Extension of the local maximum principle to abnormal optimal control problems

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.The author would like to thank Professors S. Walczak and W. Kotarski for fruitful discussions in the process of writing this paper.This research was supported by a SIUE Research Scholar Award and by NSF Grant DMS-91-009324.