High order necessary conditions for optimality.

*(English)*Zbl 0236.49005Summary: This is a survey paper. First, we review various methods of evaluating singular controls. An original proof of Kelley’s second order necessary conditions for optimality is given. High order optimality conditions are obtained by the method of Kopp and Moyer. The latter method is generalized to multidimensional singular controls (Goh’s result). A method of transformations in state space is presented (Kelley, Gurman). A method for investigating singular controls with the aid of a bundle of variations is described. A survey is made of the results which were obtained with the aid of a bundle of variations and matrix impulses in problems with closed control regions. The problem of joining extremals is singled out. The junction of singular and nonsingular extremals is considered. A survey of necessary conditions for optimality of nonsingular controls is presented. The cases of open and closed control regions are considered separately.