Optimal control of a frictional contact problem with unilateral constraints

We consider a mathematical model that describes a static contact with a nonlinear elastic body and a foundation. The contact boundary is composed of two measurable parts. In one part, the contact is frictionless with Signorini's conditions. In the other part, the normal stress is given and associated with Coulomb's friction law. We state an optimal control problem that consists of leading the stress tensor as close as possible to a given target by acting with a control on the boundary. Then, we study the penalized and regularized control problem for which we establish a convergence result.