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

TitleOne-dimensional swimmers in viscous fluids: dynamics, controllability, and existence of optimal controls
Publication TypeJournal Article
Year of Publication2013
AuthorsDal Maso, G, DeSimone, A, Morandotti, M
Abstract

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

URLhttp://hdl.handle.net/1963/6511

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