Optimal control of a frictionless contact problem with normal compliance in isotropic incompressible linear elasticity

We consider a mathematical model which describes a frictionless contact with normal compliance between an incompressible linearly elastic isotropic body and a deformable foundation. We prove the existence of a unique weak solution of the problem. We address an optimal control problem for which we prove the existence of a solution. Finally, we study a penalized and regularized control problem and prove a convergence result.