Nonlinear Volterra integrodifferential evolution inclusions and optimal control

Abstract

In this paper we examine integrodifferential evolution inclusions of the Volterra type driven by time dependent, monotone, hemicontinuous operators. We prove two existence theorems; one for convex valued perturbations and the other for nonconvex valued ones. We also establish a topological property of the solution set of the “convex” problem. Then we prove a result on the continuous dependence of the solutions on the data of the problem (sensitivity analysis). We also consider a random version of the inclusion and prove that it admits a random solution. Then we pass to optimal control problems. First we establish the existence of optimal admissible pairs and then using the notions of epigraphical and G-convergences, we obtain a variational stability result. Finally we work in detail two parabolic distributed parameter optimal control problems, illustrating the applicability of our work. © 1991 Department of Mathematics, Tokyo Institute of Technology.