Optimal control of a bilateral contact with friction

We consider a mathematical model which describes a static frictional contact between a nonlinear elastic body and a rigid foundation. The contact is modelled with Tresca’s nonlocal friction law. We derive a variational formulation and prove its unique weak solvability under a certain condition on the friction coefficient. We state an optimal control problem which consists in leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. We prove that the optimal control admits a solution, and then we study a regularized control problem and establish a convergence result.