Optimal control of multisolution ordinary differential equations in the absence of convexity

Abstract

In this paper, the authors study an optimal control problem governed by a class of multistate ordinary differential equations in the absence of convexity. To overcome the difficulty that the corresponding approximate optimal control problem may have no solution, relaxed controls are introduced. With the help of relaxation theory, Pontryagin’s maximum principle for the optimal pairs of the original control problem is obtained. In the end of this paper, the authors discuss the application of the maximum principle by an example.