Guaranteed Pursuit in a Two-Coupled Infinite-Dimensional System with Geometric Constraints on Controls

A differential pursuit game in the image space (the space of square-summable sequences) was modelled and analysed. Two players, a pursuer and an evader, controlled the state of a dynamical system governed by an infinite system of two coupled first-order ordinary differential equations. Both players were subject to geometric constraints on their control inputs, with the pursuer possessing a greater control resource bound. The pursuer was considered to have won the game by transferring the system state from an initial configuration to a predetermined target state (not necessarily the zero state) within a finite time interval, despite the evader’s resistance. The previously unexplored scenario in which the system parameters were nonnegative, resulting in unstable free dynamics, was considered. To address this challenge, an auxiliary control problem was first solved by constructing an admissible control that achieved the required state transfer. This solution was subsequently adapted to formulate an explicit admissible pursuit strategy that guaranteed completion of the pursuit. A sharp sufficient condition for pursuit completion was derived that, unlike its counterpart in the stable case, depended explicitly on the system parameters, representing a qualitative shift arising directly from the unstable dynamics. A formula for the guaranteed pursuit time was also established. A concrete example, accompanied by graphical illustrations of the convergence of the system state to the target and the behaviour of the guaranteed pursuit time, provided numerical validation of the analytical results. This work addressed a significant gap in the literature by extending pursuit game theory to unstable infinite-dimensional systems with nonzero target states. Potential applications included control design for distributed-parameter systems, robotics, and autonomous systems.

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