Relaxation in non-convex optimal control problems described by first-order evolution equations

The problem is considered of minimizing an integral functional with integrand that is not convex in the control, on solutions of a control system described by a first-order non-linear evolution equation with mixed non-convex constraints on the control. A relaxation problem is treated along with the original problem. Under appropriate assumptions it is proved that the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence for the original problem that converges to the optimal solution. Moreover, in the appropriate topologies the convergence is uniform simultaneously for the trajectory, the control, and the functional. The converse also holds. An example of a non-linear parabolic control system is treated in detail.