Interception Conditions in Multi-Agent Pursuit-Evasion Game with Integral and Geometric Constraints on Control Functions

In this paper, the authors carefully examine a pursuit-evasion differential game problem involving multiple
number of pursuers and one evader in the Hilbert space 𝑙2 . Players' motions obey certain first-order differential
equations. Control functions of the pursuers and evader are subject to integral and geometric constraints
respectively. The payoff function is the infimum of the distance between evader and closest pursuer when the
game comes to an end at a fixed time 𝜃 which is termed as duration of the game. During the course of the game,
pursuers try to minimize the payoff and the evader tries to do the opposite. The authors obtain the value of the
game and construct optimal strategies of the players. The results are established through the construction of
attainability domains and the solution of an auxiliary game involving one pursuer and one evader. Conditions are
obtained under which the pursuer can guarantee equality of the states at the terminal time, and these conditions
are then extended to the multi-pursuer setting. A positive number 𝜙 is introduced, defined as the infimum of all
shifts 𝑙 for which the evader's attainability ball is covered by the union of the pursuers' shifted attainability balls.
Under a geometric condition on the initial positions, this number is shown to coincide with the value of the game.
An illustrative example in 𝑙2 is presented to demonstrate the applicability of the established results.

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