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  • Open Access

Accuracy comes at a cost: Optimal localization against a flow

Till Welker* and Patrick Pietzonka†

  • *Contact author: t.a.welker@sms.ed.ac.uk
  • †Contact author: p.pietzonka@ed.ac.uk
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Phys. Rev. E 113, 064142 – Published 22 June, 2026

DOI: https://doi.org/10.1103/z6b5-s3z2

Abstract

How much work does it cost for a propelled particle to stay localized 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 tradeoff, 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. If the initial position is fixed, an initial jump of the particle, enabled by a sudden burst of propulsion, can be optimal. 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.

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This article appears in the following collection:

Controlling Stochastic Dynamics Across Scales

We present a Collection of papers on Controlling Stochastic Dynamics Across Scales. It seeks to highlight novel studies on controlling the dynamics of complex stochastic systems with a rich phenomenology. Guest editors of the Collection are Étienne Fodor of the University of Luxembourg and Todd Gingrich of Northwestern University.

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Phys. Rev. E 113, 064142– Published 22 June, 2026
  • Received 23 February 2026
  • Accepted 29 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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