×

Method of construction of the evasion strategy for differential games with many pursuers. (English) Zbl 0597.90106

The research described, it is the author’s dissertation, deals with evasion strategies in differential games if there are many, but a finite number of pursuers. The theory of pursuit and evasion is a difficult subject and the author refers to some of the existing literature in the introduction. The paper is rather mathematical; it is concisely written. Some more background and explanations would have been helpful to the reviewer as to why certain technical definitions and hypotheses are introduced. The central theme is a constructive method of evasion, which can be applied to many (also existing) examples. Evasion (winning the game) essentially means that the evader can maintain a positive (but otherwise arbitrarily small) distance from all pursuers. The author distinguishes three different winning situations; the evader can i) win in the game, ii) win locally in the game, and iii) win locally along each trajectory. In the literature several definitions of strategies exist and the author adds his own, which has some resemblence with the definitions of B. N. Pshenichnyj [Topics Diff. Games, 45-99 (1973; Zbl 0325.90069)]. It is not assumed that the evader knows the current decision of the pursuer(s) (as is often the case with respect to other definitions), rather he knows the current position (or state) of the pursuer. The definition of strategies of evader and pursuer(s) are not symmetric.
The paper starts with an introduction and then continues with preliminaries. Here control systems for both the evader and pursuer(s) are introduced; each player has his own dynamics, interconnections are not considered apart from a final remark at the end of the paper. The control systems introduced are by definition stationary (i.e. time does not appear explicitly) and satisfy the semi-group property. For all admissible strategies, trajectories resulting from a control system must be equicontinuous. In section 2 the main lemma is given; it deals with one evader and one pursuer. Section 3 deals with the avoidance of many pursuers. Section 4 introduces the concept of ”evasion along each trajectory” in which only small perturbations along existing trajectories are considered. Section 4 also gives some examples (among which the homicidal chauffeur) with respect to this concept. Section 5 briefly discusses differential games with many evaders and pursuers, an example of which is ”A squadron of ships, moving in a given formation, meeting a group of icebergs on its way. The task of the squadron is to pass through the dangerous area without changing the formation”. Mathematically this task can be realized. Section 6 deals with incomplete information; the evader can only approximately determine the position of the pursuer. Lastly, section 7 gives some final remarks.
Reviewer: G.J.Olsder

MSC:

91A24 Positional games (pursuit and evasion, etc.)
91A23 Differential games (aspects of game theory)
91A99 Game theory

Citations:

Zbl 0325.90069