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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 Pontryagains 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.
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 derive
In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential f
In this paper, we study the mathematical program with equilibrium constraints (MPEC) formulated as a mathematical program with a parametric generalized equation involving the regular normal cone. We derive a new necessary optimality condition which i
The bilevel program is an optimization problem where the constraint involves solutions to a parametric optimization problem. It is well-known that the value function reformulation provides an equivalent single-level optimization problem but it result
This article treats optimal sparse control problems with multiple constraints defined at intermediate points of the time domain. For such problems with intermediate constraints, we first establish a new Pontryagin maximum principle that provides firs