Differential game of one evader and multiple pursuers with exponential integral constraints

We prove the existence of a value for pursuit games with state constraints. We also prove that this value is lower semicontinuous.

A linear two player zero-sum pursuit-evasion differential game is considered. The control functions of players are subject to integral constraints. In the game, the first player, the Pursuer, tries to force the state of the system towards the origin, while the aim of the second player, the Evader, is the opposite. We construct the optimal strategies of the players when the control resource of the Pursuer is greater than that of the Evader. The case where the control resources of the Pursuer are less than or equal to that of the Evader is studied to prove the main theorem. For this case a new method for solving of the evasion problem is proposed. We assume that the instantaneous control employed by the Evader is known to the Pursuer. For construction, the strategy of the Evader information about the state of the system and the control resources of the players is used. References R. Isaacs. Differential games. John Wiley and Sons, New York, 1965. L. S. Pontryagin. Collected works. Nauka, Moscow, 1988. (Russian) L. D. Berkovitz. Necessary conditions for optimal strategies in a class of differential games and control problems. SIAM Journal on Control , 5 , 1--24, 1967. L. D. Berkovitz. A survey of differential games. Mathematical Theory of Control , Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New York, 373--385, 1967. N. N. Krasovskii and A. I. Subbotin. Game-theoretical control problems. New York, Springer, 1988. W. H. Fleming. The convergence problem for differential games. Journal of Mathematical Analysis and Applications . 3 , 102--116, 1961. W. H. Fleming. The convergence problem for differential games, Part 2. Advances in Game Theory, Annals of Mathematics Studies , (52), Princeton University Press, Princeton, New Jersey,195--210, 1964. A. Friedman. Differential games. Wiley-Interscience, New York, 1971. R. J. Elliott and N. J. Kalton. The existence of value in differential games. Memoirs of the American Mathematical Society , 126 , 1--67, 1972. L. A. Petrosyan. Differential games of pursuit. World Scientific, Singapore, London, 1993. O. Hajek. Pursuit games. Academic Press, New York, San Francisco, 1975. A. Ya. Azimov. Linear differential pursuit game with integral constraints on the control. Differentsial'nye Uravneniya, 11 (10), 1975, 1723--1731; English transl. in Differential Equations 11 , 1283--1289, 1975. A. Ya. Azimov. A linear differential evasion game with integral constraints on the controls. USSR Computational Mathematics and Mathematical Physics, 14 (6), 56--65, 1974. M. S. Nikolskii. The direct method in linear differential games with integral constraints. Controlled systems, IM, IK, SO AN SSSR, (2), 49--59, 1969. A. I. Subbotin and V. N. Ushakov. Alternative for an encounter-evasion differential game with integral constraints on the playersi controls. PMM 39 (3), 387--396, 1975. V. N. Ushakov. Extremal strategies in differential games with integral constraints. PMM , 36 (1), 15--23, 1972. B. N. Pshenichnii and Yu. N. Onopchuk. Linear differential games with integral constraints. Izvestige Akademii Nauk SSSR, Tekhnicheskaya Kibernetika , (1), 13--22, 1968. A. A. Azamov, B. Samatov. $\pi $-strategy. An elementary introduction to the theory of differential games. National University of Uzbekistan. Tashkent, Uzbekistan, 2000. G. I. Ibragimov. A game problem on a closed convex set. Siberian Advances in Mathematics . 12 (3), 16--31, 2002. G. I. Ibragimov. A problem of optimal pursuit in systems with distributed parameters. J. Appl. Math. Mech, 66 (5), 719--724, 2003. E. B. Lee and and L. Markus. Foundations of optimal control theory, John Wiley and Sons Inc., New York, 1967.

This paper considers pursuit-evasion differential games in the Euclidean plane where an evader is engaged by multiple pursuers and point capture is required. The players have simple motion (i.e., holonomic) in the manner of Isaacs, and the pursuers are faster than the evader. The attention of this paper is confined to the case where the pursuers have the same speed, and so the game’s parameter is that the evader/pursuers speed ratio is . State feedback capture strategies and an evader strategy that yields a lower bound on his/her time-to-capture are devised using a geometric method. It is shown that, in group/swarm pursuit, when the players are in general position, capture is effected by one, two, or three critical pursuers, and this is irrespective of the size of the pursuit pack. Group pursuit devolves into pure pursuit by one of the pursuers or into a pincer movement pursuit by two or three pursuers who isochronously capture the evader. The critical pursuers are identified. However, these geometric method-based pursuit and evasion strategies are optimal only in a part of the state space where a strategic saddle point is obtained and the value of the differential game is established. As such, these strategies are suboptimal. To fully explore the differential game’s high-dimensional state space and get a better understanding of group pursuit, numerical experimentation is undertaken. The state space region where the geometric solution of the group pursuit differential game is the optimal solution becomes larger the smaller the speed ratio parameter is.

In the Hilbert space l2, a differential evasion game involving multiple pursuers is considered. Integral constraints are imposed on player control functions. The pursuers are tasked with bringing the state of a system back to the origin of l2, while the evader simultaneously tries to avoid it. It is assumed that the energy of the evader is greater than the total energy of the pursuers. In this paper, we contribute to the solution of the differential evasion game with multiple pursuers by building an exact strategy for the evader.

We investigate a differential evasion game with multiple pursuers and an evader for the infinite systems of differential equations in ℓ2. 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 ℓ2, 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.

