Optimal location of a rigid inclusion for an equilibrium problem describing Kirchhoff–Love plate with nonpenetration conditions for known configurations of plate edges

A nonlinear model describing equilibrium of a cracked plate with a volume rigid inclusion is studied. It is assumed that under the action of certain given loads, plates have deformations with a certain predetermined configuration of edges near the crack. On the crack curve we impose a nonlinear boundary condition as a system of inequalities and an equality describing the nonpenetration of the opposite crack faces. For a family of variational problems, we study the dependence of their solutions on the location of the inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on a suitable Sobolev space. For this problem, the location parameter of the inclusion serves as a control parameter. We prove continuous dependence of the solutions on the location parameter and the existence of a solution to the optimal control problem.