Optimization problems for elastic contact models with unilateral constraints

The aim of this paper is to provide some results in the study of an abstract optimization problem in reflexive Banach spaces and to illustrate their use in the analysis and control of static contact problems with elastic materials. We start with a simple model problem which describes the equilibrium of an elastic body in unilateral contact with a foundation. We derive a variational formulation of the model which is in the form of minimization problem for the stress field. Then we introduce the abstract optimization problem for which we prove existence, uniqueness and convergence results. The proofs are based on arguments of lower semicontinuity, monotonicity, convexity, compactness and Mosco convergence. Finally, we use these abstract results to deduce both the unique solvability of the contact model and the existence and the convergence of the optimal pairs for an associated optimal control problem.