Necessary Conditions for Optimal Controls — The Pontryagin Maximum Principle

Abstract

In Chapter IV we described conditions which guarantee the existence of at least one optimal control — we call these sufficient conditions. However, these sufficient conditions are not very helpful in actually finding an optimal control. In this chapter we will describe one rather complicated set of conditions which any optimal control must necessarily satisfy. This set of necessary conditions is collectively known as the Pontryagin Maximum Principle (PMP). For many important problems, the conditions of the PMP will only be satisfied by a small subset of our control class (perhaps only by a single control). In this case there is a reasonable chance of our finding an optimal control if one exists. We emphasize that the PMP is a necessary set of conditions — there may be no optimal control, yet the PMP may delineate a nonempty class of candidates. To be sure that an optimal control actually exists, we must appeal to sufficiency theorems from Chapter IV.