Abstract

Abstract. This paper introduces and studies the optimal control problem with equilib-rium 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 (MPEC). 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 Fritz John type 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. Key Words. Optimal control problem with equilibrium constraints, necessary optimality condition, weak stationarity, Mordukhovich stationarity, strong stationarity.