Pursuit–evasion problems with a “life line” under repulsive forces

B. T. Samatov^a, U. B. Soyibboev<sup>b

a</sup> Andijan State University, 129 University str., Andijan, 170100 Republic of Uzbekistan
^b V.I. Romanovsky Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan, 9 University str., Tashkent, 100170 Republic of Uzbekistan

Abstract: This paper studies pursuit–evasion differential games and a game with a “life line” for the case where the inertial movements of the players are realized using controls subject to repulsive forces. For solving the pursuit problem and the problem with a “life line”, the main tool remains the parallel pursuit strategy (in short, the $\mathbf{\Pi}$-strategy). With the help of this $\mathbf{\Pi}$-strategy, necessary and sufficient conditions for the solvability of the pursuit problem are obtained, and the capture set or the players’ reachability set is constructed. For solving the problem with a “life line” in favor of the pursuer, a monotone (with respect to set inclusion) decrease over time of the players’ reachability set is proved. In solving the evasion problem, lower bounds for the distances between the players are obtained, and for a game with a “life line” in this case a set is constructed that the evading player can reach without being captured, under arbitrary control of the pursuer. The obtained results are illustrated by representative examples.

Keywords: differential game, pursuit, evasion, geometric constraint, $\mathbf{\Pi}$-strategy, evasion strategy, players’ reachability set, life line.

UDC: 517.977

Received: 18.09.2024
Revised: 30.11.2024
Accepted: 18.12.2024

DOI: 10.26907/0021-3446-2026-1-52-71