The evasion problem in a nonlinear differential game with discrete control

A two-agent differential game is considered. The game is described by the following system of differential equations: $\dot x = f(x, v) + g(x, u),$ where $x \in \mathbb R^k$, $u \in U$, $v \in V$. The evader's admissible control set is a finite subset of phase space. The pursuer's admissible control set is a compact subset of phase space. The pursuer's purpose is to avoid an encounter, that is, to ensure a system position no closer than some neighborhood of zero. Sufficient conditions for avoidance of an encounter in the class of piecewise open-loop strategies on infinite and any finite-time intervals are obtained. The conditions are superimposed on the velocity vectogram at the zero point of phase space. When the game is considered on an infinite time interval, the conditions provide the evader with some advantage. The properties of a positive basis play a major role in proving the theorems.