January  2022, 9(1): 1-12. doi: 10.3934/jdg.2021019

Game value for a pursuit-evasion differential game problem in a Hilbert space

1. 

Department of Mathematical Sciences, Bayero University, Kano, Nigeria

2. 

Federal University, Gashua, Yobe State, Nigeria

* Corresponding author: Abbas Ja'afaru Badakaya

Received  December 2020 Revised  April 2021 Published  January 2022 Early access  May 2021

We consider a pursuit-evasion differential game problem with countable number pursuers and one evader in the Hilbert space $ l_{2}. $ Players' dynamic equations described by certain $ n^{th} $ order ordinary differential equations. Control functions of the players subject to integral constraints. The goal of the pursuers is to minimize the distance to the evader and that of the evader is the opposite. The stoppage time of the game is fixed and the game payoff is the distance between evader and closest pursuer when the game is stopped. We study this game problem and find the value of the game. In addition to this, we construct players' optimal strategies.

Citation: Abbas Ja'afaru Badakaya, Aminu Sulaiman Halliru, Jamilu Adamu. Game value for a pursuit-evasion differential game problem in a Hilbert space. Journal of Dynamics and Games, 2022, 9 (1) : 1-12. doi: 10.3934/jdg.2021019
References:
[1]

J. AdamuK. MuangchooA. J. Badakaya and J. Rilwan, On pursuit-evasion differential game problem in a Hilbert space, AIMS Math., 5 (2020), 7467-7479.  doi: 10.3934/math.2020478.

[2]

A. J. Badakaya, Value of a differential game problem with multiple players in a certain Hilbert space, J. Nigerian Math. Soc., 36 (2017), 287-305. 

[3]

E. Bakolas and P. Tsiotras, Relay pursuit of a maneuvering target using dynamic Voronoi diagrams, Automatica J. IFAC, 48 (2012), 2213-2220.  doi: 10.1016/j.automatica.2012.06.003.

[4]

M. ChenZ. Zhou and C. J. Tomlin, Multiplayer reach-avoid games via pairwise outcomes, IEEE Trans. Automat. Control, 62 (2017), 1451-1457.  doi: 10.1109/TAC.2016.2577619.

[5]

G. I. Ibragimov, On a game of optimal pursuit of one evader by several pursuers, J. Appl. Math. Mech., 62 (1998), 187-192.  doi: 10.1016/S0021-8928(98)00024-0.

[6]

G. I. Ibragimov, Optimal pursuit of an evader by countably many pursuers, Differ. Equ., 41 (2005), 627-635.  doi: 10.1007/s10625-005-0198-y.

[7]

G. IbragimovN. Abd RasidA. Kuchkarov and F. Ismail, Multi pursuer differential game of optimal approach with integral constraints on control of players, Taiwanese J. Math., 19 (2015), 963-976.  doi: 10.11650/tjm.19.2015.2288.

[8]

G. IbragimovI. A. AliasU. Waziri and A. B. Ja'afaru, Differential game of optimal pursuit for an infinite system of differential equations, Bull. Malays. Math. Sci. Soc., 42 (2019), 391-403.  doi: 10.1007/s40840-017-0581-x.

[9]

G. Ibragimov and N. A. Hussin, A Pursuit-evasion differential game with many pursuers and one evader, Malaysian J. Math. Sci., 4 (2010), 183-194. 

[10]

G. I. Ibragimov and A. S. Kuchkarov, Fixed duration pursuit-evasion differential game with integral constraints, J. Physics: Conference Series, 435 (2017). doi: 10.1088/1742-6596/435/1/012017.

[11]

G. I. Ibragimov and M. Salimi, Pursuit-evasion differential game with many inertial players, Math. Probl. Eng., 2009 (2009), 15pp. doi: 10.1155/2009/653723.

[12]

A. B. Ja'afaru and G. I. Ibragimov, On some pursuit and evasion differential game problems for an infinite number of first-order differential equations, J. Appl. Math., 2012 (2012), 13pp. doi: 10.1155/2012/717124.

[13]

A. S. KuchkarovG. I. Ibragimov and M. Khakestari, On a linear differential game of optimal approach of many pursuers with one evader, J. Dyn. Control Syst., 19 (2013), 1-15.  doi: 10.1007/s10883-013-9161-z.

[14]

A. Y. Levchenkov and A. G. Pashkov, Differential game of optimal approach of two inertial pursuers to a noninertial evader, J. Optim. Theory Appl., 65 (1990), 501-518.  doi: 10.1007/BF00939563.

[15]

L. A. Petrosyan, Differential pursuit games, Izdat. Leningrad. Univ., Leningrad, 1977,222pp.

[16]

M. V. Ramana and M. Kothari, Pursuit-evasion games of high speed evader, J. Intell. Robot. Syst., 85 (2017), 293-306.  doi: 10.1007/s10846-016-0379-3.

[17]

M. V. Ramana and M. Kothari, Pursuit strategy to capture high-speed evaders using multiple pursuers, J. Guidance Control Dyn., 49 (2017), 139-149.  doi: 10.2514/1.G000584.

[18]

M. Salimi and M. Ferrara, Differential game of optimal pursuit of one evader by many pursuers, Internat. J. Game Theory, 48 (2019), 481-490.  doi: 10.1007/s00182-018-0638-6.

