New stability results for a class of countably infinite systems of differential equations are established. We consider those systems which may be viewed as an interconnection of countably infinitely many free or isolated subsystems. Throughout, the analysis is accomplished in terms of simpler subsystems and in terms of the system interconnecting structure. This approach makes it often possible to circumvent difficulties usually encountered in the application of the Lyapunov approach to complex systems with intricate structure. Both scalar Lyapunov functions and vector Lyapunov functions are used in the analysis. The applicability of the present results is demonstrated by means of several motivating examples, including a neural model.
Liapunov functions of simple form have been used for the study of stability properties of difference-differential equations. In this paper we provide necessary and sufficient conditions for the existence of such functions.
Boundedness and asymptotic behavior as t→∞ of solutions of nonlinearly perturbed Volterra equations are studied. Equations of both convolution and nonconvolution type are considered. An auxiliary equation plays an important role in the analysis of each type.