# American Institute of Mathematical Sciences

March  2017, 7(1): 89-94. doi: 10.3934/naco.2017005

## The soft landing problem for an infinite system of second order differential equations

 1 Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Malaysia 2 Department of Informatics, Tashkent University of Information Technologies, Tashkent, Uzbekistan

* Corresponding author: Gafurjan Ibragimov

Received  January 2016 Published  February 2017

We study a soft landing differential game problem for an infinite system of second order differential equations. Control functions of pursuer and evader are subject to integral constraints. The pursuer tries to obtain equations $z(τ)=0$ and $\dot z(τ)=0$ at some time $τ > 0$ and the purpose of the evader is opposite. We obtain a condition under which soft landing problem is not solvable.

Citation: Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005
##### References:
 [1] J. Albus, A. Meystel, A. A. Chikrii, A. A. Belousov and A. I. Kozlov, Analytical method for solution of the game problem of soft landing for moving objects, Cybernetics and Systems, 37 (2001), 75-91.  doi: 10.1023/A:1016620201241. [2] A. A. Chikrii and A. A. Belousov, Game problem of "soft landing" for second-order systems, Journal of Mathematical Sciences, 139 (2006), 6997-7012.  doi: 10.1007/s10958-006-0402-5. [3] K. G. Guseinov, A. A. Neznakhin and V. N. Ushakov, Approximate construction of attainability sets of control systems with integral constraints on the controls, Journal of Applied Mathematics and Mechanics, 63 (1999), 557-567.  doi: 10.1016/S0021-8928(99)00070-2. [4] G. I. Ibragimov, An optimal pursuit problem in systems with distributed parameters, Cybernetics and Prikladnaya Matematika i Mekhanika, 66 (2002), 753-759.  doi: 10.1016/S0021-8928(02)90002-X. [5] G. I. Ibragimov, F. Allahabi and A. Sh. Kuchkarov, pursuit problem in an infinite system of second-order differential equations, Ukrainian Mathematical Journal, 65 (2014), 1203-1216.  doi: 10.1007/s11253-014-0852-8. [6] G. I. Ibragimov, A problem of damping of oscillation system in presence of disturbance, Uzbek Math. Journal, Tashkent, 1 (2005), 34-45. [7] G. I. Ibragimov, The optimal pursuit problem reduced to an infinite system of differential equations, J. Appl Math Mech, 77 (2013), 470-476.  doi: 10.1016/j.jappmathmech.2013.12.002. [8] G. I. Ibragimov, Optimal pursuit time for a differential game in the hilbert space, Science Asia, 39S (2013), 25-30. [9] G. I. Ibragimov, M. Tukhtasinov, R. M. Hasim and I. A. Alias, A Pursuit problem described by infinite system of differential equations with coordinate-wise integral constraints on control functions, MJMS, 9 (2015), 67-76. [10] G. I. Ibragimov, On the optimal pursuit game of several pursuers and one evader, Prikladnaya Matematika I Mekhanika, 62 (1998), 199-205.  doi: 10.1016/S0021-8928(98)00024-0. [11] G. I. Ibragimov, A. Azamov and M. Khakestari, Solution of a linear pursuit-evasion game with integral constraints, ANZIAM J, 52 (2011), E59-E75.  doi: 10.1017/S1446181111000538. [12] A. S. Kuchkarov, G. I. Ibragimov and M. Khakestari, On a linear differential game of optimal approach of many pursuers with one evader, Journal of Dynamical and Control Systems, 19 (2013), 1-15.  doi: 10.1007/s10883-013-9161-z. [13] M. S. Nikolskii, The direct method in linear differential games with integral constraints, Controlled systems, IM, IK, SO AN SSSR, 2 (1969), 49-59. [14] N. N. Petrov and I. N. Shuravina, On the "soft" capture in one group pursuit problem}, Journal of Computer and Systems Sciences International, 48 (2009), 521-526.  doi: 10.1134/S1064230709040042. [15] N. Y. Satimov and M. Tukhtasinov, On game problems for second-order evolution equations, Russian Mathematics (Iz. VUZ), 51 (2007), 49-57.  doi: 10.3103/S1066369X07010070. [16] N. Y. Satimov and M. Tukhtasinov, On some game problems for first-order controlled evolution equations, Differentsial'nye Uravneniya, 41 (2005), 1114-1121. doi: 10.1007/s10625-005-0263-6.

