Necessary and sufficient conditions of minimum in optimal control problems with sliding modes and generalized controls (Q759994)

Some necessary and sufficient conditions for optimality of sliding modes and generalized controls (measures) in Mayer-type problems for general nonlinear control systems are studied. The optimality is understood in the sense of a weak minimum and in the sense of a non-local minimum specific for sliding modes. There are no assumptions concerning the number of basic controls which are Lipschitzian functions while the coefficients are measurable functions; the optimal sliding regime is assumed to be singular. By using some earlier results of Russian mathematicians the necessary and sufficient conditions for optimality are obtained in abstract form and next in analytical form and are illustrated by two examples. The sufficient conditions are also sufficient for minimum problems in the class of generalized control measures.