A Local Second- and Third-order Maximum Principle

In this note we announce preliminary results obtained by applying the methodology of Sussmann [4, 5, 6, 7, 8] to the problem studied by Ledzewicz and Schättler (cf. [1, 2]), of deriving high-order necessssary conditions for a minimum in optimal control theory, extending the classical Pontryagin Maximum Principle (abbr. MP) of [3]. The work of [1, 2] uses high-order generalizations of the theorems of Lyusternik and Avakov. We pursue a different approach, constructing needle variations and, at the crucial point where a “topological argument” is needed, applying the Brouwer fixed point theorem. The result is a high-order version of the MP that contains and extends the results of [1, 2]. In particular, our version makes it clear that new multipliers occur for the first time for the third-order principle. It turns out that what is called a “second-order MP” in [1, 2] is naturally a “third-order MP” in our setting. Our third-order MP contains new multipliers exactly as the result of [2] does, but in addition it also contains the classical multipliers, to which the new multipliers are coupled in a precise way, described in Theorem 6.1. We present an abstract necessary condition for set separation (Theorem 4.1), a necessary condition for a minimum in an abstract setting (Theorem 5.1), and, finally, the optimal control result (Theorem 6.1). We also present an example showing that our results apply in cases where those of [1, 2] do not. For lack of space, we omit the proofs, we limit our discussion to problems with fixed endpoints, and we only consider the second and third-order cases.