Optimal control and approximation for elliptic bilateral obstacle problems

Abstract

The aim of this paper is to study an elliptic bilateral obstacle system (EBOS, for short) involving a nonlinear and nonhomogeneous partial differential operator and a multivalued term which is described by Clarke’s generalized gradient. First, we obtain the weak formulation of (EBOS) which is a variational-hemivariational inequality, and prove the unique solvability of the bilateral obstacle problem. Second, we consider a nonlinear optimal control problem governed by (EBOS) in which the control variable is the bilateral obstacle, and establish the existence of an optimal solution to the obstacle control problem under mild conditions. Then, employing the regularization technique and penalty approach, we introduce a family of approximating problems corresponding to the optimal control problem under consideration. Finally, a convergence result that any sequence of solutions for the approximating problems converges to an optimal solution of the original optimal control problem is delivered.