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Possibility of encounter avoidance in a critical case. (English. Russian original) Zbl 0599.90146
Differ. Equations 20, 1451-1455 (1984); translation from Differ. Uravn. 20, No. 12, 2061-2066 (1984).
The differential game of pursuit-evasion is modelled by the p-order equation $$z^{(p)}+\sum^{p}_{i=1}a_ iz^{(p-i)}=\rho (-u+v)$$, where $$z\in {\mathbb{R}}^{\nu}$$, $$\nu\geq 2$$, $$p\geq 1$$, $$a_ i$$, $$\rho \in {\mathbb{R}}^ 1$$, $$\rho >0$$ and u, v are pursuer and evader controls, respectively, $$v\in S$$, the closed unit ball centred at $$0\in {\mathbb{R}}^{\nu}$$. The game terminates if $$z(\tau)=0$$ for some $$\tau\geq 0$$. Sufficient conditions for avoidance of the antitarget $$z=0$$ are introduced in quite general terms. It is proved that $$\nu\geq p$$ and the fact that u(t) is piecewise continuous secure the avoidance.

##### MSC:
 91A24 Positional games (pursuit and evasion, etc.) 91A23 Differential games (aspects of game theory)
##### Keywords:
Sufficient conditions for avoidance