Optimal radius of a rigid cylindrical inclusion in nonhomogeneous plates with a crack

We consider equilibrium problems for a cracked inhomogeneous plate with a rigid circular inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that reaches the boundary of the rigid inclusion. The boundary condition on the crack curve is given in the form of inequality and describes the mutual nonpenetration of the crack faces. For a family of corresponding variational problems, we analyze the dependence of their solutions on the radius of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the radius of the cylindrical inclusion is chosen as the control parameter. Existence of a solution to the optimal control problem and continuous dependence of the solutions with respect to the radius of the rigid inclusion are proved.