  Micron-scale swimmers move in the realm of negligible inertia, dominated by
viscous drag forces. In this paper, we formulate the leading-order dynamics of
a slender multi-link (N-link) microswimmer assuming small-amplitude undulations
about its straight configuration. The energy-optimal stroke to achieve a given
prescribed displacement in a given time period is obtained as the largest
eigenvalue solution of a constrained optimal control problem. Remarkably, the
optimal 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. For
large N, the optimal stroke is a traveling wave of bending, modulo edge
effects. If the number of shape variables is small, we can consider the same
problem when the prescribed displacement in one time period is large, and not
attainable with small variations of the joint angles. The fully nonlinear
optimal control problem is solved numerically for the cases N=3 (Purcell's
three-link swimmer) and N=5 showing that, as the prescribed displacement
becomes small, the optimal solutions obtained using the small-amplitude
assumption are recovered. We also show that, when the prescribed displacements
become large, the picture is different. For N=3 we recover the non-convex
planar loops already known from previous studies. For N=5 we obtain non-planar
loops, raising the question of characterizing the geometry of complex
high-dimensional loops.
