A general existence theorem for a multi-valued control problem (Q6155049)

In this paper, a general existence theorem for a multi-valued control problem is proposed, with the proof being based on a decomposition result of the weak\(^{*}\) closure of the set containing all the multi-valued controls and the application of the bathtub principle. The optimality condition for the optimal control is also obtained. A decomposition result of the weak\(^{*}\) closure of the admissible set \(A_{n}\) and the bathtub principle are utilized. The known example is revisited and demonstrate to which the Theorem is applied. A surprising consequence of the Theorem is that the multi-valued control problem, where the impression had been held by the authors that the most efficient way to prove the existence and uniqueness of the optimal solution is derived from the classical rearrangement optimization theory, can now be obtained by multiple applications of the bathtub principle.