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Micron-scale swimmers move in the realm of negligible inertia, dominated byviscous drag forces. In this paper, we formulate the leading-order dynamics ofa slender multi-link (N-link) microswimmer assuming small-amplitude undulationsabout its straight configuration. The energy-optimal stroke to achieve a givenprescribed displacement in a given time period is obtained as the largesteigenvalue solution of a constrained optimal control problem. Remarkably, theoptimal stroke is an ellipse lying within a two-dimensional plane in the(N-1)-dimensional space of joint angles, where N can be arbitrarily large. Forlarge N, the optimal stroke is a traveling wave of bending, modulo edgeeffects. If the number of shape variables is small, we can consider the sameproblem when the prescribed displacement in one time period is large, and notattainable with small variations of the joint angles. The fully nonlinearoptimal control problem is solved numerically for the cases N=3 (Purcell'sthree-link swimmer) and N=5 showing that, as the prescribed displacementbecomes small, the optimal solutions obtained using the small-amplitudeassumption are recovered. We also show that, when the prescribed displacementsbecome large, the picture is different. For N=3 we recover the non-convexplanar loops already known from previous studies. For N=5 we obtain non-planarloops, raising the question of characterizing the geometry of complexhigh-dimensional loops.

Energy-optimal strokes for multi-link microswimmers: Purcell's loops and Taylor's waves reconciled