Relaxation in Nonconvex Optimal Control Problems Containing the Difference of Two Subdifferentials

Abstract

We consider the problem of minimization of an integral functional with a nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system with mixed nonconvex control constraint. The right-hand side of the system contains the difference of the subdifferentials of two proper convex, lower semicontinuous functions. Along with the original problem, we also consider a relaxed problem: the problem of minimizing the integral functional with convexified with respect to the control integrand over the solution set of the same system with the convexified control constraint. By a solution of the system we mean a “trajectory-control" pair. Under appropriate assumptions, we prove that the relaxed problem has an optimal solution and for each optimal solution there exists a minimizing sequence of the original problem converging to this solution. The converse statement is also true. An example of a control parabolic system is considered.