A unified theory of deterministic two-players zero-sum differential games

Abstract

This paper is a shorter presentation of "Generalization of the optimality theory of Pontryagin to deterministic two-players zero-sum differential games" [MARCHAL, 1973] presented at the fifth IFIP conference on optimization techniques.

The very notions of zero-sum game and deterministic game are discussed in the first sections. The only interesting case is the case when there is "complete and infinitely rapid information". When the minimax assumption is not satisfied it is necessary to define 3 types of games according to ratios between time-constant of a chattering between two or several controls and delays necessary to measure adverse control and to react to that control ; it thus emphasizes the meaning of the "complete and infinitely rapid information" concept.

In the last sections the optimality theory of Pontryagin is generalized to deterministic two-players zero-sum differential games ; it leads to the notion of extremal pencil (or bundle) of trajectories.

When some canonicity conditions generalizing that of Pontryagin are satisfied the equations describing the extremal pencils are very simple but lead to many kinds of singularities already found empirically in some simple examples and called barrier, universal surfaces, dispersal surfaces, focal lines, equivocal lines etc...