Abstract
This paper is concerned with the optimality system for an optimal control problem governed by parabolic bilateral obstacle problems with Robin boundary conditions. The control-state dynamics is described by variational inequalities and the control on the boundary is considered. Our analysis relies on a regularization method of approximating the variational inequality by a series of semilinear PDEs. The Lipschitz continuity and directionally differentiability in the weak sense for the control-to-state mapping are proved. Then, the necessary optimality conditions for approximate optimal control problems are derived. Finally, the first-order optimality system including complementarity conditions for the original problem is established by a limiting procedure. The result is carefully compared with those obtained in existing literature.
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