  Copepod nauplii are larval crustaceans with important ecological functions.
Due to their small size, they experience an environment of low Reynolds number
within their aquatic habitat. Here we provide a mathematical model of a
swimming copepod nauplius with two legs moving in a plane. This model allows
for both rotation and two-dimensional displacement by periodic deformation of
the swimmer's body. The system is studied from the framework of optimal control
theory, with a simple cost function designed to approximate the mechanical
energy expended by the copepod. We find that this model is sufficiently
realistic to recreate behavior similar to those of observed copepod nauplii,
yet much of the mathematical analysis is tractable. In particular, we show that
the system is controllable, but there exist singular configurations where the
degree of non-holonomy is non-generic. We also partially characterize the
abnormal extremals and provide explicit examples of families of abnormal
curves. Finally, we numerically simulate normal extremals and observe some
interesting and surprising phenomena.
