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A pursuit problem in many person differential games. (Russian) Zbl 0662.90107
A general differential game for m pursuers and two evadors is described (the problem formulation follows L. S. Pontryagin); pursuers’ strategies have the form \(u_ i(t)=V_ i(t,v_ 1(t),v_ 2(t))\) where \(v_ 1(t)\) and \(v_ 2(t)\) are controls of the evadors. Capture of the j-th evador at time \(t_ j\) implies \(z_{ij}(t_ j)\in M_{ij}(t_ j)\) for some \(i\leq m\); here \(z_{ij}\) is a state of a system describing the interaction process for the i-th pursuer and the j-th evador, \(M_{ij}(t)=M^ 1_{ij}+M^ 2_{ij}(t)\), \(M^ 1_{ij}\) are linear subspaces, \(M^ 2_{ij}(t)\) are convex compact functions continuous in t. System equations linear in \(z_{ij}\) are considered. Sufficient algebraic conditions for two evadors to be captured within a finite time interval are given, and an upper bound for the capture time is deduced. For the case of three pursuers \(\dot x{}_ i=u_ i(t)\), \(\| u_ i(t)\| \leq 1\), and two evadors \(\dot y{}_ j=v_ j(t)\), \(\| v_ j(t)\| \leq 1\), it is shown that for any initial position three situations are possible: (a) ensured evasion, (b) capture of one evador, (c) capture of one prescribed evador.
Reviewer: A.V.Kryazhimskij

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