Differential games of pursuit in Banach space

We are given two entities: P the pursuer and E the evader. Each can choose a control function (or a strategy) from a certain set allotted to it. The situation is such that for any strategy v that E chooses, P can choose a strategy u which will bring him after some time, say Z’,,, , into a W-neighborhood of E, W being a given convex body. P would like to choose u so as to minimize T WV * E, on the other hand, would like to choose a strategy v = f? in such a way that when P, accordingly, chooses u = u” so as to minimize TUB , the time T,, is the largest, i.e.,