First-order conditions for the optimal control of the obstacle problem with state constraints

TL;DR

This paper investigates the first-order optimality conditions for a control problem governed by an obstacle problem with additional state constraints, establishing stationarity conditions under various constraint scenarios.

Contribution

It provides a rigorous analysis of necessary optimality conditions for obstacle control problems with state and control constraints, including regularization techniques and stationarity concepts.

Findings

C-stationarity is necessary under control constraints.

Local minimizers are strongly stationary without control constraints.

The paper develops primal first-order conditions for the problem.

Abstract

We consider an optimal control problem in which the state is governed by an unilateral obstacle problem (with obstacle from below) and restricted by a pointwise state constraint (from above). In the presence of control constraints, we prove, via regularization of the state constraints, that a system of C-stationarity is necessary for optimality. In the absence of control constraints, we show that local minimizers are even strongly stationary by a careful discussion of the primal first-order conditions of B-stationary type.