Control of the shape of a crack in an elastic body under the condition of possible contact of edges

Under consideration is a homogeneous three-dimensional body with a crack in the form of a smooth surface. We impose some inequality constraints on the crack edges that describe their mutual nonpenetration. According to the Griffith criterion, the crack begins to propagate when the derivative of the energy functional with respect to the virtual increment of the crack surface area reaches a certain critical value. The value of this derivative depends, in particular, on the crack shape. The crack shape is determined that minimizes the value of the derivative of the energy functional; more precisely, the existence of a solution to the corresponding optimal control problem is proved.