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

### Ana Leonor Silvestre

CEMAT and Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal### Takéo Takahashi

Inria Nancy Grand-Est, 615 rue du Jardin Botanique, 54600 Villers-lès-Nancy, France, Institut Élie Cartan, UMR CNRS 7502, Université de Lorraine, Boulevard des Aiguillettes, B.P. 70239, 54506 Vandoeuvre-lès-Nancy, Cedex, France### Toshiaki Hishida

Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan

## Abstract

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

## Cite this article

Ana Leonor Silvestre, Takéo Takahashi, Toshiaki Hishida, A boundary control problem for the steady self-propelled motion of a rigid body in a Navier–Stokes fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 6, pp. 1507–1541

DOI 10.1016/J.ANIHPC.2016.11.003