Strong solutions and optimal control for Duffing differential inclusions with Dirichlet conditions

Existence of strong solutions for a Dirichlet problem driven by a Duffing-type differential inclusion is achieved in the setting of separable reflexive Banach spaces. This general framework makes it possible to model and analyse complex phenomena such as periodic or chaotic dynamics as well as oscillatory behavior. The method used to obtain the existence results is based on the combination of a fixed point theorem and a selection theorem. By using the weak topology we avoid assumptions of compactness on the multivalued term. Our abstract results allow to deduce the existence of an admissible pair for a control problem driven by a Duffing-type differential equation. The paper ends with the study of optimal control problem involving a suitable functional.