In this paper we study a mathematical model of one-dimensional swimmersperforming a planar motion while fully immersed in a viscous fluid. Theswimmers are assumed to be of small size, and all inertial effects areneglected. Hydrodynamic interactions are treated in a simplified way, using thelocal drag approximation of resistive force theory. We prove existence anduniqueness of the solution of the equations of motion driven by shape changesof the swimmer. Moreover, we prove a controllability result showing that givenany pair of initial and final states, there exists a history of shape changessuch that the resulting motion takes the swimmer from the initial to the finalstate. We give a constructive proof, based on the composition of elementarymaneuvers (straightening and its inverse, rotation, translation), each of whichrepresents the solution of an interesting motion planning problem. Finally, weprove the existence of solutions for the optimal control problem of finding,among the histories of shape changes taking the swimmer from an initial to afinal state, the one of minimal energetic cost.
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