Optimal multi-parameter control of trapped active matter

Abstract

The realization of efficient micro-machines built from active matter requires precise thermodynamic control far from equilibrium. Despite theoretical progress, the focus on single-parameter driving, coupled with strict theoretical assumptions, limits efforts to capture modern multi-parameter control experiments. Here, guided by careful theoretical considerations, we develop a transparent computational framework based on exact-gradient descent via automatic differentiation. We derive optimal protocols for a wide range of multi-parameter problems -- involving trap stiffness, trap center, and particle activity -- to minimize the thermodynamic work or heat. We demonstrate that smoothed, experimentally plausible protocols -- obtained by assigning kinetic costs to the controls -- achieve near-optimal efficiencies comparable to discontinuous ``bang-bang'' solutions. By exploring both open- and closed-loop control, we find the dynamical coupling between parameters leads to genuinely new strategies, including symmetry breaking in optimal activity cycles and non-monotonic trap stiffness controls. Further, we identify regimes where initial measurement and multi-parameter flexibility combine to improve efficiency. Finally, we reveal that the naive simultaneous execution of independently optimized controls incurs only slightly more work than the full multi-parameter solutions. Taken together, our work elucidates the non-equilibrium physics of multi-parameter control and provides robust, scalable strategies for controlling active matter.

Figures (13)

• Figure 1: Examples of the control scenarios explored in this paper. (i) The single-control problem, where only one of the controls changes as a function of time, with the strengthening of the stiffness shown as an example. (ii) The dual-control problem, where two controls are subject to time-dependent protocols, with the other control fixed. (iii) The tri-control scenario, with dual control of the trap coupled to a time-dependent activity protocol.
• Figure 2: Comparing our method to the exact results of Schmiedl and Seifert (2007)Schmiedl2007. A Work minimizing single parameter control of the trap stiffness as a function of the normalized protocol time $t/t_p$. Two cases of numerical optimal protocols are shown: one accounting a kinetic control cost $J_R > 0$ (regularized, \ref{['eq:JRCost']}) and the other have no cost on control speed $J_R = 0$. Due to the discrete-time nature of the protocol, chattering occurs in numerical optimal protocols found using the unregularized cost function. The stars denote inner ($0^+\leq t \leq t_p^-$) discontinuities in the protocol. B Total work for the predicted optimal stiffness protocols as a function of protocol duration. from Model parameters: $\alpha_1(t=0^-) = 1$, $\alpha_1(t=t_p^+) = 5$, $\alpha_1(t) \in \mathbb{R}$, $D=1$ and $\mu=1$. Optimization parameters: $M=500$, $m_\varepsilon = 10^{-4}$.
• Figure 3: Comparing our method to the exact results of Schüttler et. al. (2025) Schuttler2025.A Work minimizing--closed-loop--trap center protocols ($\alpha_2$) as a function of the normalized protocol time, for different initial measurements of the self-propulsion $v_0$. The fanning of the protocols resembles a "pirhana". Numerical curves are for the regularized work ($J_R >0$). B Values of the closed-loop total work as a function of $v_0$ arising for both the exact and numerical optimal protocols. Model parameters: $\alpha_2(t=0^-) = 0$, $\alpha_2(t=t_p^+) = 1$, $\alpha_2(t) \in \mathbb{R}$, $t_p = 3$, $\alpha_1=1$, $D'=1$, and $\alpha_3 = 0.525$. Optimization parameters: $M=500$ and $m_\varepsilon = 10^{-4}$.
• Figure 4: Dual open-loop control of the trap stiffness and center at minimal work.A Optimal stiffness protocols for various protocol durations, where the the stiffness is bounded by its end-point values. B Optimal center protocols with the protocols at fixed stiffness shown as an inset. C Control phase portrait for the protocols shown in (A,B). D Averaged total work for the optimal protocols in (A,B) compared to the open-loop single parameter (fixed $\alpha_1$) case. Model parameters: $\alpha_1(t=0^-) = 1$, $\alpha_1(t=t_p^+) = 5$, $\alpha_2(t=0^-) = 0$, $\alpha_2(t=t_p^+) = 1$, $\alpha_2(t) \in (0,1]$, $\alpha_1(t) \in (0,5]$, $t_p = 3$, $D'=2$, and $D=\mu=\alpha_3=1$. Optimization parameters: $M=1000$ and $m_\varepsilon = 10^{-4}$.
• Figure 5: Dual open-loop control of the trap stiffness and center at minimal heat dissipation.A Optimal protocols for $\alpha_1$ as a function of the normalized protocol time $t/t_p$, for the case of no enforcement of a state-to-state transformation (No STS). B Optimal protocols for $\alpha_2$. Inset shows optimal $\alpha_2$ protocols at fixed trap stiffness. C Control phase portrait for the protocols shown in (A,B). D Averaged total heat for the optimal protocols, showing both the dual control and the single control costs. Open-loop dual control becomes more efficient at slower driving. E-H Same as (A-D) but for the case of enforcing an STS transformation. Model parameters: $\alpha_1(t=0^-) = 1$, $\alpha_1(t=t_p^+) = 5$, $\alpha_2(t=0^-) = 0$, $\alpha_2(t=t_p^+) = 1$, $\alpha_2(t) \in (0,1]$, $\alpha_1(t) \in (0,5]$, $t_p = 3$, $D'=2$, and $D=\mu=\alpha_3=1$. Optimization parameters: $M=1000$ and $m_\varepsilon = 10^{-4}$.