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

Abstract

Consider a rigid body $\mathcal{S}\subset {\mathbb{R}}^3$ immersed in an infinitely extended Navier–Stokes fluid. We are interested in self-propelled motions of $\mathcal{S}$ 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 $\partial \mathcal{S}$. If the velocity $V$ of $\mathcal{S}$ 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 $\mathrm{Supp}{v}_{⁎}\subset \mathrm{\Gamma }$, an arbitrary nonempty open subset of $\partial \Omega$, or $v_{⁎}\cdot n{|}_{\partial \mathrm{\Omega }}=0$. We also show that one of the self-propelled conditions implies a better summability of the fluid velocity.