Consideration was given to the problem of optimal pursuit of one object by multiple objects. The player's moves obey the ordinary differential equations. Geometrical constraints are imposed on the player controls. Sufficient conditions were obtained for optimality of the pursuit time, and the optimal player strategies were constructed.

We study a differential evasion game of multiple pursuers and an evader governed by several infinite systems of two-block differential equations in the Hilbert space l2. Geometric constraints are imposed on the players’ control functions. If the state of a controlled system falls into the origin of the space l2 at some finite time, then pursuit is said to be completed in a differential game. The aim of the pursuers is to transfer the state of at least one of the systems into the origin of the space l2, while the purpose of the evader is to prevent it. A sufficient evasion condition is obtained from any of the players’ initial states and an evasion strategy is constructed for the evader.

In this paper, we extend the well-studied results of the two-pursuer, single-evader differential game to any number of pursuers. The main objective of this investigation is to exploit the benefits of cooperation amongst the pursuers in order to reduce the capture time of the evader. Computational complexity is a chief concern as this problem would need to be solved in an online fashion, e.g., in the case of autonomous unmanned aerial vehicles. A new geometric approach to solving the game is introduced and analyzed, which changes the problem of optimizing over continuous domains to a discrete combinatoric optimization. While past efforts at solving multiple pursuer problems have suffered from the curse of dimensionality, the geometric algorithms put forth here are shown to be scalable. Categorization and removal of redundant pursuers is the primary means by which scalability is achieved. The solution of this problem serves as a stepping stone to more complex problems such as the M-pursuer N-evader differential game.

This paper considers a pursuit‐evasion game with multiple pursuers and a superior evader. A novel cooperative pursuit strategy is proposed to capture a faster evader while maintaining a formation. First, the initial states including position distribution and the minimum required number of pursuers for ensuring capture are obtained based on the idea of Apollonius circle. Second, a cost function is designed, and the cooperative hunting strategy is developed using the distances between the centers of multiple Apollonius circles. Finally, numerical simulation and UAVs flying tests are provided to demonstrate the effectiveness of the proposed cooperative hunting strategy.

This paper investigates a multiplayer reach–avoid differential game in 3-dimensional (3D) space, which involves multiple pursuers, multiple evaders, and a designated target region. The evaders aim to reach the target region, while the pursuers attempt to guard the target region by capturing the evaders. This class of research holds significant practical value. However, the complexity of the problem escalates substantially with the growing number of players, rendering its solution extremely challenging. In this paper, the multiplayer game is divided into many subgames considering the cooperation among pursuers, reducing the computational burden, and obtaining numerically tractable strategies for players. First, the Apollonius sphere, a fundamental geometric tool for analyzing the 3D differential game, is formulated, and its properties are proved. Based on this, the optimal interception point for the pursuer to capture the evader is derived and the winning conditions for the pursuer and evader are established. Then, based on the Apollonius sphere, the optimal state feedback strategies of players are designed, and simultaneously, the optimal one-to-one pairings are obtained. Meanwhile, the Value function of the multiplayer reach–avoid differential game is explicitly given and is proved to satisfy Hamilton–Jacobi–Isaacs (HJI) equation. Moreover, the matching algorithm for the case with pursuers outnumbered evaders is provided through constructing a weighted bipartite graph, and the cooperative tactics for multiple pursuers are proposed, inspired by the Harris’ Hawks intelligent cooperative hunting tactics. Finally, numerical simulations are conducted to illustrate the effectiveness of the theoretical results for both cases where the number of adversary players is equal and unequal between the 2 groups.

Matching-based capture strategies for 3D heterogeneous multiplayer reach-avoid differential games

A modified continuous-time pursuit-evasion game with multiple pursuers and a single evader is studied. The game has been played in an obstacle-free convex environment which consists an exit gate through which the evader may escape. The geometry of the convex is unknown to all players except pursuers know the location of the exit gate and they can communicate with each other. All players have equal maximum velocities and identical sensing range. An evader is navigating inside the environment and seeking the exit gate to win the game. A novel sweep-pursuit-capture strategy for the pursuers to search and capture the evader under some necessary and sufficient conditions is presented. We also show that three pursuers are sufficient to finish the operation successfully. Non-holonomic wheeled mobile robots of the same configurations have been used as the pursuers and the evader. Simulation studies demonstrate the performance of the proposed strategy in terms of interception time and the distance traveled by the players.

In this article, a class of the relay pursuit–evasion problem is dealt with. Multiple pursuers and a single evader take part in the considered game. We propose a region-based relay pursuit scheme for the pursuers to capture a single evader that adopts the pure evasion strategy. First, a round pursuit strategy, which is a quickest decent control law, is proposed as an alternative strategy to the pure evasion strategy. Next, a partition of the planar space is given by using the Voronoi diagram and some circular barriers to indicate the dominance region of each pursuer and each pursuit strategy. Then, the region-based relay pursuit scheme is put forward. Finally, two simulation examples are provided to verify the effectiveness and superiority of the proposed methodology.

In this paper, a reachability-based approach is adopted to deal with the pursuit–evasion differential game between one evader and multiple pursuers in the presence of dynamic environmental disturbances (for example, winds or sea currents). Conditions for the game to be terminated are given in terms of reachable set inclusions. Level set equations are defined and solved to generate the forward reachable sets of the pursuers and the evader. The time-optimal trajectories and the corresponding optimal strategies are subsequently retrieved from these level sets. The pursuers are divided into active pursuers, guards, and redundant pursuers according to their respective roles in the pursuit–evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flowfields, with and without obstacles.

Plane Pursuit with Curvature Constraints