We study a pursuit differential game problem for an infinite system of second-order differential equations. The control functions of players, i.e., a pursuer and an evader are subject to integral constraints. The pursuit is completed if z(τ) = \( \dot{z} \) (τ) = 0 at some τ > 0, where z(t) is the state of the system. The pursuer tries to complete the pursuit and the evader tries to avoid this. A sufficient condition is obtained for completing the pursuit in the differential game when the control recourse of the pursuer is greater than the control recourse of the evader. To construct the strategy of the pursuer, we assume that the instantaneous control used by the evader is known to the pursuer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Isaacs, Differential Games, Wiley, New York (1965).
L. S. Pontryagin, Collected Works, Nauka, Moscow (1988).
N. N. Krasovskii and A. I. Subbotin, Game-Theoretical Control Problems, Springer, New York (1988).
L. A. Petrosyan, Differential Games of Pursuit, St. Petersburg State Univ. (1993).
V. A. Il’in, “Boundary control of a string oscillating at one end with the other end fixed and under the condition of existence of finite energy,” Dokl. RAN, 378, No. 6, 743–747 (2001).
Yu. S. Osipov, “Positional control in parabolic systems,” Prikl. Mat. Mekh., 41, No. 2, 195–201 (1977).
S. G. Mikhlin, Linear Partial Differential Equations, Vyssh. Shkola, Moscow (1977).
A. G. Butkovskiy, Control Methods in Systems with Distributed Parameters, Nauka, Moscow (1975).
F. L. Chernous’ko, “Bounded controls in systems with distributed parameters,” Prikl. Mat. i Mekh., 56, No. 5, 810–826 (1992).
M. Tukhtasinov, “On some problems in the theory of differential pursuit games in systems with distributed parameters,” Prikl. Mat. Mekh., 59, No. 6, 979–984 (1995).
N. Yu. Satimov and M. Tukhtasinov, “On some game problems for the first-order controlled evolution equations,” Differents. Uravn., 41, No. 8, 1114–1121 (2005).
N. Yu. Satimov and M. Tukhtasinov, “On game problems for the second-order evolution equations,” Russ. Math. (Izv. Vuzov), 51, No. 1, 49–57 (2007).
G. I. Ibragimov, “A problem of optimal pursuit in systems with distributed parameters,” J. Appl. Math. Mech., 66, No. 5, 719–724 (2003).
M. Tukhtasinov and M. Sh. Mamatov, “On transfer problems in control systems,” Different. Equat., 45, No. 3, 439–444 (2009).
G. I. Ibragimov, “A damping problem for oscillation system in the presence of disturbance,” Uzbek. Math. J., 1, 34–45 (2005).
A. A. Chikrii and A. A. Belousov, “Game problem of “soft landing” for second-order systems,” J. Math. Sci., 139, No. 5, 6997–7012 (2006).
J. Albus, A. Meystel, A. A. Chikrii, A. A. Belousov, and A. I. Kozlov, “Analytic method for the solution of the game problem of soft landing for moving objects,” Cybernet. Syst., 37, Issue 1, 75–91 (2001).
N. N. Petrov and I. N. Shuravina, “On the “soft” capture in one group pursuit problem,” J. Comput. Systems Sci. Int., 48, No. 4, 521–526 (2009).
G. I. Ibragimov, A. Azamov, and R. M. Hasim, “Existence and uniqueness of the solution for an infinite system of differential equations,” J. KALAM. Int. J. Math. Statist., 2, 9–14 (2008).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 8, pp. 1080–1091, August, 2013.
Rights and permissions
About this article
Cite this article
Ibragimov, G., Allahabi, F. & Kuchkarov, A. A Pursuit Problem in an Infinite System of Second-Order Differential Equations. Ukr Math J 65, 1203–1216 (2014). https://doi.org/10.1007/s11253-014-0852-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0852-8