Stability analysis of differential inclusions in Banach space with applications to nonlinear systems with time delays

Abstract

We present new stability results for dynamical systems determined by differential inclusions on Banach space. Specifically, under some rather weak assumptions, we establish a relation between the stability properties of nonconvex problems and corresponding convexified problems. Also, we present a new result for differential inclusions which corresponds to invariance theory type results for differential equations. We show that the above results constitute significant improvements over existing results. The approach for the stability analysis of dynamical systems determined by differential inclusions developed in the present paper differs significantly from existing results which make use of relaxation theorems. Rather than impose conditions on the set-valued functions, we hypothesize the existence of a continuously differentiable Lyapunov function with reasonable properties. To demonstrate applicability, we use the above results in the analysis of the absolute stability problem of regulator systems with multinonlinearities and time delays. Also, we apply the above results in the stability analysis of operating points (equilibria) of a class of integrated circuits with time delays, with an emphasis on a class of artificial neural networks for associative memories.