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We constructed optimal strategies for players of a differential pursuit-evasion game in sequence space \(\ell_2\) under point-wise contraints. The players are one evader, and a countable number of pursuers who evolve under differential equations of first order for pursuers and of second order for the evader. At the end of a fixed time \(\theta\), the payoff is the distance between the evader and the closest pursuer. The pursuers aim to minimize the payoff while the evader seeks to maximize it. (with Mehdi Salimi)