L-Catch Guaranteed Pursuit Time

We consider a simple motion differential game of one pursuer and one evader. The dynamic equation of the pursuer and evader is describe by first order and second order differential equation respectively. Control functions of the players are subject to integral constraints. We show that pursuit is completed in L-catch sense, that is for some distance L , the difference in positions of the pursuit and evader x(t) and y(t) respectively is smaller than L that is || y(t) - x(t) ||< L at some time t . We construct a formula for guaranteed pursuit time and prove that pursuit is possible at that time.