Regularity of multipliers for multiobjective optimal control problems governed by evolution equations

Abstract

This article is concerned with multiobjective optimal control problems, driven by evolution equations, and involving implicit control constraints and mixed pointwise control-state constraints in infinite dimensional separable Banach spaces. We consider bounded controls and inclusion-type mixed pointwise constraints, which are given in terms of measurable set-valued mappings whose images are closed convex sets with nonempty interior. In this case, the multiplier associated with mixed constraints is an element of the dual space of the Banach space-valued essentially bounded functions. Exploiting the combination of the Ioffe-Levin decomposition theorem for this dual and a Lagrange multiplier theorem obtained for an abstract multi-criteria optimization framework, we set up Fritz-John necessary optimality conditions, in the presence of integrable functions and singular measures as multipliers, for local weak Pareto solutions of the problems under investigation. Moreover, we give some conditions under which the multipliers are regular. An example of application of the main result is also provided to show the existence of a nontrivial singular measure.