Optimal location and shape of a rigid inclusion in a contact problem for inhomogeneous two-dimensional body

We analyze a well-known mathematical nonlinear model describing equilibrium of an elastic body with single volume (bulk) rigid inclusion. A possible frictionless contact of the body with a non-deformable obstacle by the Signorini condition on a part of the body boundary is assumed. On the remaining part of the boundary we impose a clamping condition. For a family of corresponding variational problems, we analyze the dependence of their solutions on location and shape of the rigid inclusion. External volume forces depend on the parameters defining location and shape of the inclusion. Continuous dependency of the solutions on location and shape parameters of the inclusion is established. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional on the Sobolev space of sought solutions, while the control is given by three real-valued parameters describing location and shape of the rigid inclusion.