Stability Of Recurrent Neural Networks Research Articles

Exponential stability of recurrent neural networks with time-varying discrete and distributed delays

This paper is concerned with the problems of determining the global exponential stability and estimating the exponential-convergence rate for a class of recurrent neural networks (RNNs) with mixed discrete and distributed delays. By constructing an appropriate Lyapunov–Krasovskii functional and employing linear matrix inequality (LMI) technique, new delay-dependent exponential-stability criteria are derived in term of LMIs and the exponential-convergence rate is estimated. Numerical examples are given to show the effectiveness and improvement of the obtained results.

Delay-interval-dependent stability of recurrent neural networks with time-varying delay

This paper studies the delay-interval-dependent stability of the equilibrium point of a general class of recurrent neural networks with time-varying delays that may exclude zero. By constructing the appropriate Lyapunov–Krasovskii functional, two sufficient conditions ensuring the global asymptotic stability of the equilibrium point of such networks with interval-time-varying delays are established. The present results, together with two numerical examples, show that the equilibrium points of the considered networks may be globally asymptotically stable in some delay interval(s) even though the equilibrium points of the corresponding delay-free recurrent neural networks are not globally asymptotically stable.

An Improved Algebraic Criterion for Global Exponential Stability of Recurrent Neural Networks With Time-Varying Delays

This brief paper presents an M-matrix-based algebraic criterion for the global exponential stability of a class of recurrent neural networks with decreasing time-varying delays. The criterion improves some previous criteria based on M-matrix and is easy to be verified with the connection weights of the recurrent neural networks with decreasing time-varying delays. In addition, the rate of exponential convergence can be estimated via a simple computation based on the criterion herein.

Noise-Induced Stabilization of the Recurrent Neural Networks With Mixed Time-Varying Delays and Markovian-Switching Parameters

The stabilization of recurrent neural networks with mixed time-varying delays and Markovian-switching parameters by noise is discussed. First, a new result is given for the existence of unique states of recurrent neural networks (NNs) with mixed time-varying delays and Markovian-switching parameters in the presence of noise, without the need to satisfy the linear growth conditions required by general stochastic Markovian-switching systems. Next, a delay-dependent condition for stabilization of concerned recurrent NNs is derived by applying the ltd formula, the Gronwall inequality, the law of large numbers, and the ergodic property of Markovian chain. The results show that there always exists an appropriate white noise such that any recurrent NNs with mixed time-varying delays and Markovian-switching parameters can be exponentially stabilized by noise if the delays are sufficiently small.

Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach

In this paper, the problem on exponential stability analysis of recurrent neural networks with both time-varying delays and general activation functions is considered. Neither the boundedness and the monotony on these activation functions nor the differentiability on the time-varying delays are assumed. By employing Lyapunov functional and the free-weighting matrix method, several sufficient conditions in linear matrix inequality form are obtained to ensure the existence, uniqueness and global exponential stability of equilibrium point for the neural networks. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. The proposed stability results are less conservative than some recently known ones in the literature, which is demonstrated via an example with simulation.

Delayed Standard Neural Network Models for Control Systems

In order to conveniently analyze the stability of recurrent neural networks (RNNs) and successfully synthesize the controllers for nonlinear systems, similar to the nominal model in linear robust control theory, the novel neural network model, named delayed standard neural network model (DSNNM) is presented, which is the interconnection of a linear dynamic system and a bounded static delayed (or nondelayed) nonlinear operator. By combining a number of different Lyapunov functionals with S-procedure, some useful criteria of global asymptotic stability and global exponential stability for the continuous-time DSNNMs (CDSNNMs) and discrete-time DSNNMs (DDSNNMs) are derived, whose conditions are formulated as linear matrix inequalities (LMIs). Based on the stability analysis, some state-feedback control laws for the DSNNM with input and output are designed to stabilize the closed-loop systems. Most RNNs and neurocontrol nonlinear systems with (or without) time delays can be transformed into the DSNNMs to be stability-analyzed or stabilization-synthesized in a unified way. In this paper, the DSNNMs are applied to analyzing the stability of the continuous-time and discrete-time RNNs with or without time delays, and synthesizing the state-feedback controllers for the chaotic neural-network-system and discrete-time nonlinear system. It turns out that the DSNNM makes the stability conditions of the RNNs easily verified, and provides a new idea for the synthesis of the controllers for the nonlinear systems.

