Global stability analysis for delayed neural networks via an interval matrix approach

C. Li, X. Liao and T. HuangAbstract: Global asymptotic stability for a general class of neural networks with delays is reducedto that for interval linear delayed differential equations under the assumption of Lipschitz continu-ity. By employing Lyapunov–Krasovskii theory, the problem is further reduced to that of Hurwitzstability of interval matrices. Based on the later theory, several new sets of stability criteria forneural networks with constant delays are derived. This demonstration and comparison withrecent results show that the present results are new stability criteria for the investigated neuralnetwork model.1 IntroductionArtificial neural networks have been generally classified asHopfield neural networks (HNNs), cellular neural networks(CNNs), bidirectional associated memory (BAM) neuralnetworks and Lotka–Volterra neural networks, with orwithout delays. In investigating the stability properties ofneural networks, stability results that impose constraint con-ditions on the network parameters depend on the intendedapplications. In order for a neural network to function asan associative memory, the designed network must con-verge to a set of stable equilibrium points depending onthe initial conditions. However, when a neural network isused for solving an optimisation problem, it is necessarythat the network have a unique and globally asymptoticallystable (GAS) equilibrium point independent of the initialconditions. In recent years, global asymptotic stability ofvarious types of neural networks with time delays hasbeen extensively studied and many stability conditionshave been obtained for various classes of delayed neuralnetwork models: See [1–10] for CNN, [11–17] for HNNand [18–21] for the BAM model. The analytical approachesemployed to investigate the global asymptotic stability ofcontinuous-time neural networks with time delays may bedivided mainly into two types: the Lyapunov function/functional method [22] and the differential inequalitymethod. The form of presentation includes various matrix(vector) norm forms, algebraic inequalities, Raccatiequations/inequalities, M-matrices and linear matrixinequalities [LMIs] [23]. However, due to the limitationsof these analytical approaches, one can only derive somesufficient conditions with some strong constraints on thecoefficients of the systems. Hence, researchers in this fieldhave been working to find new stability criteria.In this paper, a new idea for investigating this problem ispresented. Under the assumption of bounded andLipchitz-continuous activation functions, we reduce theoriginal systems to linear and variable-coefficient delayedsystems. Then, employing a combination of the Lyapunovfunction method and the stability theory of interval matrices,we derive some new global asymptotic stability criteria fordelayed neural networks. Finally, comparisons are drawnwith recent results through numerical examples.2 Problem formulation and preliminariesThe delayed neural network model that we consider isdefined by the following state equations_x