On the equivalence of some relaxations of optimal control problems on unbounded time domains

Abstract

Relaxation methods are a general concept for solving problems that lack convexity. There are several such methods, and we consider three of them: Γ-regularization by [3], Young measures by [4], and convex combinations by [2]. For bounded time domains, the comparisons are mostly done by Roubíček in [8], considering different generalizations of Young measures. We consider the relaxations for unbounded time domains and/or unbounded control sets. We establish sufficient conditions under which all these three types of relaxations are equivalent to each other. Furthermore, we give an example showing that in some cases the relaxations differ. The equivalence to the relaxation of the problem via convex combinations is convenient for computations. This type of formulation does not introduce any new mathematical objects such as Radon measures or bipolars, but rather involves no more that functions, derivatives, and so on. In the scenario where two problems (PR1), (PR2) are equivalent, one can establish the existence of an optimal solution for the first by proving the existence for the other, and vice versa. In the subject “Existence Theorem for Relaxed Control Problems on Infinite Time Horizon Utilizing Weight Functions” on the conference (FGS2024, Gijón), we present existence results for relaxed optimal control problems utilizing Young measures technique. In this manner, one can automatically derive existence results for other equivalent relaxations. In the following, we present only the proofs that are not contained in the cited works, or that need modification.