Bogolyubov-Type Theorem with Constraints Generated by a Fractional Control System

Abstract

Abstract We first study the existence results and properties of the solution set of a control system described by fractional differential equations with nonconvex control constraint. Then a problem of minimizing an integral functional over the solution set of the control system is considered. Along with the original minimizing problem, we also consider the problem of minimizing the integral functional whose integrand is the bipolar (with respect to the control variable) of the original integrand over the solution set of the same system but with the convexified control constraint. We prove that the relaxed problem has an optimal solution and obtain some relationships between these two minimizing problems. Finally, an example is given to illustrate the results.