JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 138, 43-51 (1989) Optimality Principles of Dynamic Programming in Differential Games LESZEK S. ZAREMBA Associate Editor, Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor. Michigan 48107 Submitted by Leonard D. Berkovitz Received May 7, 1987 1. INTRODUCTION A two-person zero-sum differential game is a dynamical system whose dynamics are described in the general case by a system of differential inclusions i.(t) EF,(t, x(t)), x(r*) = x*, XER”, (1.1) Jxt) EF,(c Y(f))7 Y(t*) = Y,, y E Rk, (1.2) with phase constraints of the form x(t) EN,(f), Y(l) EN,(f)? t> t,, (1.3) where N,( .), N,( .) are set-valued maps with closed graphs. By a solution of ( 1.1) (resp. ( 1.2)) we mean an absolutely continuous function x( ) (resp. y( . )) satisfying ( 1.1) (resp. ( 1.2)) almost everywhere. The game stops when, for the first time t > t, (henceforth abbreviated by T, = T, {t*, x,, y,; x( ), y( )}) the triplet (t, x(t), y(t)) hits a prescribed terminal set Mc R”+kf’. At time T, player I receives from player II the payoff = AT,, x(T,wM), Y(T,)) + lTM NC 4th y(f)) df. (1.4) ‘* Concerning the information available to both players during the course of the game, we assume they can employ any lower strategy; this concept will be introduced in this paper (Definition 2.1) and is more general than the notion of a lower n-strategy introduced in [4, p. 4001. Under Axioms (Al)-(A6) we prove the optimality principle of dynamic programming for 43 0022-241X/89 53.00 Copyright $: 1989 by Academic Press, Inc 411 rights of reproductmn ,” any lorm rcaerved.