Next Article in Journal
Price and Quantity Competition under Vertical Pricing
Previous Article in Journal
Power Indices under Specific Multicriteria Status
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in 2

1
Department of General and Exact Subjects, Tashkent State University of Economics, Tashkent 100006, Uzbekistan
2
Faculty of Mathematics, University of Vienna, Oskar-Morgnstern Platz 1, 1090 Wien, Austria
3
Faculty of Physics-Mathematics, Andijan State University, Andijon 170100, Uzbekistan
4
Department of Law, Economics and Umane Sciences & Decisions_lab, University Mediterranea of Reggio Calabria, 89125 Reggio Calabria, Italy
*
Author to whom correspondence should be addressed.
Games 2023, 14(4), 52; https://doi.org/10.3390/g14040052
Submission received: 23 May 2023 / Revised: 14 June 2023 / Accepted: 24 June 2023 / Published: 29 June 2023

Abstract

:
We investigate a differential evasion game with multiple pursuers and an evader for the infinite systems of differential equations in 2. The control functions of the players are subject to geometric constraints. The pursuers’ goal is to bring the state of at least one of the controlled systems to the origin of 2, while the evader’s goal is to prevent this from happening in a finite interval of time. We derive a sufficient condition for evasion from any initial state and construct an evasion strategy for the evader.

1. Introduction

The research on differential games started with the pioneering works of Isaacs and Pontryagin. Since then the field has been a topic for extensive research studies. Numerous monographs and collections, such as [1,2,3,4,5,6,7,8,9,10,11,12], have compiled the results in this field. In recent years, there has been a growing interest in differential games, although most of the research has focused on games in finite-dimensional spaces.
Many real-world problems can be modeled as control problems for partial differential equations (PDEs), which is a very active field that remains dynamic. The study of control problems for PDEs began with [13] on the time-optimal control problem for the parabolic equation; an up-to-date account of the theory can be found, for example, in [14]. Differential game problems for processes described by PDEs have been considered, for example, in refs. [15,16]. It turns out that studying differential game problems for PDE is convenient, if it is reduced to an infinite system of ordinary differential equations. This approach is called the decomposition method and has been used extensively, see for example, [17,18,19,20,21,22,23,24,25,26].
Concerning evasion problems in the finite-dimensional case, the works [27,28] have shown that one evader can avoid multiple slow pursuers in a finite-dimensional Euclidean space. These results have been extended to cases where the pursuers’ control set is a subset of the interior of the evader’s control set. In [29], a multi-pursuer–single-evader simple motion differential game in R n was studied. It was proved that if the evader’s initial position lay in the interior of the convex hull of the pursuers’ initial positions, the evader could be captured; otherwise, evasion was possible. Since then, several approaches have been employed in the research of multi-pursuer–single-evader problems. For instance, in [30], the authors assumed that the evader possessed complete information about the pursuers’ positions and strategies, and they utilized optimal control theory to analyze the problem. In [31], the authors considered a differential game involving an evader and multiple pursuers moving in an external dynamic flow field, and they derived a simplified capturing condition when the evader’s maximum speed was lower than that of each pursuer. In a different context, [32] focused on high-speed-pursuit–evasion games with multiple pursuers and a single evader in an open domain with holonomic constraints and proposed an escape strategy for the evader based on the concept of the Apollonius circle. Problems with integral constraints on the control functions are considerably difficult. In [33], the authors considered such a problem and proved that evasion was possible from multiple pursuers regardless of the initial positions of players, when the evader had an advantage in energy over the pursuers. These works naturally introduced the question of evasion problem studies in infinite dimensions. However, a simple motion pursuit differential game in 2 with a finite number of pursuers, when players have identical capabilities has not been studied, and no initial position has been identified from which the pursuit can be completed. The pursuit differential game of one pursuer and one evader has been studied in [34] for an infinite system of binary differential equations in the Hilbert space 2. The general case of this problem was studied in [35]. Moreover, the papers [36,37,38] relate to differential games with an infinite system.
In this paper, we investigate an evasion differential game involving multiple pursuers and one evader in an infinite system of differential equations with a special operator (refer to (3)), which gives a coupled system. In the this case, in contrast to [34,35] the dynamics cannot be reduced to finite dimensional subsystems. We establish that if the evader’s control set contains or overlaps with the control set of any pursuer, then evasion is feasible regardless of the number of pursuers and the initial position of the game. Additionally, we construct an evasion strategy for the evader. Our main result highlights that the infinite dimensionality of the space 2 confers an advantage to the evader, enabling evasion from any finite number of pursuers.

