One-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls

TL;DR

This paper models one-dimensional viscous fluid swimmers, proving their controllability, existence and uniqueness of solutions, and establishing the existence of energy-minimizing shape change strategies for their motion.

Contribution

It introduces a mathematical framework for controlling and optimizing the motion of 1D swimmers in viscous fluids, including constructive motion planning methods.

Findings

Proved existence and uniqueness of solutions for shape-driven motion equations.

Established controllability between any two states via shape change sequences.

Demonstrated the existence of energy-minimizing control strategies.

Abstract

In this paper we study a mathematical model of one-dimensional swimmers performing a planar motion while fully immersed in a viscous fluid. The swimmers are assumed to be of small size, and all inertial effects are neglected. Hydrodynamic interactions are treated in a simplified way, using the local drag approximation of resistive force theory. We prove existence and uniqueness of the solution of the equations of motion driven by shape changes of the swimmer. Moreover, we prove a controllability result showing that given any pair of initial and final states, there exists a history of shape changes such that the resulting motion takes the swimmer from the initial to the final state. We give a constructive proof, based on the composition of elementary maneuvers (straightening and its inverse, rotation, translation), each of which represents the solution of an interesting motion planning problem. Finally, we prove the existence of solutions for the optimal control problem of finding, among the histories of shape changes taking the swimmer from an initial to a final state, the one of minimal energetic cost.