Results concerning the absolute stability of delayed neural networks

Continuous neural identifier for uncertain nonlinear systems with time delays in the input signal

Time-delay systems have been successfully used to represent the complexity of some dynamic systems. Time-delay is often used for modeling many real systems. Among others, biological and chemical plants have been described using time-delay terms with better results than those models that have not consider them. However, getting those models represented a challenge and sometimes the results were not so satisfactory. Non-parametric modeling offered an alternative to obtain suitable and usable models. Continuous neural networks (CNN) have been considered as a real alternative to provide models over uncertain non-parametric systems. This article introduces the design of a specific class of non-parametric model for uncertain time-delay system based on CNN considering the so-called delayed learning laws analysis. The convergence analysis as well as the learning laws were produced by means of a Lyapunov-Krasovskii functional. Three examples were developed to demonstrate the effectiveness of the modeling process forced by the identifier proposed in this study. The first example was a simple nonlinear model used as benchmark example. The second example regarded the human immunodeficiency virus dynamic behavior is used to show the performance of the suggested non-parametric identifier based on CNN for no fictitious neither academic models. Finally, a third example describing the evolution of hepatitis B virus served to test the identifier presented in this study and was also useful to provide evidence of its superior performance against a non-delayed identifier based on CNN.

RELAXED STABILITY CONDITIONS FOR DELAYED RECURRENT NEURAL NETWORKS WITH POLYTOPIC UNCERTAINTIES

This paper investigates the problem of stability analysis for recurrent neural networks with time-varying delays and polytopic uncertainties. Parameter-dependent Lypaunov functionals are employed to obtain sufficient conditions that guarantee the robust global exponential stability of the equilibrium point of the considered neural network. The derived stability criteria are expressed in terms of a set of relaxed linear matrix inequalities, which can be easily tested by using commercially available software. Two numerical examples are provided to demonstrate the effectiveness of the proposed results.

DELAY-DEPENDENT ROBUST EXPONENTIAL STABILITY FOR UNCERTAIN RECURRENT NEURAL NETWORKS WITH TIME-VARYING DELAYS

This paper considers the problem of robust exponential stability for a class of recurrent neural networks with time-varying delays and parameter uncertainties. The time delays are not necessarily differentiable and the uncertainties are assumed to be time-varying but norm-bounded. Sufficient conditions, which guarantee that the concerned uncertain delayed neural network is robustly, globally, exponentially stable for all admissible parameter uncertainties, are obtained under a weak assumption on the neuron activation functions. These conditions are dependent on the size of the time delay and expressed in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness and less conservatism of the proposed stability results.

Complete Stability in Multistable Delayed Neural Networks

We investigate the complete stability for multistable delayed neural networks. A new formulation modified from the previous studies on multistable networks is developed to derive componentwise dynamical property. An iteration argument is then constructed to conclude that every solution of the network converges to a single equilibrium as time tends to infinity. The existence of 3n equilibria and 2n positively invariant sets for the n-neuron system remains valid under the new formulation. The theory is demonstrated by a numerical illustration.

State estimation for jumping recurrent neural networks with discrete and distributed delays

This paper is concerned with the state estimation problem for a class of Markovian neural networks with discrete and distributed time-delays. The neural networks have a finite number of modes, and the modes may jump from one to another according to a Markov chain. The main purpose is to estimate the neuron states, through available output measurements, such that for all admissible time-delays, the dynamics of the estimation error is globally asymptotically stable in the mean square. An effective linear matrix inequality approach is developed to solve the neuron state estimation problem. Both the existence conditions and the explicit characterization of the desired estimator are derived. Furthermore, it is shown that the traditional stability analysis issue for delayed neural networks with Markovian jumping parameters can be included as a special case of our main results. Finally, numerical examples are given to illustrate the applicability of the proposed design method.

Almost periodic dynamics of a class of delayed neural networks with discontinuous activations

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.

