Multi-pursuer pursuit differential game for an infinite system of second order differential equations

R. Yu. Kazimirova   ; G. I. Ibragimov   ; R. M. Hasim

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 34 (2024) no. 1, pp. 48-64

Résumé

We study a pursuit differential game of many pursuers and one evader. The game is described by the infinite systems of $m$ inertial equations. By definition, pursuit in the game is completed if the state and its derivative of one of the systems are equal to zero at some time. In the literature, such a condition of completion of pursuit is also called soft landing. We obtain a condition in terms of energies of players which is sufficient for completion of pursuit in the game. The pursuit strategies are also constructed.

Détail

Keywords: differential game, control, strategy, many pursuers, infinite system of differential equations, integral constraint

Bibliographie

[1] Albeverio S., Alimov Sh., “On a time-optimal control problem associated with the heat exchange process”, Applied Mathematics and Optimization, 57:1 (2008), 58–68 | DOI | MR | Zbl

[2] Avdonin S.A., Ivanov S.A., Families of exponentials. The method of moments in controllability problems for distributed parameter systems, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[3] Azamov A.A., Ruziboyev M.B., “The time-optimal problem for evolutionary partial differential equations”, Journal of Applied Mathematics and Mechanics, 77:2 (2013), 220–224 | DOI | MR | Zbl

[4] Bannikov A.S., Petrov N.N., “On a nonstationary problem of group pursuit”, Proceedings of the Steklov Institute of Mathematics, 271:suppl. 1 (2010), S41–S52 | DOI | MR

[5] Blagodatskikh A.I., Petrov N.N., “Simultaneous multiple capture of rigidly coordinated evaders”, Dynamic Games and Applications, 9:3 (2019), 594–613 | DOI | MR | Zbl

[6] Butkovskiy A.G., Theory of optimal control of distributed parameter systems, Elsevier, New York, 1969 | MR

[7] Chaves-Silva F.W., Zhang Xu, Zuazua E., “Controllability of evolution equations with memory”, SIAM Journal on Control and Optimization, 55:4 (2017), 2437–2459 | DOI | MR | Zbl

[8] Chernous’ko F.L., “Bounded controls in distributed-parameter systems”, Journal of Applied Mathematicsand Mechanics, 56:5 (1992), 707–723 | DOI | MR | Zbl

[9] Chikrii A.A., Chikrii G.Ts., “Game problems of approach for quasilinear systems of general form”, Proceedings of the Steklov Institute of Mathematics, 304:suppl. 1 (2019), S44–S58 | DOI | DOI | MR | Zbl

[10] Chikrii A.A., Belousov A.A., “Game problem of “soft landing” for second-order systems”, Journal of Mathematical Sciences, 139:5 (2006), 6997–7012 | DOI | MR | Zbl

[11] Garcia E., Casbeer D.W., von Moll A., Pachter M., “Multiple pursuer multiple evader differential games”, IEEE Transactions on Automatic Control, 66:5 (2021), 2345–2350 | DOI | MR

[12] Grinikh A.L., Petrosyan L.A., “An effective punishment for an n-person Prisoner’s Dilemma on a network”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 27:3 (2021), 256–262 | DOI | MR

[13] Ibragimov G., Ferrara M., Ruziboev M., Pansera B.A., “Linear evasion differential game of one evader and several pursuers with integral constraints”, International Journal of Game Theory, 50:3 (2021), 729–750 | DOI | MR | Zbl

[14] Ibragimov G., Allahabi F., Kuchkarov A., “A pursuit problem in an infinite system of second-order differential equations”, Ukrainian Mathematical Journal, 65:8 (2014), 1203–1216 | DOI | MR

[15] Ibragimov G., Kazimirova R., Pansera B.A., “Evasion differential game of multiple pursuers and one evader for an infinite system of binary differential equations”, Mathematics, 10:23 (2022), 4448 | DOI

