-varying delays is investigated. Combining the Lyapunov-Krasovskii functional with linear matrix inequality (LMI) techniques and integral inequality approach (IIA), delay-dependent robust stability conditions for RNNs with time-varying delay, expressed in terms of quadratic forms of state and LMI, are derived. The proposed methods contain the least numbers of computed variables while maintaining the effectiveness of the stability conditions. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. Numerical examples are included to show that the proposed method is effective and can provide less conservative results.

Stability of Discrete Recurrent Neural Networks with Interval Delays

A global exponential stability method for a class of discrete time recurrent neural networks with interval time-varying delays and norm-bounded time-varying parameter uncertainties is developed in this paper. The method is derived based on a new Lyapunov-Krasovskii functional to exhibit the delay-range-dependent dynamics and to compensate for the enlarged time-span. In addition, it eliminates the need for over bounding and utilizes smaller number of LMI decision variables. Effective solutions to the global stability problem are provided in terms of feasibility-testing of parameterized linear matrix inequalities (LMIs). Numerical examples are presented to demonstrate the potential of the developed technique.

Theoretic design of differential minimax controllers for stochastic cellular neural networks

This paper presents a theoretical design of how a minimax equilibrium of differential game is achieved in stochastic cellular neural networks. In order to realize the equilibrium, two opposing players are selected for the model of stochastic cellular neural networks. One is the vector of external inputs and the other is the vector of internal noises. The design procedure follows the nonlinear H infinity optimal control methodology to accomplish the best rational stabilization in probability for stochastic cellular neural networks, and to attenuate noises to a predefined level with stability margins. Three numerical examples are given to demonstrate the effectiveness of the proposed approach.

Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions and time-varying delays

The paper presents theoretical results on the global exponential periodicity and global exponential stability of a class of recurrent neural networks with various general activation functions and time-varying delays. The general activation functions include monotone nondecreasing functions, globally Lipschitz continuous and monotone nondecreasing functions, semi-Lipschitz continuous mixed monotone functions, and Lipschitz continuous functions. For each class of activation functions, testable algebraic criteria for ascertaining global exponential periodicity and global exponential stability of a class of recurrent neural networks are derived by using the comparison principle and the theory of monotone operator. Furthermore, the rate of exponential convergence and bounds of attractive domain of periodic oscillations or equilibrium points are also estimated. The convergence analysis based on the generalization of activation functions widens the application scope for the model design of neural networks. In addition, the new effective analytical method enriches the toolbox for the qualitative analysis of neural networks.

An LMI approach to global asymptotic stability of the delayed Cohen–Grossberg neural network via nonsmooth analysis

In this paper, a linear matrix inequality (LMI) to global asymptotic stability of the delayed Cohen-Grossberg neural network is investigated by means of nonsmooth analysis. Several new sufficient conditions are presented to ascertain the uniqueness of the equilibrium point and the global asymptotic stability of the neural network. It is noted that the results herein require neither the smoothness of the behaved function, or the activation function nor the boundedness of the activation function. In addition, from theoretical analysis, it is found that the condition for ensuring the global asymptotic stability of the neural network also implies the uniqueness of equilibrium. The obtained results improve many earlier ones and are easy to apply. Some simulation results are shown to substantiate the theoretical results.

New Delay-Dependent Stability Criteria for Neural Networks With Time-Varying Delay

In this letter, a new method is proposed for stability analysis of neural networks (NNs) with a time-varying delay. Some less conservative delay-dependent stability criteria are established by considering the additional useful terms, which were ignored in previous methods, when estimating the upper bound of the derivative of Lyapunov functionals and introducing the new free-weighting matrices. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method.

