24. Inductive methods

Sometimes we suspect that we have identified a solution of an optimal control problem, but the deductive reasoning that would allow us to assert its optimality is unavailable. This might happen because no existence theorem applies, or because the applicability of necessary conditions is uncertain. In this situation, we may seek to use an inductive method to confirm the optimality of the suspect. We describe three such methods in this chapter, the first of which is based on a strengthening of the conditions that appear in the maximum principle. We show that, in the convex case of the problem (properly interpreted), the maximum principle (in normal form, and somewhat strengthened) is indeed a sufficient condition for optimality. The next method considered is that of verification functions, studied in detail in the calculus of variations. We show that it carries over to the optimal control setting. Finally, we illustrate in the last section how a uniqueness theorem for generalized solutions of the Hamilton-Jacobi equation leads to an inductive method allowing us to confirm conjectured optimality.