Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

Abstract

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm–Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad–Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm–Liouville operator.