Convergence of nonautonomous Cohen-Grossberg-type neural networks with variable delays

This paper is concerned with the global convergence of the solutions of a nonautonomous system with variable delays, arising from the description of the states of neurons in delayed Cohen-Grossberg type in a time-varying situation. By exploring intrinsic features between nonautonomous system and its asymptotic equation, several novel sufficient conditions are established to ensure that all solutions of the networks converge to a periodic function or a constant vector for delayed Cohen-Grossberg-type neural network (NN) models in time-varying situation. The results can be applied directly to group of NNs models including Hopfield NNs, bidirectional association memory NNs, and cellular NNs. Our results are not only presented in terms of system parameters and can be easily verified but also are less restrictive than previously known criteria. Numerical simulations have also been presented to demonstrate the theoretical analysis.

New necessary and sufficient conditions for absolute stability of neural networks

This paper presents new necessary and sufficient conditions for absolute stability of asymmetric neural networks. The main result is based on a solvable Lie algebra condition, which generalizes existing results for symmetric and normal neural networks. An exponential convergence estimate of the neural networks is also obtained. Further, it is demonstrated how to generate larger sets of weight matrices for absolute stability of the neural networks from known normal weight matrices through simple procedures. The approach is nontrivial in the sense that non-normal matrices can possibly be contained in the resulting weight matrix set. And the results also provide finite checking for robust stability of neural networks in the presence of parameter uncertainties.

Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions

In this letter, without assuming the boundedness of the activation functions, we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions. A relaxed set of sufficient conditions is derived, guaranteeing the existence, uniqueness, and global stability of the equilibrium point. Convergence behaviors for both state and output are discussed. The constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique. We also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high-slope continuous activation functions.

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.

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.

State Estimation for Delayed Neural Networks

In this letter, the state estimation problem is studied for neural networks with time-varying delays. The interconnection matrix and the activation functions are assumed to be norm-bounded. The problem addressed is to estimate the neuron states, through available output measurements, such that for all admissible time-delays, the dynamics of the estimation error is globally exponentially stable. An effective linear matrix inequality approach is developed to solve the neuron state estimation problem. In particular, we derive the conditions for the existence of the desired estimators for the delayed neural networks. We also parameterize the explicit expression of the set of desired estimators in terms of linear matrix inequalities (LMIs). Finally, it is shown that the main results can be easily extended to cope with the traditional stability analysis problem for delayed neural networks. Numerical examples are included to illustrate the applicability of the proposed design method.

Criteria for exponential stability of Cohen–Grossberg neural networks

In this paper, the Cohen-Grossberg neural network models without and with time delays are considered. By constructing several novel Lyapunov functionals, some sufficient criteria for the existence of a unique equilibrium and global exponential stability of the network are derived. These results are fairly general and can be easily verified. Besides, the approach of the analysis allows one to consider different types of activation functions, including piecewise linear, sigmoids with bounded activations as well as C1-smooth sigmoids. In the meantime, our approach does not require any symmetric assumption of the connection matrix. It is believed that these results are significant and useful for the design and applications of the Cohen-Grossberg model.

Some sufficient conditions for global exponential stability of delayed Hopfield neural networks

In this paper, we have derived some sufficient conditions for existence and uniqueness of equilibrium and global exponential stability in delayed Hopfield neural networks by using a different approach from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps, rather we first prove global exponential convergence to 0 of the difference between any two solutions of the original neural networks, the existence and uniqueness of equilibrium is the direct results of this procedure. We obtain the conditions by suitable construction of Lyapunov functionals and estimation of derivates of the Lyapunov functionals by the well-known Young's inequality and Holder's inequality. The proposed conditions are related to p-norms of vector or matrix, p in [1, infinity] and thus unify and generalize some results in the literature.

Absolutely exponential stability of a class of neural networks with unbounded delay

In this paper, the existence and uniqueness of the equilibrium point and absolute stability of a class of neural networks with partially Lipschitz continuous activation functions are investigated. The neural networks contain both variable and unbounded delays. Using the matrix property, a necessary and sufficient condition for the existence and uniqueness of the equilibrium point of the neural networks is obtained. By constructing proper vector Liapunov functions and nonlinear integro-differential inequalities involving both variable delays and unbounded delay, using M-matrix theory, sufficient conditions for absolutely exponential stability are obtained.