Robust Exponential Stability of Recurrent Neural Networks With Multiple Time-Varying Delays

New criteria for the uniqueness and global robust exponential stability are established for the equilibrium point of interval recurrent neural networks with multiple time-varying delays via a decomposition method and analysis technique. Results are presented in the form of linear matrix inequality, which can be solved efficiently. Two numerical examples are employed to show the effectiveness of the present results.

Global Asymptotical Stability of Recurrent Neural Networks With Multiple Discrete Delays and Distributed Delays

By employing the Lyapunov-Krasovskii functional and linear matrix inequality (LMI) approach, the problem of global asymptotical stability is studied for recurrent neural networks with both discrete time-varying delays and distributed time-varying delays. Some sufficient conditions are given for checking the global asymptotical stability of recurrent neural networks with mixed time-varying delay. The proposed LMI result is computationally efficient as it can be solved numerically using standard commercial software. Two examples are given to show the usefulness of the results.

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.

Novel criteria for global exponential periodicity and stability of recurrent neural networks with time-varying delays

In this paper, the global exponential periodicity and stability of recurrent neural networks with time-varying delays are investigated by applying the idea of vector Lyapunov function, M-matrix theory and inequality technique. We assume neither the global Lipschitz conditions on these activation functions nor the differentiability on these time-varying delays, which were needed in other papers. Several novel criteria are found to ascertain the existence, uniqueness and global exponential stability of periodic solution for recurrent neural network with time-varying delays. Moreover, the exponential convergence rate index is estimated, which depends on the system parameters. Some previous results are improved and generalized, and an example is given to show the effectiveness of our method.

Improved conditions for global exponential stability of recurrent neural networks with time-varying delays

This paper presents new theoretical results on global exponential stability of recurrent neural networks with bounded activation functions and time-varying delays. The stability conditions depend on external inputs, connection weights, and time delays of recurrent neural networks. Using these results, the global exponential stability of recurrent neural networks can be derived, and the estimated location of the equilibrium point can be obtained. As typical representatives, the Hopfield neural network (HNN) and the cellular neural network (CNN) are examined in detail.

On Stability of Recurrent Neural Networks—An Approach From Volterra Integro-Differential Equations

The uniform asymptotic stability of recurrent neural networks (RNNs) with distributed delay is analyzed by comparing RNNs to linear Volterra integro-differential systems under Lipschitz continuity of activation functions. The stability criteria obtained have unified and extended many existing results on RNNs.

New results on global exponential stability of recurrent neural networks with time-varying delays

This Letter provides new sufficient conditions for the existence, uniqueness and global exponential stability of the equilibrium point of recurrent neural networks with time-varying delays by employing Lyapunov functions and using the Halanay inequality. The time-varying delays are not necessarily differentiable. Both Lipschitz continuous activation functions and monotone nondecreasing activation functions are considered. The derived stability criteria are expressed in terms of linear matrix inequalities (LMIs), which can be checked easily by resorting to recently developed algorithms solving LMIs. Furthermore, the proposed stability results are less conservative than some previous ones in the literature, which is demonstrated via some numerical examples.

