Relaxation in an Optimal Control Problem Described by a Coupled System with Maximal Monotone Operators
摘要
We study the minimization problem for an integral functionalon the solutions to a coupled system.The system consists of an evolutionary inclusionin a separable Hilbert space with maximal monotone operatorsand an ordinary differential equationin a separable Banach space containing the control.The control constraint isa 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 functionalwith the integrand convexified with respect to the controlon 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 systemand for the values of the convexified functionalon the solutions to the convexified system,we consider questions of approximationby the solutions to the original systemand the values of the original functionalon the solutions to the original system(a relaxation theorem).We prove that an optimal control exists in the relaxed system.