Topological Derivatives for Optimal Control Problems

In this chapter, the topological derivative is obtained for the optimal value of the cost functional for a class of optimal control problems. We obtain the closed form of the topological derivative. Such an approach would allow to consider simultaneous structure design and control modifications. In order to introduce the topological derivative of the shape functional \(\varOmega \rightarrow J(\varOmega )\) we usually consider the mapping \(\varOmega \rightarrow y\) for the boundary value problem which gives the function y. In the case of optimal control problems the couple (u, y) of the control u and the state y is given by the optimality system.