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

In this paper, we obtain new sufficient conditions ensuring existence, uniqueness, and global asymptotic stability (GAS) of the equilibrium point for a general class of delayed neural networks (DNNs) via nonsmooth analysis, which makes full use of the Lipschitz property of functions defining DNNs. Based on this new tool of nonsmooth analysis, we first obtain a couple of general results concerning the existence and uniqueness of the equilibrium point. Then those results are applied to show that existence assumptions on the equilibrium point in some existing sufficient conditions ensuring GAS are actually unnecessary; and some strong assumptions such as the boundedness of activation functions in some other existing sufficient conditions can be actually dropped. Finally, we derive some new sufficient conditions which are easy to check. Comparison with some related existing results is conducted and advantages are illustrated with examples. Throughout our paper, spectral properties of the matrix (A + Atau) play an important role, which is a distinguished feature from previous studies. Here, A and Atau are, respectively, the feedback and the delayed feedback matrix defining the neural network under consideration.

Back-propagation learning of infinite-dimensional dynamical systems

This paper presents numerical studies of applying back-propagation learning to a delayed recurrent neural network (DRNN). The DRNN is a continuous-time recurrent neural network having time delayed feedbacks and the back-propagation learning is to teach spatio-temporal dynamics to the DRNN. Since the time-delays make the dynamics of the DRNN infinite-dimensional, the learning algorithm and the learning capability of the DRNN are different from those of the ordinary recurrent neural network (ORNN) having no time-delays. First, two types of learning algorithms are developed for a class of DRNNs. Then, using chaotic signals generated from the Mackey-Glass equation and the Rössler equations, learning capability of the DRNN is examined. Comparing the learning algorithms, learning capability, and robustness against noise of the DRNN with those of the ORNN and time delay neural network, advantages as well as disadvantages of the DRNN are investigated.

Global convergence of delayed neural network systems

In this paper, without assuming the boundedness, strict monotonicity and differentiability of the activation functions, we utilize a new Lyapunov function to analyze the global convergence of a class of neural networks models with time delays. A new sufficient condition guaranteeing the existence, uniqueness and global exponential stability of the equilibrium point is derived. This stability criterion imposes constraints on the feedback matrices independently of the delay parameters. The result is compared with some previous works. Furthermore, the condition may be less restrictive in the case that the activation functions are hyperbolic tangent.

New conditions on global stability of Cohen-Grossberg neural networks

In this letter, we discuss the dynamics of the Cohen-Grossberg neural networks. We provide a new and relaxed set of sufficient conditions for the Cohen-Grossberg networks to be absolutely stable and exponentially stable globally. We also provide an estimate of the rate of convergence.

Delay-dependent exponential stability analysis of delayed neural networks: an LMI approach

For neural networks with constant or time-varying delays, the problems of determining the exponential stability and estimating the exponential convergence rate are studied in this paper. An approach combining the Lyapunov-Krasovskii functionals with the linear matrix inequality is taken to investigate the problems, which provide bounds on the interconnection matrix and the activation functions, so as to guarantee the systems' exponential stability. Some criteria for the exponentially stability, which give information on the delay-dependence property, are derived. The results obtained in this paper provide one more set of easily verified guidelines for determining the exponentially stability of delayed neural networks, which are less conservative and less restrictive than the ones reported so far in the literature.

Dynamic stability conditions for Lotka-Volterra recurrent neural networks with delays

The Lotka-Volterra model of neural networks, derived from the membrane dynamics of competing neurons, have found successful applications in many "winner-take-all" types of problems. This paper studies the dynamic stability properties of general Lotka-Volterra recurrent neural networks with delays. Conditions for nondivergence of the neural networks are derived. These conditions are based on local inhibition of networks, thereby allowing these networks to possess a multistability property. Multistability is a necessary property of a network that will enable important neural computations such as those governing the decision making process. Under these nondivergence conditions, a compact set that globally attracts all the trajectories of a network can be computed explicitly. If the connection weight matrix of a network is symmetric in some sense, and the delays of the network are in L2 space, we can prove that the network will have the property of complete stability.

Exponential stability of Cohen-Grossberg neural networks

Exponential stabilities of the Cohen-Grossberg neural network with and without delays are analyzed. By Liapunov functions/functionals, sufficient conditions are obtained for general exponential stability, while by using a comparison result from the theory of monotone dynamical systems, componentwise exponential stability is also discussed. All results are established without assuming any symmetry of the connection matrix, and the differentiability and monotonicity of the activation functions.