Necessary optimality conditions for optimal control problems with nonsmooth mixed state and control constraints

TL;DR

This paper develops new necessary optimality conditions for control problems with nonsmooth mixed constraints, relaxing smoothness assumptions and using advanced constraint qualification concepts.

Contribution

It introduces necessary optimality conditions under pseudo-Lipschitz and calmness assumptions, extending existing results to nonsmooth constraint scenarios.

Findings

Derived stratified optimality conditions for nonsmooth constraints

Established conditions under pseudo-Lipschitz continuity and calmness

Provided explicit multiplier form Euler inclusion for specific cases

Abstract

In this paper we study an optimal control problem with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality condition for optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition and under the assumption that the state and control constraint functions are smooth. In this paper we derive necessary optimality conditions for problems with nonsmooth mixed state and control constraints under constraint qualifications based on pseudo-Lipschitz continuity and calmness of certain set-valued maps. The necessary conditions are stratified, in the sense that they are asserted on precisely the domain upon which the hypotheses (and the optimality) are assumed to hold. Moreover necessary optimality conditions with an Euler inclusion taking an explicit multiplier form are derived for certain cases.