  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.
