Optimal Control and Closed-Loop Stability of Droplet Transport in a Microchannel
Abstract
Understanding the efficient transport of fluid droplets in confined geometries has been a domain of interest for industrial applications in recent times. In the present work, we focus on designing control strategies that optimally steer droplet motion. Here, we apply optimal control framework to the droplet transport problem in a microchannel based on lubrication theory that minimizes viscous dissipation. Two complementary modeling routes are adopted: a reduced-order ordinary differential equation (ODE) model optimized via Pontryagin's Maximum Principle, and a full nonlinear partial differential equation (PDE) model optimized using a Covariance Matrix Adaptation-Evolutionary Strategy. By parameterizing target displacement, droplet size, and capillary number, we uncover two distinct optimal transport regimes: a "translate-relax" strategy for short distances and a "compact-translate-relax" strategy for longer targets. In continuum mechanics, the competition between surface forces and cumulative viscous dissipation decides the optimal transport strategies. We further show that the displacement range over which the reduced-order controller transfers to the continuum model is governed by capillary stiffness. We finally address closed-loop stability by adapting a control Lyapunov function (CLF) framework to the reduced-order dynamics. We demonstrate that the terminal cost which penalizes deviation from the target, serves as a cost-compatible CLF on the physical domain, and prove that the CLF-compatible feedback exponentially stabilizes the target state.
Related
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BibTeX
@misc{anand2026optimal,
title = {{Optimal Control and Closed-Loop Stability of Droplet Transport in a Microchannel}},
author = {Rajneesh Anand and Mayuresh V. Kothare},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19320},
eprint = {2609.19320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19320},
}
RIS
TY - UNPB
TI - Optimal Control and Closed-Loop Stability of Droplet Transport in a Microchannel
AU - Rajneesh Anand
AU - Mayuresh V. Kothare
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19320
UR - https://arxiv.org/abs/2609.19320
AB - Understanding the efficient transport of fluid droplets in confined geometries has been a domain of interest for industrial applications in recent times. In the present work, we focus on designing control strategies that optimally steer droplet motion. Here, we apply optimal control framework to the droplet transport problem in a microchannel based on lubrication theory that minimizes viscous dissipation. Two complementary modeling routes are adopted: a reduced-order ordinary differential equation (ODE) model optimized via Pontryagin's Maximum Principle, and a full nonlinear partial differential equation (PDE) model optimized using a Covariance Matrix Adaptation-Evolutionary Strategy. By parameterizing target displacement, droplet size, and capillary number, we uncover two distinct optimal transport regimes: a "translate-relax" strategy for short distances and a "compact-translate-relax" strategy for longer targets. In continuum mechanics, the competition between surface forces and cumulative viscous dissipation decides the optimal transport strategies. We further show that the displacement range over which the reduced-order controller transfers to the continuum model is governed by capillary stiffness. We finally address closed-loop stability by adapting a control Lyapunov function (CLF) framework to the reduced-order dynamics. We demonstrate that the terminal cost which penalizes deviation from the target, serves as a cost-compatible CLF on the physical domain, and prove that the CLF-compatible feedback exponentially stabilizes the target state.
ER -