Accuracy Comes at a Cost: Optimal Localisation Against a Flow

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.

Figures (5)

• Figure 1: System and trade-off.\ref{['Fig:schematic']}, Swimmer with propulsion velocity $v(t)$ and diffusivity $D(t)$ in a flow of velocity $-u$ trying to stay close to a target at $x=0$. \ref{['Fig:pareto']}, Optimal trade-off between work $W$ and inaccuracy $V$ represented by the Pareto front for static (orange line) and for dynamic (green line) protocols for $v(t)$ and $D(t)$. The trade-off parameter $\alpha$ controls the emphasis on reducing $W$ during optimisation. The shaded area represents the region accessible for suboptimal protocols. The naive protocol (orange dashed line) describes swimmers staying on average on the target. Protocols with dynamic velocity but fixed diffusivity lead to the Pareto front shown as the green dotted line.
• Figure 2: Static Protocol.\ref{['Fig:constant_velocity']}, Optimal swim velocity and, \ref{['Fig:constant_diffusivity']}, optimal diffusivity as a function of the trade-off parameter $\alpha$. \ref{['Fig:constant_schematic']}, schematic illustration of expensive $\bigstar$, intermediate $\bullet$ and free $\blacksquare$ protocols. The solid black particles indicate the mean at the end of the protocol, while the transparent black particles illustrate the diffusive spread. The different sizes of particles and flagella represent the changed mobility (diffusivity) and swim velocity, respectively.
• Figure 3: Dynamic protocol.\ref{['Fig:optimal_protocol']}, Optimal protocol for trade-off parameter $\alpha = 0.001$. The grey line is the optimal mean $\bar{x}^*(t)$, the grey area is the standard deviation caused by the optimal diffusivity $D^*(t)$, and the golden line is the target. The drifting phase with zero mobility is coloured in blue and illustrated by a bacterium with spread flagella. The swimming phase is coloured in white and illustrated by a bacterium with bundled flagella. \ref{['Fig:glue_times']}, Optimal switching times $t_1$ and $t_2$ over trade-off parameters $\alpha$. When $\alpha$ is increased, the cost is reduced by shortening the swimming period. At the critical value $\alpha_\mathrm{c}$, the start and end of the swim period meet at $t_\mathrm{c} = 2/3$. Beyond $\alpha_\mathrm{c}$, it is optimal to always drift along.
• Figure 4: Dynamic velocity and fixed diffusivity.\ref{['Fig:fixedD_optimal_D']}, Optimal diffusivity $D^*$ as a function of the trade-off parameter $\alpha$. \ref{['Fig:fixedD_trajectory']}, Optimal protocol with dynamic velocity $v(t)$ but constant diffusivity $D$ for $\alpha =0.001$. As in \ref{['Fig:optimal_protocol']}, the grey line is the optimal mean, the grey area is the standard deviation caused by the optimal diffusivity, and the golden line is the target.
• Figure 5: Hamiltonian Description.\ref{['Fig:hamilton_potential']}, The Lagrangian optimisation problem can be mapped to a ball with momentum $p' = \bar{x}$ in a time-dependent potential. The linear part of the potential corresponds to the drift protocol, and the non-linear region between the linear region and the location of the maximal potential corresponds to the swim protocol. The abyss beyond drift and swim protocols results in divergences for $\epsilon \to 0$. \ref{['Fig:hamilton_trajectory']}, Swim, drift, and abyss region. The black line is the solution to the Hamiltonian problem with $p'(0)=x_0$ and $q'(1)=0$ (for which $q'(0)\approx0.00357$). The solution is sensitive to initial conditions, as shown by the red curves, generated with $q'(0)=0.00355,0.00360$. Here we chose $\alpha = 0.001$ and $\epsilon =0.1$ and integrate the dynamics using the Julia package DifferentialEquationsrackauckas2017differentialequations.