Global asymptotic and robust stability of recurrent neural networks with time delays

In this paper, two related problems, global asymptotic stability (GAS) and global robust stability (GRS) of neural networks with time delays, are studied. First, GAS of delayed neural networks is discussed based on Lyapunov method and linear matrix inequality. New criteria are given to ascertain the GAS of delayed neural networks. In the designs and applications of neural networks, it is necessary to consider the deviation effects of bounded perturbations of network parameters. In this case, a delayed neural network must be formulated as a interval neural network model. Several sufficient conditions are derived for the existence, uniqueness, and GRS of equilibria for interval neural networks with time delays by use of a new Lyapunov function and matrix inequality. These results are less restrictive than those given in the earlier references.

Delay-independent exponential stability of recurrent neural networks

In this Letter, the authors investigate further the recurrent neural networks model with delays, and give some sufficient criteria ensuring existence, uniqueness and global exponential stability (GES) of the equilibrium point by employing the inequality a ∏ k = 1 m b k q k ⩽ 1 r ∑ k = 1 m q k b k r + 1 r a r ( a ⩾ 0 , b k ⩾ 0 , q k > 0 with ∑ k = 1 m q k = r − 1 , and r > 1 ), constructing a new Lyapunov functional, and applying the homeomorphism theory. These criteria do not require the signal functions are differentiable, bounded and monotone nondecreasing. Thus the criteria obtained have highly important significance in solving optimization problems and reducing the neural computing time. Furthermore, we extend or improve the previously known results.

ON PERIODIC DYNAMICAL SYSTEMS

The authors investigate the existence and the global stability of periodic solution for dynamical systems with periodic interconnections, inputs and self-inhibitions. The model is very general, the conditions are quite weak and the results obtained are universal.

Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays

This paper investigates the absolute exponential stability of a general class of delayed neural networks, which require the activation functions to be partially Lipschitz continuous and monotone nondecreasing only, but not necessarily differentiable or bounded. Three new sufficient conditions are derived to ascertain whether or not the equilibrium points of the delayed neural networks with additively diagonally stable interconnection matrices are absolutely exponentially stable by using delay Halanay-type inequality and Lyapunov function. The stability criteria are also suitable for delayed optimization neural networks and delayed cellular neural networks whose activation functions are often nondifferentiable or unbounded. The results herein answer a question: if a neural network without any delay is absolutely exponentially stable, then under what additional conditions, the neural networks with delay is also absolutely exponentially stable.

Boundedness and stability for recurrent neural networks with variable coefficients and time-varying delays

In this Letter, the problems of boundedness and stability for a general class of non-autonomous recurrent neural networks with variable coefficients and time-varying delays are analyzed via employing Young inequality technique and Lyapunov method. Some simple sufficient conditions are given for boundedness and stability of the solutions for the recurrent neural networks. These results generalize and improve the previous works, and they are easy to check and apply in practice. Two illustrative examples and their numerical simulations are also given to demonstrate the effectiveness of the proposed results.

On global asymptotic stability of recurrent neural networks with time-varying delays

In this paper, by constructing a new Lyapunov functional, and using M-matrix and topological degree tool, problem of the global asymptotic stability (GAS) is discussed for a class of recurrent neural networks with time-varying delays. Some simple and new sufficient conditions are obtained ensuring existence, uniqueness of the equilibrium point and its GAS of the neural networks. Some previous works are improved. In addition, this condition does not require the activation functions to be differentiable, bounded and monotone nondecreasing and the weight-connected matrices to be symmetric. The neural network model considered in this paper include the delayed Hopfield neural networks, bidirectional associative memory networks and delayed cellular neural networks as its special cases.

Local stability of recurrent networks with time-varying weights and inputs

We present local conditions for stability of recurrent neural networks with time-varying parameters. Our constructive approach guarantees that a recurrent network implements a proper mapping from time-varying input to time-varying output functions using a local equilibrium as point of operation. The corresponding bounds on the allowed inputs and time-variation are defined by linear matrix inequalities and can be obtained numerically efficient by means of interior point optimisation schemes. We compare the method to the standard Lyapunov approach and apply it to an example of learning an input–output map implied by the chaotic Roessler attractor.