On Abnormal Optimal Control Problems with Mixed Equality and Inequality Constraints

In this paper an optimal control problem with mixed equality and inequality constraints is considered in the so-called abnormal case, i.e., in the case when the classical Pontryagin-type Maximum Principle has a degenerate form which does not depend on the minimized functional. Some extension of the Dubovitskii-Milyutin method to the case of nonregular operator Constraints obtained by using Avakov*s generalization of the Lusternik theorem is applied and the extension of the local Maximum Principle (LMP) for the optimal control problem with mixed equality and inequality constraints is proved. The extended version of LMP presented here has a nondegenerate form for abnormal optimal controls and reduces to the classical LMP under classical regularity conditions.