In this paper, the multiagent pursuit-evasion (MPE) games are solved in order to obtain optimal strategic policies for all players. In these games, multiple pursuers attempt to intercept multiple evaders who try to avoid capture. A graph-theoretic approach is employed to study the interactions of the agents with limited sensing capabilities, such that distributed control policies are obtained for every agent. Furthermore, the minimization of performance indices associated with the goals of the agents is guaranteed. Nash equilibrium among the players is obtained by means of optimal policies that use the solutions of the Hamilton–Jacobi–Isaacs (HJI) equations of the game. Minmax strategies are also proposed to guarantee a security-level performance when the solutions of the HJI equations for Nash equilibrium do not exist. Scenarios for finite-time capture and for asymptotic rendezvous are analyzed, and emergent behaviors are obtained by means of modifications of the proposed general-case performance indices. The containment control results are shown to be special cases of the solutions of the MPE games.

Solutions for Multiagent Pursuit-Evasion Games on Communication Graphs: Finite-Time Capture and Asymptotic Behaviors

If a pursuit game with many persons can be formalized in the framework of zero-sum differential games, then general methods can be applied to solve it. But difficulties arise connected with very high dimension of the phase vector when there are too many objects. Just due to this problem, special formulations and methods have been elaborated for conflict interaction of groups of objects. This paper is a survey of publications and results on group pursuit games.

Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications

This paper addresses a particular pursuit-evasion game, called as “fishing game” where a faster evader attempts to pass the gap between two pursuers. We are concerned with the conditions under which the evader or pursuers can win the game. This is a game of kind in which an essential aspect, barrier, separates the state space into disjoint parts associated with each player’s winning region. We present a method of explicit policy to construct the barrier. This method divides the fishing game into two subgames related to the included angle and the relative distances between the evader and the pursuers, respectively, and then analyzes the possibility of capture or escape for each subgame to ascertain the analytical forms of the barrier. Furthermore, we fuse the games of kind and degree by solving the optimal control strategies in the minimum time for each player when the initial state lies in their winning regions. Along with the optimal strategies, the trajectories of the players are delineated and the upper bounds of their winning times are also derived.

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Construction of Barrier in a Fishing Game With Point Capture

Pursuit–evasion problems involving multiple pursuers and evaders are studied in this paper. The pursuers and the evaders are all assumed to be identical, and the pursuers are assumed to follow either a constant bearing or a pure pursuit strategy, giving rise to two distinct cases. The problem is simplified by adopting a dynamic divide and conquer approach, where at every time instant each evader is assigned to a set of pursuers based on the instantaneous positions of all the players. In this regard, the corresponding multi-pursuer single-evader problem is analyzed first. Assuming that the evader knows the positions of all the pursuers and their pursuit strategy, the time-optimal evading strategies are derived for both constant bearing and pure pursuit cases for the pursuers using tools from optimal control theory. In the case of a constant bearing strategy, and assuming that the evader can follow any strategy, a dynamic task allocation algorithm is proposed for the pursuers. The algorithm is based on the well-known Apollonius circle and allows the pursuers to allocate their resources in an intelligent manner while guaranteeing the capture of the evader in minimum time. For the case of pure pursuit, the algorithm is modified using the counterpart of the Apollonius circle leading to an “Apollonius closed curve.” Finally, the proposed algorithms are extended to assign pursuers in the case of a problem with multiple pursuers and multiple evaders. Numerical simulations are included to demonstrate the performance of the proposed algorithms.

Optimal Evading Strategies and Task Allocation in Multi-player Pursuit–Evasion Problems

Pursuit–evasion problems involving two pursuers and one evader are presented and analyzed. Two distinct scenarios differing in the type of information that the pursuers have about the evader’s stra...

Optimal Evading Strategies for Two-Pursuer/One-Evader Problems

An algorithm for numerical constructing level sets of the value function is shortly described for one class of linear differential games with fixed termination instant. Some model interception problems with one target and two interceptors are considered; all objects are weak maneuverable.

Level Sets of the Value Function in Differential Games with Two Pursuers and One Evader. Interval Analysis Interpretation

A three-dimensional reachable set “at the instant” for a controlled object called the Dubins car is considered. The turn angular velocity of the linear velocity vector is a control variable. Along with the case when, according to the statement of the problem, rotation is possible in both directions, the cases of one-sided rotation are studied. Three-dimensional images of reachable sets are presented. Sufficiency of the Pontryagin maximum principle conditions for the control leading to the boundary of the reachable set is analyzed. Possibility of using reachable sets in the problem of motion observation under conditions of inaccurate measurements of the geometric position is considered.

Reachable set for Dubins car and its application to observation problem with incomplete information

We study a model differential zero-sum game, which can be regarded as an idealized variant of the final stage of a space pursuit, in which two pursuing objects and one evader are involved. Results of numeric constructions of level sets of the value function for qualitatively different cases of the game parameters and results of simulation of optimal motions are presented.

Valerii S. Patsko

Papers

Zero-Sum Pursuit-Evasion Differential Games with Many Objects: Survey of Publications

Level Sets of the Value Function in Differential Games with Two Pursuers and One Evader. Interval Analysis Interpretation

Reachable set for Dubins car and its application to observation problem with incomplete information

Model Formulation of Pursuit Problem with Two Pursuers and One Evader