2. Statement of the Problem

It is remembered that 2 is the set of all sequences of real numbers
x = ( x 1 , x 2 , ) , | k = 1 x k 2 <
with the inner product and norm given by
x , y = k = 1 x k y k , | | x | | = x , x , x , y l 2 .
We consider a differential game described by the following infinite system of differential equations:
x ˙ i k = λ x i k + a 1 x i , k + 1 + a 2 x i , k + 2 + + a M x i , k + M + u i k , x i k ( 0 ) = x i k 0 , y ˙ k = λ y k + a 1 y k + 1 + a 2 y k + 2 + + a M y k + M + v k , y k ( 0 ) = y k 0 ,
where x i k , y k R , i = 1 , 2 , , m , k = 1 , 2 , , λ R and M N are given constants, j = 1 M a j = 1 for a j 0 , and u i k , v k are the control parameters of the i-th pursuer and the evader, respectively. Moreover, it is considered that
x i 0 = ( x i 1 0 , x i 2 0 , . . . ) l 2 , y 0 = ( y 1 0 , y 2 0 , . . . ) l 2 , i = 1 , 2 , , m .
Further, it is assumed that x i 0 y 0 for all i = 1 , 2 , , m .
With the notation η i k = x i k y k for i = 1 , 2 , , m , k = 1 , 2 , we can write (1) in the form
η ˙ i = Q η i + u i v , η i ( 0 ) = η i 0 , i = 1 , 2 , . . . , m ,
where η i = ( η i 1 , η i 2 , ) , u i = ( u i 1 , u i 2 , ) , v = ( v 1 , v 2 , ) , and η i 0 = x i 0 y 0 for i = 1 , 2 , m , and η i 0 0 is assumed as well. The linear operator Q : l 2 l 2 is defined by the following equation:
Q η = { λ η k + a 1 η k + 1 + a 2 η k + 2 + . . . + a M η k + M } k N .
Further, we continue as
Q = λ I + j = 1 M a j E j ,
where I is the identity map, and E : l 2 l 2 is the shift map whose action is defined as [ E x ] k = x k + 1 . It is easy to see that E j = 1 for all j N ; hence, Q is a bounded linear operator. Indeed,
Q = | λ | + j = 1 M a j E j | λ | + 1 .
This implies that e t Q is a continuous semi-group, and the solution of (2) can be written as
η i ( t ) = e t Q η i 0 + 0 t e ( t s ) Q ( u i ( s ) v ( s ) ) d s .
Definition 1. 
A function u i ( t ) = ( u i 1 ( t ) , u i 2 ( t ) , ) , i { 1 , 2 , , m } with measurable coordinates u i k ( t ) , k = 1 , 2 , , which satisfy the constraint | | u i ( t ) | | ρ i , 0 t T , is called the admissible control of the i-th pursuer, where ρ i > 0 and T > 0 are given numbers.
Definition 2. 
A function v ( t ) = ( v 1 ( t ) , v 2 ( t ) , ) with measurable coordinates v k ( t ) , k = 1 , 2 , , which satisfy the constraint | | v ( t ) | | σ , 0 t T , is called the admissible control of the evader, where σ > 0 and T > 0 are given numbers.
Assume that σ ρ i for all i = 1 , , m .
Definition 3. 
If there exists a control v ¯ ( t ) of the evader, such that η i ( t ) 0 , i = 1 , , m , 0 t T , for any admissible controls u i ( t ) , i = 1 , , m , 0 t T of the pursuers, then we say that evasion is possible on the time interval [ 0 , T ] .
The problem here is that to find a strategy for the evader, such that evasion is possible in game (2).

