Relaxation in an optimal control problem described by a coupled system with maximal monotone operators

We study the minimization problem for an integral functional on the solutions to a coupled system. The system consists of an evolutionary inclusion in a separable Hilbert space with maximal monotone operators and an ordinary differential equation in a separable Banach space containing the control. The control constraint is a multivalued mapping with closed nonconvex values, while the integrand is a nonconvex function of the control. Along with the original problem, we consider the minimization problem for an integral functional with the integrand convexified with respect to the control on the solutions to the system with convexified control constraint (a relaxation problem). An existence theorem for solutions to our systems is established. Both for the solutions to the convexified system and for the values of the convexified functional on the solutions to the convexified system, we consider questions of approximation by the solutions to the original system and the values of the original functional on the solutions to the original system (a relaxation theorem). We prove that an optimal control exists in the relaxed system.