Necessary optimality conditions for a new type of singular controls in optimal control problems

The paper is devoted to the study of the optimality of the Pontryagin extremal in the Mayer-type optimal control problem. To do this, a new type of singular control is introduced. The research method in this paper is based on the introduction of special variations of the considered singular control. To obtain more subtle equality and inequality type necessary optimality conditions, the variation introduced here is used in two forms: the variation on the right and left with respect to the fixed point. The necessary optimality conditions obtained in this paper also depend on the numerical parameter. Next, applying one of the necessary optimality conditions obtained here, the second variation of the objective functional is studied, and the well-known Kelley necessary condition is obtained. The effectiveness of the main results of this work is illustrated by specific examples.