A boundary control problem for the steady self-propelled motion of a rigid body in a Navier-Stokes fluid

Consider a rigid body S⊂R3 immersed in an infinitely extended Navier-Stokes fluid. We are interested in self-propelled motions of S in the steady state regime of the system rigid body-fluid, assuming that the mechanism used by the body to reach such a motion is modeled through a distribution of velocities v⁎ on ∂S. If the velocity V of S is given, can we find v⁎ that generates V? We show that this can be solved as a control problem in which v⁎ is a six-dimensional control such that either Suppv⁎⊂Γ, an arbitrary nonempty open subset of ∂Ω, or v⁎⋅n|∂Ω=0. We also show that one of the self-propelled conditions implies a better summability of the fluid velocity.