[16] Ibragimov G., Rakhmanov A., Alias I.A., Mohd Jaffar M.Z.A., “The soft landing problem for an infinite system of second order differential equations”, Numerical Algebra, Control and Optimization, 7:1 (2017), 89–94 | DOI | MR | Zbl

[17] Ibragimov G., “Optimal pursuit time for a differential game in the Hilbert space $l_2$”, ScienceAsia, 39S:1 (2013), 25–30 | DOI | MR

[18] Ibragimov G.I., “A problem of optimal pursuit in systems with distributed parameters”, Journal of Applied Mathematics and Mechanics, 66:5 (2002), 719–724 | DOI | MR | Zbl

[19] Makkapati V.R., Tsiotras P., “Optimal evading strategies and task allocation in multi-player pursuit–evasion problems”, Dynamic Games and Applications, 9:4 (2019), 1168–1187 | DOI | MR | Zbl

[20] Martin Philippe, Rosier Lionel, Rouchon Pierre, “Null controllability of the structurally damped wave equation with moving control”, SIAM Journal on Control and Optimization, 51:1 (2013), 660–684 | DOI | MR | Zbl

[21] Petrov N.N., “Conflict controlled processes by interaction of controlled object groups”, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2005, no. 4(34), 81–102 (in Russian) | DOI | DOI

[22] Petrov N.N., Shuravina I.N., “On the “soft” capture in one group pursuit problem”, Journal of Computer and Systems Sciences International, 48:4 (2009), 521–526 | DOI | MR | Zbl

[23] Petrov N.N., “Simple group pursuit subject to phase constraints and data delay”, Journal of Computer and Systems Sciences International, 57:1 (2018), 37–42 | DOI | MR | Zbl

[24] Ramana M.V., Kothari M., “Pursuit–evasion games of high speed evader”, Journal of Intelligent and Robotic Systems, 85:2 (2017), 293–306 | DOI

[25] Rilwan J., Kumam P., Badakaya A.J., Ahmed I., “A modified dynamic equation of evasion differential game problem in a Hilbert space”, Thai Journal of Mathematics, 18:1 (2020), 199–211 https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/993 | MR | Zbl

[26] Salimi M., Ferrara M., “Differential game of optimal pursuit of one evader by many pursuers”, International Journal of Game Theory, 48:2 (2019), 481–490 | DOI | MR | Zbl

[27] Satimov N.Yu., Tukhtasinov M., “Game problems on a fixed interval in controlled first-order evolution equations”, Mathematical Notes, 80:3–4 (2006), 578–589 | DOI | DOI | MR | Zbl

[28] Satimov N. Yu., Tukhtasinov M., “On some game problems for first-order controlled evolution equations”, Differential Equations, 41:8 (2005), 1169–1177 | DOI | MR | Zbl

[29] Satimov N.Yu., Tukhtasinov M., “On game problems for second-order evolution equations”, Russian Mathematics, 51:1 (2007), 49–57 | DOI | MR | Zbl

[30] Sun Wei, Tsiotras P., Lolla T., Subramani D.N., Lermusiaux P.F.J., “Multiple-pursuer/one-evader pursuit–evasion game in dynamic flowfields”, Journal of Guidance, Control, and Dynamics, 40:7 (2017), 1627–1637 | DOI | MR

[31] Tukhtasinov M., Ibragimov G., Kuchkarova S., Mat Hasim R., “Differential games for an infinite 2-systems of differential equations”, Mathematics, 9:13 (2021), 1467 | DOI

[32] Tukhtasinov M., “Some problems in the theory of differential pursuit games in systems with distributed parameters”, Journal of Applied Mathematics and Mechanics, 59:6 (1995), 935–940 | DOI | MR | Zbl

[33] Tukhtasinov M., Mamatov M.Sh., “On pursuit problems in controlled distributed systems”, Mathematical Notes, 84:2 (2008), 256–262 | DOI | DOI | MR | Zbl