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The High Order Maximal Principle and Its Application to Singular Extremals

The high order maximal principle (HMP) which was announced in [11] is a generalization of the familiar Pontryagin maximal principle. By using the higher derivatives of a large class of control variations, one is able to construct new necessary conditions for optimal control problems with or without terminal constraints. In particular, we show how the HMP can be used to prove the generalized Legendre–Clebsch condition of Kelley, Kopp, Moyer.and Goh. The principle advantage of this derivation is that, unlike previous ones, it remains valid even when there are terminal constraints.