An optimal control approach to ciliary locomotion

We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.