Optimal control of a fine structure

An optimal control problem for a multivalued system governed by a nonconvex variational problem, involving a regularization parameter >0, is proposed and studied. The solution to the variational problem exhibits typically rapid oscillations (a so-called fine structure) corresponding to a multiphase state of the material. We want to control only this fine structure. Existence of an optimal control is proved. Its convergence with 0 is studied by means of an optimal control problem for a relaxed variational problem involving (suitably generalized) Young measures. The uniqueness of the solution to the relaxed variational problem, which is nontrivial but is very important in the context of optimal control, is studied in special cases. A finite-element approximation is proposed.The second author gratefully acknowledges support for this research by the Alexander von Humboldt Foundation during his stay at the Institute for Mathematics of the University of Augsburg.