Input-to-State Stability Analysis Of Hopfeld Recurrent Neural Networks

Recently, some dynamical properties such as stability, boundedness and robust-ness are widely investigated in control systems. In practice, control systems are veryoften affected by noise, for instance perturbations on controls and errors on obser-vations. Thus, it is desirable for a system not only to be stable, but also to displayso-called”input-to-state” stability properties. Intuitively, this means that the behaviorof the system should remain bounded when its inputs are bounded, and should tend toequilibrium when inputs tend to zero.ISS, which was frstly introduced in nonlinear systems by E.D. Sontag in late1980s, has been extensively and deeply studied due to its wide applications in variousareas. Since then, it becomes one of the most important concepts in the analysis andcomprehensive study of stability, especially in nonlinear systems.In this paper, we study the input-to-state stability properties of delayed Hopfeldrecurrent neural networks.In the frst part, a class of dynamical neural network models with time-varyingdelays is considered. By employing the Lyapunov-Krasovskii functional method andlinear matrix inequalities (LMIs) technique, some new suffcient conditions ensuringthe input-to-state stability property of the nonlinear network systems are obtained. Itis shown that the ISS can be determined by solving a set of LMIs, which can be easilychecked by using some standard numerical packages. Finally, Numerical examplesare provided to illustrate the effciency of the derived results.In the second part, a class of recurrent neural networks model with multipletime-varying delays is investigated. Firstly, we introduce the concepts of the ISS andexp-ISS of the considered neural networks. By utilizing the Lyapunov-Krasovskiifunctional method and linear matrix inequalities techniques, some new suffcient con-ditions ensuring the exponential input-to-state stability property of delayed networksystems are obtained. Two numerical examples are illustrated to check the effciencyof the derived results.