The existence of value in differential games

Abstract

Communicated by James Serrin, October 21, 19711. Introduction. Two person zero-sum differential games can be considered as control problems with two opposing controllers or players. One player seeks to maximize and one to minimize the pay-off function. The greatest pay-off that the maximizing player can force is termed the lower value of the game and similarly the least value which the minimizing player can force is called the upper value. Our objective is to determine conditions under which these values coincide.In the case of two person zero-sum matrix games, von Neumann showed that if the players are allowed "mixed strategies," i.e., probability measures over the pure strategies, then the values of the game will coincide. By analogy, the authors in collaboration with L. Markus [1] introduced relaxed controls into differential game theory; for a full discussion of relaxed controls the reader is referred to [1].In this announcement, we define notions of strategy and values and relate these to the approaches adopted by Fleming [3], [4] and Friedman [5]. Using relaxed controls we are able to show that if the "Isaacs condition" (3) is satisfied then the upper and lower values are equal, so that the game has value. In particular, if the players are allowed relaxed controls then the game always has value. Detailed proofs of these results will appear in a later publication [2].

Some algorithms for differential games with two players and one target

“…We refer to [7] for the interprétation of Discy(AT) in term of Victor's discriminating victory set. These two sets are equal for KrassovskiSubbotin positional stratégies defined in [13] and for nonanticipative stratégies used in [9] (see also [21] and [8]). The goal of this paper is not to explain this interprétation but, knowing it, the main aim of this paper is to compute the set of initial conditions such that Victor can win by discriminating victory.…”

Cauchy problems for noncoercive Hamilton–Jacobi–Isaacs equations with discontinuous coefficients

“…This is a technical point, since, roughly, relaxation implies existence of minima in integral functionals under appropriate assumptions. The reader can consult Warga [24] for relaxed controls in optimal control, Elliott-Kalton [11] for the case of differential games and also [1]. Thus we consider the following control system:…”

“…The notion of causality is intriguing; it has been introduced in this form by Varaya, Roxin, Elliott and Kalton (see [11] and the references therein for the history of this and other notions), and roughly means that the player a at time T only knows its opponent's choices up to time T . Causality is the key idea needed to extend to differential games Bellman's Dynamic Programming Principle and the connection of value functions to Hamilton-Jacobi equations (see also [1]).…”

“…Therefore, by choosing atx an optimal strategy αx in (4.3), which exists by the compactness of ∆ and the continuity of u (see [11]), we have…”

Robust optimality of linear saturated control in uncertain linear network flows

“…The measurable control α ∈ A = {α : [0, +∞[→ A, measurable} is governed by the first player who wants to minimize the cost, whereas and the second player, by choosing the measurable control β ∈ B = {β : [0, +∞[→ B, measurable}, wants to maximize the cost. We define the non-anticipative strategies (see, e.g., [9]) for the first player…”