The control of particles in the Stokes limit

There are numerous ways to control objects in the Stokes regime, with
microscale examples ranging from the use of optical tweezers to the application
of external magnetic fields. In contrast, there are relatively few explorations
of theoretical controllability, which investigate whether or not refined and
precise control is indeed possible in a given system. In this work, seeking to
highlight the utility and broad applicability of such rigorous analysis, we
recount and illustrate key concepts of geometric control theory in the context
of multiple particles in Stokesian fluids interacting with each other, such
that they may be readily and widely applied in this largely unexplored
fluid-dynamical setting. Motivated both by experimental and abstract questions
of control, we exemplify these techniques by explicit and detailed application
to multiple problems concerning the control of particles, such as the motion of
tracers in flow and the guidance of one sphere by another. Further, we showcase
how this analysis of controllability can directly lead to the construction of
schemes for control, in addition to facilitating explorations of mechanical
efficiency and contributing to our overall understanding of non-local
hydrodynamic interactions in the Stokes limit.