Controllability of a Stokes system with a diffusive boundary condition

Abstract

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation

“…In this section, we eliminate the last term in (48) by showing high frequency estimates for u and p. The method used here is the same as the one used in [11]. We conjugate the system (34) with e sϕ 0 , using that the spatial derivatives of ϕ 0 involve only powers of µ 0 that is fixed, instead of powers of λ for the spatial derivatives of ϕ.…”

“…Let us point out that due to the structural damping in the beam equation (−∂ t ∂ 2 x 1 ζ) the corresponding beam equation becomes a parabolic equation (see, for instance, [14]). In a previous work [11], we have replaced the damped beam equation by a heat equation and we have shown the corresponding controllability result. The proof done here is inspired by our previous work, and in particular, in the proof of the observability, we first apply results on the heat equations to the fluid velocity by considering the pressure as a source term, (in the spirit of [18]).…”

“…One of the main differences with [11] is that we work here directly with the time variable whereas in the previous work we show a spectral inequality and then use an abstract method ( [31,32]) to deduce the corresponding observability inequality. Here we do not follow the same approach since it uses that the main operator of our system is self-adjoint, and here our main operator is not self-adjoint or even a perturbation of a self-adjoint operator as in the framework considered in [31].…”

“…Remark 3. With respect to [11] or to the stabilization result [1], one should expect to obtain the controllability of (2) or of ( 5) without any control on the beam equation (g ≡ 0). However, with our present approach, it seems difficult to handle the elastic displacement without any observation on the beam equation.…”