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Approximate controllability of a hydro-elastic coupled system. (English) Zbl 0878.93034

The authors prove approximate controllability for the Stokes system controlled by a moving boundary \[ \begin{aligned} u_t-\mu \Delta u= -\text{grad} (p) \quad & \text{in } \Omega \times(0,T) \\ \text{div} (u)=0 \quad & \text{in } \Omega \times(0,T) \\ u=0 \quad & \text{on } \Gamma_1\times (0,T) \\ u= \varphi_tn \quad & \text{on } \Gamma_0 \times (0,T) \\ \varphi_{tt} +\Delta^2_{\Gamma_0} \varphi+ \mu n. {\partial u\over \partial n} =h+p+c \quad & \text{in }\Gamma_0\times(0,T)\\ \int_{\Gamma_0}\varphi_td\Gamma_0=0 \quad & \text{for } t\in(0,T) \\ \varphi= \Delta_{\Gamma_0} \varphi=0 \quad & \text{on } \partial \Gamma_0 \times (0,T) \\ u(0)=0 \quad & \text{in } \Omega \\ \varphi(0) =\varphi_t(0) =0 \quad & \text{on } \Gamma_0. \end{aligned} \] Here, \(\Omega \subset \mathbb{R}^3\) is a bounded domain such that \(\partial \Omega= \Gamma_0\cup \Gamma_1\) is analytic. \(\varphi\) is the displacement of the controlling boundary \(\Gamma_0\) along the normal \(n\). \(p\) is the pressure, \(h\in L^2 (\Gamma_0 \times (0,T))\) is the control and \(c\) is a function of time only. \(\Delta_{\Gamma_0}\) is the Laplace-Beltrami operator on \(\Gamma_0\). It is assumed that the spectrum of the Laplacian \(\Delta\) in \(H^1_0 (\Omega)\) is simple.
The idea of the proof consists in reducing the controllability problem to a uniqueness result concerning the solution to the Stokes equation, via an associated evolution problem (which reexpresses the parabolic and hyperbolic coupled systems together) whose solution’s existence and uniqueness are justified by the construction of a Galerkin basis, in this way, the dynamics on \(\Gamma_0\) is handled. Next, the Stokes system is expressed in an eigenbasis; the eigenpressures and the eigenfunctions vanish on \(\partial\Omega\). Using analyticity of \(\partial\Omega\), and the fact that the pressures are harmonic functions, one obtains that they vanish on \(\Omega\). This allows, using the simplicity of the spectrum of the Laplacian, to show that the eigenfunctions vanish on \(\Omega\).
An example when \(\Omega\) is a ball shows that the uniqueness result for the Stokes system does not hold when one drops the assumption concerning the spectrum of the Laplacian.

MSC:

93C20 Control/observation systems governed by partial differential equations
93B05 Controllability
76D07 Stokes and related (Oseen, etc.) flows
35Q30 Navier-Stokes equations
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References:

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