Solvable models of learning to pursue a moving target

Agents tasked with intercepting a moving target must learn how to reach it, ideally as efficiently as possible. As observers of animal pursuit behavior have noticed, different strategies are possible: on one extreme, the agent reactively moves toward the target's current location; on the other, the agent predicts the target's future location and moves directly there. Motivated by the desire to understand how such strategies might be learned, we introduce a continuous-time linear-quadratic model of open-loop target pursuit whose optimal control strategies interpolate between these possibilities. Usefully, a small number of interpretable parameters control which type of strategy is optimal, and it is possible to derive closed-form solutions for optimal strategies, policy learning dynamics, and value learning dynamics. Exploiting our model's linear structure, we find that the time scales of learning precisely correspond to the eigenvalues of certain matrices, and that relevant eigenvalue spectra generically have a gap. We show that this gap indicates that agents tend to learn how to reach their target before optimizing their movement. Our results provide a detailed mathematical characterization of pursuit behavior and its learning dynamics, which can both serve as a benchmark for empirical work, and as a foundation for more elaborate theoretical treatments of pursuit.