Time-dependent subdifferential evolution inclusions and optimal control


The purpose of this paper is to study from many different viewpoints evolution inclusions and optimal control problems involving time dependent subdifferential operators. Throughout this work we take a special interest in the t-dependence of the functional φ(t,x), involved in the subdifferential. We employ a condition that allows the domain domφ(t, ·) to vary regularly without precluding the possibility that domφ(t, ·) ∩ dom(s, ·) = ∅ for t ≠ s. Hence our formulation is general enough to incorporate problems with time varying constraints (obstacles). In section 3, we deal with evolution inclusions. In § 3.1 we prove two existence theorems; one for a nonconvex valued orientor field F and the other for a convex valued one. In § 3.2 we look for extremal solution. In § 3.3 we relate the nonconvex and the convexified 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 version of the relaxation result which is done using a parametrized analogue of the "Filippov-Gronwall" inequality. In § 3.6 we establish the path-connectedness of the solution set. In section 4, we focus our attention 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 comparisons. In § 4.3 we investigate the well-posedness of the optimal control 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 abstract results.

Document Type





Differential inclusions, Extremal solutions, Multivalued mappings, Optimal control, Relaxability, Subdifferentials, Variational inequalities

Publication Date


Journal Title

Memoirs of the American Mathematical Society