[19]

N. SatimovB. B. Rikhsiev and A. A. Khamdamov, A pursuit problem for linear differential and discrete $n$-person games with integral constraints, Mat. Sb. (N.S.), 118(160) (1982), 456-469. 

[20]

A. I. Subbotin and A. G. Chentsov, Guaranteed Optimization in Control Problems, Nauka, Moscow, 1981,288pp.

[21] A.-M. Wazwaz, Linear and Nonlinear Integral Equations. Mathods and Applications, Higher Education Press, Beijing; Springer, Heidelberg, 2011.  doi: 10.1007/978-3-642-21449-3.

show all references

References:
[1]

J. AdamuK. MuangchooA. J. Badakaya and J. Rilwan, On pursuit-evasion differential game problem in a Hilbert space, AIMS Math., 5 (2020), 7467-7479.  doi: 10.3934/math.2020478.

[2]

A. J. Badakaya, Value of a differential game problem with multiple players in a certain Hilbert space, J. Nigerian Math. Soc., 36 (2017), 287-305. 

[3]

E. Bakolas and P. Tsiotras, Relay pursuit of a maneuvering target using dynamic Voronoi diagrams, Automatica J. IFAC, 48 (2012), 2213-2220.  doi: 10.1016/j.automatica.2012.06.003.

[4]

M. ChenZ. Zhou and C. J. Tomlin, Multiplayer reach-avoid games via pairwise outcomes, IEEE Trans. Automat. Control, 62 (2017), 1451-1457.  doi: 10.1109/TAC.2016.2577619.

[5]

G. I. Ibragimov, On a game of optimal pursuit of one evader by several pursuers, J. Appl. Math. Mech., 62 (1998), 187-192.  doi: 10.1016/S0021-8928(98)00024-0.

[6]

G. I. Ibragimov, Optimal pursuit of an evader by countably many pursuers, Differ. Equ., 41 (2005), 627-635.  doi: 10.1007/s10625-005-0198-y.

[7]

G. IbragimovN. Abd RasidA. Kuchkarov and F. Ismail, Multi pursuer differential game of optimal approach with integral constraints on control of players, Taiwanese J. Math., 19 (2015), 963-976.  doi: 10.11650/tjm.19.2015.2288.

[8]

G. IbragimovI. A. AliasU. Waziri and A. B. Ja'afaru, Differential game of optimal pursuit for an infinite system of differential equations, Bull. Malays. Math. Sci. Soc., 42 (2019), 391-403.  doi: 10.1007/s40840-017-0581-x.

[9]

G. Ibragimov and N. A. Hussin, A Pursuit-evasion differential game with many pursuers and one evader, Malaysian J. Math. Sci., 4 (2010), 183-194. 

[10]

G. I. Ibragimov and A. S. Kuchkarov, Fixed duration pursuit-evasion differential game with integral constraints, J. Physics: Conference Series, 435 (2017). doi: 10.1088/1742-6596/435/1/012017.

[11]

G. I. Ibragimov and M. Salimi, Pursuit-evasion differential game with many inertial players, Math. Probl. Eng., 2009 (2009), 15pp. doi: 10.1155/2009/653723.

[12]

A. B. Ja'afaru and G. I. Ibragimov, On some pursuit and evasion differential game problems for an infinite number of first-order differential equations, J. Appl. Math., 2012 (2012), 13pp. doi: 10.1155/2012/717124.

[13]

A. S. KuchkarovG. I. Ibragimov and M. Khakestari, On a linear differential game of optimal approach of many pursuers with one evader, J. Dyn. Control Syst., 19 (2013), 1-15.  doi: 10.1007/s10883-013-9161-z.

[14]

A. Y. Levchenkov and A. G. Pashkov, Differential game of optimal approach of two inertial pursuers to a noninertial evader, J. Optim. Theory Appl., 65 (1990), 501-518.  doi: 10.1007/BF00939563.

[15]

L. A. Petrosyan, Differential pursuit games, Izdat. Leningrad. Univ., Leningrad, 1977,222pp.

[16]

M. V. Ramana and M. Kothari, Pursuit-evasion games of high speed evader, J. Intell. Robot. Syst., 85 (2017), 293-306.  doi: 10.1007/s10846-016-0379-3.

[17]

M. V. Ramana and M. Kothari, Pursuit strategy to capture high-speed evaders using multiple pursuers, J. Guidance Control Dyn., 49 (2017), 139-149.  doi: 10.2514/1.G000584.

[18]

M. Salimi and M. Ferrara, Differential game of optimal pursuit of one evader by many pursuers, Internat. J. Game Theory, 48 (2019), 481-490.  doi: 10.1007/s00182-018-0638-6.

[19]

N. SatimovB. B. Rikhsiev and A. A. Khamdamov, A pursuit problem for linear differential and discrete $n$-person games with integral constraints, Mat. Sb. (N.S.), 118(160) (1982), 456-469. 

[20]

A. I. Subbotin and A. G. Chentsov, Guaranteed Optimization in Control Problems, Nauka, Moscow, 1981,288pp.

[21] A.-M. Wazwaz, Linear and Nonlinear Integral Equations. Mathods and Applications, Higher Education Press, Beijing; Springer, Heidelberg, 2011.  doi: 10.1007/978-3-642-21449-3.
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