Exact controllability of a fluid–rigid body system

This paper is devoted to the controllability of a 2D fluid–structure system. The fluid is viscous and incompressible and its motion is modelled by the Navier–Stokes equations whereas the structure is a rigid ball which satisfies Newton's laws. We prove the local null controllability for the velocities of the fluid and of the rigid body and the exact controllability for the position of the rigid body. An important part of the proof relies on a new Carleman inequality for an auxiliary linear system coupling the Stokes equations with some ordinary differential equations.