Necessary Optimality Conditions for Optimal Control Problems with Equilibrium Constraints

TL;DR

This paper develops optimality conditions for control problems with equilibrium constraints, providing a theoretical framework that extends classical optimal control principles to more complex, complementarity-based models.

Contribution

It introduces new stationarity concepts and sufficient conditions for optimality in control problems with equilibrium constraints, expanding the theoretical understanding of OCPECs.

Findings

Defined weak, Clarke, Mordukhovich, and strong stationarities for OCPECs.

Provided conditions ensuring local minimizers are stationary.

Identified circumstances under which multipliers for complementarity constraints coincide.

Abstract

This paper introduces and studies the optimal control problem with equilibrium constraints (OCPEC). The OCPEC is an optimal control problem with a mixed state and control equilibrium constraint formulated as a complementarity constraint and it can be seen as a dynamic mathematical program with equilibrium constraints. It provides a powerful modeling paradigm for many practical problems such as bilevel optimal control problems and dynamic principal-agent problems. In this paper, we propose weak, Clarke, Mordukhovich and strong stationarities for the OCPEC. Moreover, we give some sufficient conditions to ensure that the local minimizers of the OCPEC are Fritz John type weakly stationary, Mordukhovich stationary and strongly stationary, respectively. Unlike Pontryagain's maximum principle for the classical optimal control problem with equality and inequality constraints, a counter example shows that for general OCPECs, there may exist two sets of multipliers for the complementarity constraints. A condition under which these two sets of multipliers coincide is given.