Continuous dependence results for a class of evolution inclusions

Abstract

In this paper we examine the dependence of the solutions of an evolution inclusion on a parameter X. We prove two dependence theorems. In the first the parameter appears only in the orientor field and we show that the solution set depends continuously on it for both the Vietoris and Hausdorff topologies. In the second the parameter appears also in the monotone operator. Using the notion of G-convergence of operators we prove that the solution set is upper semicontinuous with respect to the parameter. Both results make use of a general existence theorem which we also prove in this paper. Finally, we present two examples. One from control theory and the other from partial differential inclusions.

On the “bang-bang” principle for nonlinear evolution inclusions

“…In [11], under generally mild hypotheses on the data, we proved that S(xo)¢ C5. Here we turn our attention to the solution set S~(xo) (extremal solutions).…”

“…We know (see the proof of theorem 3.1 in [11]), that p(-) is sequentially continuous from Lq(H) equipped with the weak topology into C(T, H). Then combining this with the L~ (H)-continuity of g(-) and Lemma 2.1, we get that Vg(.…”

On an existence result for nonlinear evolution inclusions

“…In [5] the present author has delivered an example which shows that Theorem 2 of Nagy [6] is false. For this reason, the proof of existence of solutions to (1) can not arise directly from the reasoning given in [10].…”

“…We follow the procedure that closely resembles the one of Papageorgiou [10]. A crucial point in our approach is the fact that the solution map of the associated evolution equation is continuous in suitable topologies (see Proposition 1).…”

Infinite dimensional parametric optimal control problems

“…For details we refer to Pieri [24], Stassinopoulos-Vinter [29] and Zolezzi [33] who considered finite dimensional control systems and to Benatti [4], Carja [7], Papageorgiou [19], [20], [21], [22] and Przyluski [25], who examined inŸ dimensional control systems. In particular, Benatti [4], Carja [7], Papageorgiou [20], [21] and Przyluski [25] investigated linear systems, using rather strong hypotheses on the data and their modes of convergence, while Papageorgiou [19], [22], studied 1 Revised version. 2 PartiaUy supported by the National Science Foundation under grant DMS-91-11794.…”