3. The Main Result

In this section, we prove the following theorem.
Theorem 1. 
In game (2), evasion is possible for any initial states η i 0 , i = 1 , 2 , , m .
Proof. 
We can write (4) in the form
η i ( t ) = e t Q ξ i ( t ) , ξ i ( t ) = η i 0 + 0 t e s Q ( u i ( s ) v ( s ) ) d s .
For any i = 1 , 2 , , m , the equation η i ( t ) = 0 is equivalent to ξ i ( t ) = 0 since the matrix e t Q is not singular. Furthermore, proving the theorem only requires constructing an admissible control v ( t ) for the evader during the time interval of 0 t T , which means that ξ i ( t ) = ( ξ i 1 ( t ) , ξ i 2 ( t ) , ) 0 holds for every i = 1 , 2 , , m and 0 t T .
Since η i 0 0 , we can conclude that there exists a unit vector α = ( α 1 , α 2 , . . . ) 2, where α = 1 , such that the inner product η i 0 , α 0 for all i = 1 , , m . We can choose the vector α to be an orthonormal vector to the hyperplane that passes through the points η i 0 , where i = 1 , 2 , , m .
Now, we set the evader’s control as follows
v ( t ) = e t Q * α σ e t Q * α ,
where Q * stands for the transpose of the matrix Q. We show that evasion is possible by using the control (6) of the evader on the time interval [ 0 , T ] . Note that since α = ( α 1 , α 2 , . . . ) is a unit vector, the denominator in (6) is not equal to zero, i.e.,
e t Q * α 0 , t 0 .
To obtain a contradiction, suppose that there exist admissible controls of pursuers, such that ξ p ( θ ) = 0 for some p { 1 , , m } and θ > 0 , while the evader applies the control (6). From η i 0 , α 0 , i = 1 , 2 , , m , we then have
η p 0 , α 0 , i = 1 , 2 , , m .
Thus,
ξ p ( θ ) , α = η p 0 + 0 t e s Q ( u i ( s ) v ( s ) ) d s , α
= η p 0 , α + 0 θ e s Q ( u p ( s ) v ( s ) ) , α d s
0 θ e s Q u p ( s ) , α d s 0 θ e s Q v ( s ) , α d s .
Using the Cauchy–Schwartz inequality, the first term of the right side of Equation (8) can be bounded as follows
0 θ e s Q u p ( s ) , α d s = 0 θ u p ( s ) , e s Q * α d s
ρ p 0 θ e s Q * α d s .
In (11), equality holds at
u p ( t ) = e t Q * α ρ p e s Q * α , 0 t θ .
From (6) and (11), we have ξ p ( θ ) , α 0 for σ ρ p . Indeed,
ξ p ( θ ) , α 0 θ u p ( s ) , e s Q * α d s 0 θ v ( s ) , e s Q * α d s
ρ p 0 θ e s Q * α d s 0 θ e t Q * α σ e t Q * α , e s Q * α d s
= ρ p 0 θ e s Q * α d s + σ 0 θ e s Q * α d s
= ( σ ρ p ) 0 θ e s Q * α d s 0 .
The assumption ξ p ( θ ) = 0 shows that ξ p ( θ ) , α = 0 . However, in (14), the equality sign is only true for ρ p = σ . By comparing (6) and (13), we can conclude that u p ( t ) = v ( t ) , for 0 t θ . By substituting this into (4), we have
ξ p ( θ ) = η p 0 .
This implies that
ξ p ( θ ) = ( ξ p 1 ( θ ) , ξ p 2 ( θ ) , . . . ) = ( η p 1 0 , η p 2 0 , . . . ) 0 .
This contradicts our assumption ξ p ( θ ) = 0 . From this, we conclude that ξ i ( t ) 0 , for all t [ 0 , T ] , and i = 1 , 2 , , m . This completes the proof. □

4. Conclusions

In the present paper work, we studied a multi-pursuer evasion differential game problem for an infinite system of differential equations with a special operator Q = λ I + i = 1 M a i E i on the right hand side, where a i 0 and i = 1 M a i = 1 .
We considered a differential game with geometric constraints and showed that evasion is possible from any initial position under natural conditions in 2. In the construction of the evasion strategy, the fact that a finite number of points lies on a hyperplane played the key role.
A similar result can be obtained in the case of an infinite (countable) number of pursuers, if the initial states of all objects lie on a hyperplane. In general, for the countable number of pursuers, this problem is open.

