Guaranteed Pursuit and Evasion Times in Hilbert Space Differential Games

Abstract

We analyze a differential game involving two competitors - a pursuing player and an evading player - who control the state of a dynamic system evolving in the Hilbert space ℓ₂. The system dynamics are governed by an infinite system of first-order ordinary differential equations (ISFODE), and the players’ controls are geometrically constrained, with the pursuing player holding a significant control resource advantage over the evading player. The pursuing player aims to drive the system’s state to the zero state of ℓ₂ after some time, whereas the evading player’s objective is to keep the state away from the zero state indefinitely. We derive an explicit
expression for the guaranteed pursuit time and construct a strategy that ensures pursuit completion. Similarly, we establish a formula for the guaranteed evasion time and identify a corresponding evasion strategy. Moreover, we provide necessary and sufficient conditions on the players’ control resources limit under which these guaranteed times exist.