Optimality principles of dynamic programming in differential games (Q1823885)

A two-person zero-sum differential game for a system described by differential inclusions \(\dot x\in F_ 1(t,x)\), \(\dot y\in F_ 2(t,x)\), and phase constraints of the form \(x(t)\in N_ 1(t)\), \(y(t)\in N_ 2(t)\) is considered. The payoff of the game is given by a functional of the type \[ P=g(T_ M,x(T_ M),y(T_ M))+\int^{T_ M}_{t_ 0}h(t,x(t),y(t))dt, \] where \(T_ M\) is the first moment of time \(t>t_ 0\), when the triplet (t,x(t),y(t)) hits a prescribed terminal set. The game is formulated in a class of lower strategies which are introduced in the paper. Under some assumptions the author proves for each player the optimality principle, which can be considered a generalization of that for control problems originally formulated by P. Bellman.