Author Contributions

Investigation, G.I., M.R., I.Z. and B.A.P.; methodology, G.I., M.R., I.Z. and B.A.P.; project administration, M.R. and I.Z.; validation, G.I., M.R., I.Z. and B.A.P.; visualization, G.I.; writing—original draft preparation, G.I., M.R., I.Z. and B.A.P.; writing—review and editing, G.I., M.R., I.Z. and B.A.P. All authors have contributed equally at all stages of research. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Acknowledgments

The authors thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Berkovitz, L.D. A Survey of Differential Games, Mathematical Theory of Control; Academic Press: New York, NY, USA, 1967; pp. 373–385. [Google Scholar]
  2. Chikrii, A.A. Conflict-Controlled Processes; Kluwer: Dordrecht, The Netherlands, 1997. [Google Scholar]
  3. Elliot, R.J.; Kalton, N.J. The Existence of Value for Differential Games; American Mathematical Soc.: Providence, RI, USA, 1972. [Google Scholar]
  4. Fleming, W.H. A note on differential games of prescribed duration. Contrib. Theory Games 1957, 3, 407–416. [Google Scholar]
  5. Fleming, W.H. The convergence problem for differential games. J. Math. Anal. Appl. 1961, 3, 102–116. [Google Scholar] [CrossRef] [Green Version]
  6. Friedman, A. Differential Games; Wiley Interscience: New York, NY, USA, 1971. [Google Scholar]
  7. Hajek, O. Pursuit Games; Academic Press: New York, NY, USA, 1975. [Google Scholar]
  8. Ho, Y.; Bryson, A.; Baron, S. Differential games and optimal pursuit-evasion strategies. IEEE Trans. Autom. Control 1965, 10, 385–389. [Google Scholar] [CrossRef]
  9. Isaacs, R. Differential Games; John Wiley & Sons: New York, NY, USA, 1965. [Google Scholar]
  10. Krasovskii, N.N.; Subbotin, A.I. Game-Theoretical Control Problems; Springer: New York, NY, USA, 1988. [Google Scholar]
  11. Petrosyan, L.A. Differential Games of Pursuit; World Scientific: Singapore, 1993. [Google Scholar]
  12. Pontryagin, L.S. Selected Works; Nauka: Moscow, Russia, 1988. [Google Scholar]
  13. Fattorini, H.O. Time-Optimal control of solutions of operational differential equations. SIAM J. Control 1964, 2, 54–59. [Google Scholar]
  14. Fursikov, A.V. Optimal Control of Distributed Systems, Theory and Applications; Translations of Math. Monographs, 187; American Mathematical Society: Providence, RI, USA, 2000. [Google Scholar]
  15. Lions, J.L. Contrôle Optimal de Systémes Gouvernées par des Equations aux Dérivées Partielles; Dunod: Paris, France, 1968. [Google Scholar]
  16. Osipov, Y.S. The theory of differential games in systems with distributed parameters. Dokl Akad Nauk SSSR 1975, 223, 1314–1317. [Google Scholar]
  17. Avdonin, S.A.; Ivanov, S.A. Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems; Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
  18. Azamov, A.A.; Ruziboev, M.B. The time-optimal problem for evolutionary partial differential equations. J. Appl. Math. Mech. 2013, 77, 220–224. [Google Scholar] [CrossRef]
  19. Butkovskiy, A.G. Theory of Optimal Control of Distributed Parameter Systems; Elsevier: New York, NY, USA, 1969. [Google Scholar]
  20. Chernous’ko, F.L. Bounded controls in distributed-parameter systems. J. Appl. Math. Mech. 1992, 56, 707–723. [Google Scholar] [CrossRef]
  21. Ibragimov, G.I. A Problem of Optimal Pursuit in Systems with Distributed Parameters. J. Appl. Math. Mech. 2002, 66, 719–724. [Google Scholar] [CrossRef]
  22. Satimov, N.Y.; Tukhtasinov, M. Game problems on a fixed interval in controlled first-order evolution equations. Math. Notes 2006, 80, 578–589. [Google Scholar] [CrossRef]
