Optimal control of the shape of a layer shape in the equilibrium problem of elastic bodies with overlapping domains

We consider the equilibrium problem for a two-layer elastic body. One of the plates contains a crack. The second is a disk centered at one of the crack tips. The spherical layer is glued by its edge to the first plate. The unique solvability is proved of the problem in the nonlinear setting. An optimal control problem is also considered. The radius of the second layer $a$ is chosen as the control function. It is assumed that $a$ is positive and takes values in a closed interval. We show that there exist a value of $a$ minimizing the functional that characterizes the change of the potential energy as the crack length increases and a value of $a$ that characterizes the opening of the crack.