Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.