  23. Satimov, N.Y.; Tukhtasinov, M. On Some Game Problems for First-Order Controlled Evolution Equations. Differ. Equ. 2005, 41, 1169–1177. [Google Scholar] [CrossRef]
  24. Satimov, N.Y.; Tukhtasinov, M. On Game Problems for Second-Order Evolution Equations. Russ. Math. 2007, 51, 49–57. [Google Scholar] [CrossRef]
  25. Tukhtasinov, M. Some problems in the theory of differential pursuit games in systems with distributed parameters. J. Appl. Math. Mech. 1995, 59, 979–984. [Google Scholar] [CrossRef]
  26. Tukhtasinov, M.; Mamatov, M.S. On Pursuit Problems in Controlled Distributed Parameters Systems. Math. Notes 2008, 84, 256–262. [Google Scholar] [CrossRef]
  27. Chernous’ko, F.L. A Problem of Evasion of Several Pursuers. Prikl. Mat. Mekh. 1976, 40, 14–24. [Google Scholar]
  28. Chernous’ko, F.L.; Zak, V.L. On differential games of evasion from many pursuers. J. Optim. Theory Appl. 1985, 46, 461–470. [Google Scholar] [CrossRef]
  29. Pshenichnii, B.N. Simple pursuit by several objects. Cybern. Syst. Anal. 1976, 12, 484–485. [Google Scholar] [CrossRef]
  30. Makkapati, V.R.; Tsiotras, P. Optimal evading strategies and task allocation in multi-player pursuit-evasion problems. Dyn. Games Appl. 2019, 9, 1168–1187. [Google Scholar] [CrossRef]
  31. Sun, W.; Tsiotras, P.; Lolla, T.; Subramani, D.N.; Lermusiaux, P.F.J. Multiple-pursuer/one-evader pursuit-evasion game in dynamic flowfields. JGCD 2017, 40. [Google Scholar] [CrossRef]
  32. Ramana, M.V.; Kothari, M. Pursuit-Evasion Games of High Speed Evader. J. Intell. Robot. Syst. 2017, 85, 293–306. [Google Scholar] [CrossRef]
  33. Ibragimov, G.I.; Ferrara, M.; Ruziboev, M.; Pansera, B.A. Linear evasion differential game of one evader and several pursuers with integral constraints. Int. J. Game Theory 2021, 50, 729–750. [Google Scholar] [CrossRef]
  34. Tukhtasinov, M.; Ibragimov, G.; Kuchkarova, S.; Mat Hasim, R. Differential Games for an Infinite 2-Systems of Differential Equations. Mathematics 2021, 9, 1467. [Google Scholar] [CrossRef]
  35. Ruziboev, M.; Mamayusupov, K.; Ibragimov, G.; Khaitmetov, A. On a linear differential game in the Hilbert space 2. arXiv 2023, arXiv:2302.01632. [Google Scholar]
  36. Ibragimov, G.I.; Allahabi, F.; Kuchkarov, A.S. A pursuit problem in an infinite system of second-order differential equations. Ukr. Math. J. 2014, 65, 1203–1216. [Google Scholar] [CrossRef]
  37. Ibragimov, G.I. The optimal pursuit problem reduced to an infinite system of differential equation. J. Appl. Math. Mech. 2013, 77, 470–476. [Google Scholar] [CrossRef]
  38. Ibragimov, G.I. Optimal pursuit time for a differential game in the Hilbert space l2. Sci. Asia 2013, 39S, 25–30. [Google Scholar] [CrossRef] [Green Version]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Ibragimov, G.; Ruziboev, M.; Zaynabiddinov, I.; Pansera, B.A. Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in 2. Games 2023, 14, 52. https://doi.org/10.3390/g14040052

AMA Style

Ibragimov G, Ruziboev M, Zaynabiddinov I, Pansera BA. Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in 2. Games. 2023; 14(4):52. https://doi.org/10.3390/g14040052

Chicago/Turabian Style

Ibragimov, Gafurjan, Marks Ruziboev, Ibroximjon Zaynabiddinov, and Bruno Antonio Pansera. 2023. "Evasion Differential Game of Multiple Pursuers and a Single Evader with Geometric Constraints in 2" Games 14, no. 4: 52. https://doi.org/10.3390/g14040052

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop