Controlling non-controllable scallops

Any swimmer embedded on a inertialess fluid must perform a non-reciprocal motion to swim forward. The archetypal demonstration of this unique motion-constraint was introduced by Purcell with the so-called "scallop theorem". Scallop here is a minimal mathematical model of a swimmer composed by two arms connected via a hinge whose periodic motion (of opening and closing its arms) is not sufficient to achieve net displacement. Any source of incongruence on the motion or in the forces/torques experienced by such time-reversible scallop will break the symmetry imposed by the Stokes linearity and lead to subsequent propulsion of the scallop. However, little is known about the controllability of time-reversible scalloping systems. Here, we consider two individually non-controllable scallops swimming together. Under a suitable geometric assumption on the configuration of the system, it is proved that non-zero net displacement can be achieved as a consequence of their hydrodynamic interaction. A detailed analysis of the control system of equations is carried out analytically by means of geometric control theory. We obtain an analytic expression for the the displacement after a prescribed sequence of controls in function of the phase difference of the two scallops. Numerical validation of the theoretical results is presented with model predictions in further agreement with the literature.