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##### References:
 [1] J. Albus, A. Meystel, A. A. Chikrii, A. A. Belousov and A. I. Kozlov, Analytical method for solution of the game problem of soft landing for moving objects, Cybernetics and Systems, 37 (2001), 75-91.  doi: 10.1023/A:1016620201241. [2] A. A. Chikrii and A. A. Belousov, Game problem of "soft landing" for second-order systems, Journal of Mathematical Sciences, 139 (2006), 6997-7012.  doi: 10.1007/s10958-006-0402-5. [3] K. G. Guseinov, A. A. Neznakhin and V. N. Ushakov, Approximate construction of attainability sets of control systems with integral constraints on the controls, Journal of Applied Mathematics and Mechanics, 63 (1999), 557-567.  doi: 10.1016/S0021-8928(99)00070-2. [4] G. I. Ibragimov, An optimal pursuit problem in systems with distributed parameters, Cybernetics and Prikladnaya Matematika i Mekhanika, 66 (2002), 753-759.  doi: 10.1016/S0021-8928(02)90002-X. [5] G. I. Ibragimov, F. Allahabi and A. Sh. Kuchkarov, pursuit problem in an infinite system of second-order differential equations, Ukrainian Mathematical Journal, 65 (2014), 1203-1216.  doi: 10.1007/s11253-014-0852-8. [6] G. I. Ibragimov, A problem of damping of oscillation system in presence of disturbance, Uzbek Math. Journal, Tashkent, 1 (2005), 34-45. [7] G. I. Ibragimov, The optimal pursuit problem reduced to an infinite system of differential equations, J. Appl Math Mech, 77 (2013), 470-476.  doi: 10.1016/j.jappmathmech.2013.12.002. [8] G. I. Ibragimov, Optimal pursuit time for a differential game in the hilbert space, Science Asia, 39S (2013), 25-30. [9] G. I. Ibragimov, M. Tukhtasinov, R. M. Hasim and I. A. Alias, A Pursuit problem described by infinite system of differential equations with coordinate-wise integral constraints on control functions, MJMS, 9 (2015), 67-76. [10] G. I. Ibragimov, On the optimal pursuit game of several pursuers and one evader, Prikladnaya Matematika I Mekhanika, 62 (1998), 199-205.  doi: 10.1016/S0021-8928(98)00024-0. [11] G. I. Ibragimov, A. Azamov and M. Khakestari, Solution of a linear pursuit-evasion game with integral constraints, ANZIAM J, 52 (2011), E59-E75.  doi: 10.1017/S1446181111000538. [12] A. S. Kuchkarov, G. I. Ibragimov and M. Khakestari, On a linear differential game of optimal approach of many pursuers with one evader, Journal of Dynamical and Control Systems, 19 (2013), 1-15.  doi: 10.1007/s10883-013-9161-z. [13] M. S. Nikolskii, The direct method in linear differential games with integral constraints, Controlled systems, IM, IK, SO AN SSSR, 2 (1969), 49-59. [14] N. N. Petrov and I. N. Shuravina, On the "soft" capture in one group pursuit problem}, Journal of Computer and Systems Sciences International, 48 (2009), 521-526.  doi: 10.1134/S1064230709040042. [15] N. Y. Satimov and M. Tukhtasinov, On game problems for second-order evolution equations, Russian Mathematics (Iz. VUZ), 51 (2007), 49-57.  doi: 10.3103/S1066369X07010070. [16] N. Y. Satimov and M. Tukhtasinov, On some game problems for first-order controlled evolution equations, Differentsial'nye Uravneniya, 41 (2005), 1114-1121. doi: 10.1007/s10625-005-0263-6.
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