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Relaxation in nonconvex optimal control problems governed by evolution inclusions with the difference of two Clarke's subdifferentials

This paper considers the minimisation problem of an integral functional with an integrand and a mixed nonconvex control constraint under some suitable conditions. We minimise the integral functional on the solution set of a control system described by nonlinear evolutionary inclusions with the mixed nonconvex control constraint, in which the left-hand side of the system contains the difference of two Clarke's Subdifferentials. We discuss a relaxation problem along with the original problem and show that the relaxation problem has an optimal solution, and for every optimal solution, there exists a minimising sequence of the original problem that converges to this solution concerning the trajectory, the control, and the functional.