TIME-DEPENDENT SUBDIFFERENTIAL EVOLUTION INCLUSIONS AND OPTIMAL-CONTROL

The purpose of this paper is to study from many different viewpoints e volution inclusions and optimal control problems involving time depend ent subdifferential operators. Throughout this work we take a special interest in the t-dependence of the functional phi(t, x), involved in the subdifferential. We employ a condition that allows the domain dom phi(t,.) to vary regularly without precluding the possibility that dom phi(t,.) boolean AND dom(s,.) = 0 for t not equal s. Hence our formul ation is general enough to incorporate problems with time varying cons traints (obstacles), In section 3, we deal with evolution inclusions. In 3.1 we prove two existence theorems; one for a nonconvex valued ori entor field F and the other for a convex valued one, In 3.2 we look fo r extremal solution, In 3.3 we relate the nonconvex and the convexifie d evolution inclusions. In 3.4 we study the dependence of the solution set in all the data of the problem. In 3.5 we prove a parametrized ve rsion of the relaxation result which is done using a parametrized anal ogue of the ''Filippov-Gronwall'' inequality. In 3.6 we establish the path-connectedness of the solution set. In section 4, we focus our att ention to the optimal control of systems monitored by subdifferential evolution inclusions, In 4.1 we develop an existence theory. In 4.2 we study three different formulations of the relaxed problem and make co mparisons. In 4.3 we investigate the well-posedness of the optimal con trol problem. In 4.4 we compare the concepts of relaxability and well- posedness and show that under mild conditions on the data they are in fact equivalent. In section 5, we present several examples of systems monitored by p.d.e's which illustrate the applicability of our abstrac t results.