An existence and uniqueness result for the motion of self-propelled micro-swimmers

We present an analytical framework to study the motion of microswimmers in a viscous fluid. Our main result is that, under very mild regularity assumptions, the change of body shape uniquely determines the motion of the swimmer. We assume that the Reynolds number is very small, so that the velocity field of the surrounding infinite fluid is governed by the Stokes system and all inertial effects can be neglected. Moreover, we enforce the self propulsion constraint (no external forces and torques). Therefore, Newton\u2019s equations of motion reduce to the vanishing of the viscous force and torque acting on the body. By exploiting an integral representation of viscous force and torque, the equations of motion can be reduced to a system of six ordinary differential equations. Variational techniques are used to prove the boundedness and measurability of this system\u2019s coefficients, so that classical results on ordinary differential equations can be invoked to prove existence and uniqueness of the solution

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

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