A Generalized Filippov-like Existence Theorem for Optimal Control Problems with Constraints

Abstract

As is known, an optimal control problem may not have a solution. A.F. Filippov in [1] obtained his well-known theorem under the assumption of the convexity of the velocity set. Further, this convexity-based approach was significantly improved in the work of R.V. Gamkrelidze and J. Warga, see in [2,3], while a more general existence theorem was proposed for which, by introducing the so-called generalized controls and the concept of convexification of the problem, the existence of a solution in an extended control problem was asserted. In this paper, such an approach is developed on discontinuous arcs. More precisely, our work combines the two approaches – the one based on the Lebesgue discontinuous time variable change, and the other, based on the convexification of the optimal control problem by virtue of the generalized controls proposed by Gamkrelidze. This leads to a general impulsive extension of the optimal control problem based on the concept of generalized impulsive control. A generalized Filippov-like existence theorem for a solution is obtained. Within the framework of the proposed approach, a few classic examples taken from the calculus of variations are examined, in which discontinuities of optimal arcs inevitably arise.