Global exponential stability of recurrent neural networks with time-varying delays in the presence of strong external stimuli

This paper presents new theoretical results on the global exponential stability of recurrent neural networks with bounded activation functions and bounded time-varying delays in the presence of strong external stimuli. It is shown that the Cohen-Grossberg neural network is globally exponentially stable, if the absolute value of the input vector exceeds a criterion. As special cases, the Hopfield neural network and the cellular neural network are examined in detail. In addition, it is shown that criteria herein, if partially satisfied, can still be used in combination with existing stability conditions. Simulation results are also discussed in two illustrative examples.

Delay-dependent state estimation for delayed neural networks

In this letter, the delay-dependent state estimation problem for neural networks with time-varying delay is investigated. A delay-dependent criterion is established to estimate the neuron states through available output measurements such that the dynamics of the estimation error is globally exponentially stable. The proposed method is based on the free-weighting matrix approach and is applicable to the case that the derivative of a time-varying delay takes any value. An algorithm is presented to compute the state estimator. Finally, a numerical example is given to demonstrate the effectiveness of this approach and the improvement over existing ones.

Global point dissipativity of neural networks with mixed time-varying delays

By employing the Lyapunov method and some inequality techniques, the global point dissipativity is studied for neural networks with both discrete time-varying delays and distributed time-varying delays. Simple sufficient conditions are given for checking the global point dissipativity of neural networks with mixed time-varying delays. The proposed linear matrix inequality approach is computationally efficient as it can be solved numerically using standard commercial software. Illustrated examples are given to show the usefulness of the results in comparison with some existing results.

An Improved Global Asymptotic Stability Criterion for Delayed Cellular Neural Networks

A new Lyapunov-Krasovskii functional is constructed for delayed cellular neural networks, and the S-procedure is employed to handle the nonlinearities. An improved global asymptotic stability criterion is also derived that is a generalization of, and an improvement over, previous results. Numerical examples demonstrate the effectiveness of the criterion.

A new approach to exponential stability analysis of neural networks with time-varying delays

This paper considers the problem of exponential stability analysis of neural networks with time-varying delays. The activation functions are assumed to be globally Lipschitz continuous. A linear matrix inequality (LMI) approach is developed to derive sufficient conditions ensuring the delayed neural network to have a unique equilibrium point, which is globally exponentially stable. The proposed LMI conditions can be checked easily by recently developed algorithms solving LMIs. Examples are provided to demonstrate the reduced conservativeness of the proposed results.

Improved Global Robust Asymptotic Stability Criteria for Delayed Cellular Neural Networks

This paper considers the problem of global robust stability analysis of delayed cellular neural networks (DCNNs) with norm-bounded parameter uncertainties. In terms of a linear matrix inequality, a new sufficient condition ensuring a nominal DCNN to have a unique equilibrium point which is globally asymptotically stable is proposed. This condition is shown to be a generalization and improvement over some previous criteria. Based on the stability result, a robust stability condition is developed, which contains an existing robust stability result as a special case. An example is provided to demonstrate the reduced conservativeness of the proposed results.

Deriving Sufficient Conditions for Global Asymptotic Stability of Delayed Neural Networks via Nonsmooth Analysis—II

Following our recent approach of nonsmooth analysis, we report a new set of sufficient conditions and its implications for the global asymptotic stability of delayed cellular neural networks (DCNN). The new conditions not only unify a string of previous stability results, but also yield strict improvement over them by allowing the symmetric part of the feedback matrix positive definite, hence enlarging the application domain of DCNNs. Advantages of the new results over existing ones are illustrated with examples. We also compare our results with those related results obtained via LMI approach.

Global Asymptotic Stability Analysis of Bidirectional Associative Memory Neural Networks With Time Delays

This paper presents a sufficient condition for the existence, uniqueness and global asymptotic stability of the equilibrium point for bidirectional associative memory (BAM) neural networks with distributed time delays. The results impose constraint conditions on the network parameters of neural system independently of the delay parameter, and they are applicable to all continuous nonmonotonic neuron activation functions. It is shown that in some special cases of the results, the stability criteria can be easily checked. Some examples are also given to compare the results with the previous results derived in the literature.