<p>The paper addresses the analysis of equilibrium problem for an elastic body with incorporated thin inclusion. The inclusion is assumed to be partially delaminated from the surrounding elastic body forming therefore a crack. We impose inequality type boundary conditions at the crack faces to prevent a mutual penetration. Assuming that the rigidity parameter is changing at the part of the inclusion we investigate a passage to the limit, and prove a solution existence of the limit problem describing an equilibrium of the elastic body with the inclusion consisting of elastic and rigid parts. For this limit model, an existence of a solution to optimal control problem with the cost functional describing the derivative of the energy functional with respect to the crack length is established. In this case, Lame parameters of the elastic body serve as control functions.</p>