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A linear two player zero-sum pursuit-evasion differential game is considered. The control functions of players are subject to integral constraints. In the game, the first player, the Pursuer, tries to force the state of the system towards the origin, while the aim of the second player, the Evader, is the opposite. We construct the optimal strategies of the players when the control resource of the Pursuer is greater than that of the Evader. The case where the control resources of the Pursuer are less than or equal to that of the Evader is studied to prove the main theorem. For this case a new method for solving of the evasion problem is proposed. We assume that the instantaneous control employed by the Evader is known to the Pursuer. For construction, the strategy of the Evader information about the state of the system and the control resources of the players is used.   References R. Isaacs. Differential games. John Wiley and Sons, New York, 1965. L. S. Pontryagin. Collected works. Nauka, Moscow, 1988. (Russian) L. D. Berkovitz. Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM Journal on Control , 5 , 1--24, 1967. L. D. Berkovitz. A survey of differential games. Mathematical Theory of Control , Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, 373--385, 1967. N. N. Krasovskii and A. I. Subbotin. Game-theoretical control problems. New York, Springer, 1988. W. H. Fleming. The convergence problem for differential games. Journal of Mathematical Analysis and Applications . 3 , 102--116, 1961. W. H. Fleming. The convergence problem for differential games, Part 2. Advances in Game Theory, Annals of Mathematics Studies , (52), Princeton University Press, Princeton, New Jersey,195--210, 1964. A. Friedman. Differential games. Wiley-Interscience, New York, 1971. R. J. Elliott and N. J. Kalton. The existence of value in differential games. Memoirs of the American Mathematical Society , 126 , 1--67, 1972. L. A. Petrosyan. Differential games of pursuit. World Scientific, Singapore, London, 1993. O. Hajek. Pursuit games. Academic Press, New York, San Francisco, 1975. A. Ya. Azimov. Linear differential pursuit game with integral constraints on the control. Differentsial'nye Uravneniya, 11 (10), 1975, 1723--1731; English transl. in Differential Equations 11 , 1283--1289, 1975. A. Ya. Azimov. A linear differential evasion game with integral constraints on the controls. USSR Computational Mathematics and Mathematical Physics, 14 (6), 56--65, 1974. M. S. Nikolskii. The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, (2), 49--59, 1969. A. I. Subbotin and V. N. Ushakov. Alternative for an encounter-evasion differential game with integral constraints on the playersi controls. PMM 39 (3), 387--396, 1975. V. N. Ushakov. Extremal strategies in differential games with integral constraints. PMM , 36 (1), 15--23, 1972. B. N. Pshenichnii and Yu. N. Onopchuk. Linear differential games with integral constraints. Izvestige Akademii Nauk SSSR, Tekhnicheskaya Kibernetika , (1), 13--22, 1968. A. A. Azamov, B. Samatov. $\pi $-strategy. An elementary introduction to the theory of differential games. National University of Uzbekistan. Tashkent, Uzbekistan, 2000. G. I. Ibragimov. A game problem on a closed convex set. Siberian Advances in Mathematics . 12 (3), 16--31, 2002. G. I. Ibragimov. A problem of optimal pursuit in systems with distributed parameters. J. Appl. Math. Mech, 66 (5), 719--724, 2003. E. B. Lee and and L. Markus. Foundations of optimal control theory, John Wiley and Sons Inc., New York, 1967.

Solution of a linear pursuit-evasion game with integral constraints