Extension of the local maximum principle to abnormal optimal control problems

Abstract

In this paper, an optimal control problem with terminal data is considered in the so-called abnormal case, i.e., when the classical Pontryagin-type maximum principle has a degenerate form which does not depend on the functional being minimized. An extension of the Dubovitskii-Milyutin method to the nonregular case, obtained by applying Avakov's generalization of the Lusternik theorem, is presented. By using this extension, a local maximum principle which has a nondegenerate form also in the abnormal case is proved. An example which supports the theory is given.