Abstract
The aim of this work is to show the local null controllability of a fluid–solid interaction system by using a distributed control located in the fluid. The fluid is modeled by the incompressible Navier–Stokes system with Navier slip boundary conditions and the rigid body is governed by the Newton laws. Our main result yields that we can drive the velocities of the fluid and of the structure to 0 and we can control exactly the position of the rigid body. One important ingredient consists in a new Carleman estimate for a linear fluid–rigid body system with Navier boundary conditions. This work is done without imposing any geometrical conditions on the rigid body.
Highlights
Let Ω be a bounded, non empty open subset of R2 with a regular boundary
We assume that Ω contains a rigid body and an incompressible viscous fluid
At each time t > 0, the domain of the rigid body is denoted by S(t) ⊂ Ω that is assumed to be compact with non empty interior and regular
Summary
Let Ω be a bounded, non empty open subset of R2 with a regular boundary. We assume that Ω contains a rigid body and an incompressible viscous fluid. Concerning controllability results of fluid-structure systems with Dirichlet boundary conditions, in dimension 2, we mention the paper [7], where the authors proved the null controllability in velocity and the exact controllability for the position of the rigid body assuming some geometric properties for the solid and provided that the initial conditions are small enough, more precisely a condition of smallness on the H3 norm of the initial fluid velocity is needed. We prove the local null controllability of the system (1.1), (1.3), (1.4), (1.5), that is the case of the Navier slip boundary conditions in the presence of a rigid structure of arbitrary shape.
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