Accuracy Comes at a Cost: Optimal Localisation Against a Flow

TL;DR

This paper investigates the optimal control strategies for a self-propelled particle to maintain localization near a target in a flow, balancing accuracy and energetic cost through time-dependent propulsion and diffusivity.

Contribution

It introduces a framework for optimizing particle control without feedback, revealing a trade-off between accuracy and energy expenditure, and highlights the role of time-dependent diffusivity.

Findings

Optimal protocols involve switching between passive and active states.

Accuracy beyond a threshold requires active propulsion, increasing cost.

Time-dependent diffusivity improves control efficiency.

Abstract

How much work does it cost for a propelled particle to stay localised near a stationary target, defying both thermal noise and a constant flow that would carry it away? We study the control of such a particle in finite time and find optimal protocols for time-dependent swim velocity and diffusivity, without feedback. Accuracy, quantified via the mean squared deviation from the target, and energetic cost turn out to be related by a trade-off, which complements the one between precision and cost known in stochastic thermodynamics. We show that accuracy better than a certain threshold requires active driving, which comes at a cost that increases with accuracy. The optimal protocols have discontinuous swim velocity and diffusivity, switching between a passive drift state with vanishing diffusivity and an active propulsion state. This study highlights how a time-dependent diffusivity enhances optimal control and sets benchmarks for cost and accuracy of artificial self-propelled particles navigating noisy environments.