A Game Problem on a Closed Convex Set

The movements of Pursuer $P$ and Evader $E$ in ${\mathbb R}^n$ are described by the equations $P:\,\dot{x}=a(t)u$ and $E:\,\dot{y}=a(t)v$, where $u$ and $v$ are control parameters of $P$ and $E$. A closed convex subset $S$ of ${\mathbb R}^n$ is given. The players $P$ and $E$ must not leave $S$. Integral restrictions are imposed on the controls of the players. For arbitrary initial locations $x_0,y_0\in S$ of the players, the optimal time of pursuit is found and optimal strategies for the players are constructed.