Optimality conditions for feedback controls in problems of control under disturbance

Concerns the question whether it be possible to find infinitesimal conditions characterizing optimal feedback control in problems of control under disturbance and differential games which are analogous to Pontryagin minimum principle in optimal control. The positive answer to this question is given in terms of some minimum principles in integral and pointwise forms for optimal feedback. The derivation of pointwise minimum principle from integral one requires special technique based on the theorem on disintegration of measure. This technique which is especially suited for the consideration of nonanticipating strategies is applied to the construction of the duality theory for linear-convex differential games. We discuss here the application of the pointwise minimum principle to the solution of the long standing problem of optimality of some “natural” pursuit strategy in the game problem “Lion and Man” of Rado (1928)