Asymptotic behavior and control of a "guidance by repulsion" model

TL;DR

This paper studies a herding control model where drivers influence evaders' trajectories through repulsion, analyzing its long-term behavior, controllability, and developing strategies for effective guidance in multi-agent scenarios.

Contribution

It extends the guidance-by-repulsion model to multi-driver and multi-evader cases, providing well-posedness, long-term analysis, and feedback control strategies.

Findings

Exact controllability in a long time horizon for one-driver, one-evader case.

Drivers adopt a pattern of positioning behind the target for effective control.

A feedback strategy stabilizes evaders' direction, enhancing control effectiveness.

Abstract

We model and analyze a herding problem, where the drivers try to steer the evaders' trajectories while the evaders always move away from the drivers. This problem is motivated by the guidance-by-repulsion model [Escobedo, R., Iba\~nez, A. and Zuazua, E. COMMUN NONLINEAR SCI 39 (2016) 58-72], where the authors answer how to control the evaders' positions and what is the optimal maneuver of the drivers. First, we obtain the well-posedness and the long-time behavior of the one-driver and one-evader model, assuming of the same friction coefficients. In this case, the exact controllability is proved in a long enough time horizon. We extend the model to the multi-driver and multi-evader case, and develop numerical simulations to systematically explore the nature of controlled dynamics in various scenarios. The optimal strategies turn out to share a common pattern to the one-driver and one-evader case: the drivers rapidly occupy the position behind the target, and want to pursuit evaders in a straight line for most of the time. Inspired by this, we build a feedback strategy which stabilizes the direction of evaders.