Optimal control of a thin rigid stiffener for a model describing equilibrium of a Timoshenko plate with a crack

We consider a family of variational problems describing equilibrium of plates containing a crack and rigid thin stiffener on the outer boundary. Nonlinear conditions of the Signorini type on the crack faces are imposed. For this family of problems, we formulate an optimal problem with a control parameter determining the length of the thin rigid stiffener. Meanwhile, a cost functional is specified with the help of an arbitrary continuous functional in the solution space. The existence of the solution to the optimal control problem is proved. We state the continuous dependence of the solutions with respect to the stiffener's size parameter.