Differential games with many pursuers when evader moves on the surface of a cylinder

We study a pursuit differential game with many Pursuers when the Evader moves on the surface of a given cylinder. Maximal speeds of all players are equal. We consider two cases: in the first case, the Pursuers move arbitrarily without phase constraints; and in the second case, the Pursuers move on the surface of the cylinder. In both cases, we give necessary and sufficient conditions to complete the pursuit. In addition, in the second case, we show that pursuit differential game on a cylinder are equivalent to a differential game on the plane with many groups of Pursuers where each group consists of infinite number of pursuers having the same control parameter. References Isaacs, R. Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control, and Optimization. New York: Wiley, 1963. Petrov, N. N. A problem of group pursuit with phase constraints. J. Appl. Math. Mech. 1988, 52, No 6, 1030--1033. Petrosyan, L. A. Survival differential game with many participants. Dokl. Akad. Nauk USSR. 1965, 161, No 2, 285--287. Petrosyan, L. S. Differentsial'nye igry presledovaniya (Pursuit Differential Games). Leningrad (SPb): Leningr.State Univ(SPbSU), 1977. Pshennichnyi, B. N. Simple pursuit of several targets. Kibernetika. 1976, No 3, 145--146. Chernous'ko, F. L. A problem of evasion of several pursuers. J. Appl. Maths Mechs.1976, 40, No 1, 14--24. Ivanov R. P. Simple pursuit-evasion on a compact. Dokl. Akad. Nauk SSSR. 1980, 254, No 6, 1318--1321. Melikyan, A. A. and Ovakimyan, N. V. Singular trajectories in the problem of simple pursuit on a manifold. J. Appl. Math. Mech. 1991, 55, No 1, 42--48. Melikyan, A. A. and Ovakimyan, N. V. Differential Games of Simple Pursuit and Approach on Manifolds. Institute of Mechanics, National Academy of Sciences of Armenia. Yerevan. Preprint, 1993. Kuchkarov, A. Sh. The problem of optimal approach in locally euclidean spaces. Automation and Remote Control. 2007, 68, No 6, 974-978. Kuchkarov, A. Sh. A simple pursuit--evasion problem on a ball of a Riemannian manifold. Mathematical Notes. 2009, 85, No 2, 190--197. Azamov, A. On a problem of escape along a prescribed curve. J. Appl. Math. Mech. 1982, 46, No 4, 553--555. Kuchkarov, A. Sh. and Rikhsiev, B. B. on the solution of a pursuit problem with phase constraints. Automation and Remote Control. 2001, 62, No 8, 1259--1262. Ibragimov G. I. A game problem on a closed convex set. Siberian advances in mathematics., 2002, 12, No 3, 16--31. Nikulin, V. V. and Shafarevich, I. R. Geometriya i Gruppy (Geometric and Groups). Moscow: Nauka, 1983