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

Lions, Jacques-Louis; Zuazua, Enrique

ESAIM, Control Optim. Calc. Var. 1, 1-15 (1995).

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.

Reviewer: A.Akutowicz (Berlin)

MSC:

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

Keywords:

Hydroelasticity; approximate controllability; coupled system; spectral decomposition; generic spectral properties; Stokes system

References:

[1]	J.H. Albert: Genericity of simple eigenvalues for elliptic pde’s, Proc. AMS, 48(2), 1975, 413-418. Zbl0302.35071 MR385934 · Zbl 0302.35071
[2]	H. Cohen and S. I. Rubinow: Some mathematical topics in Biology, Proc. Symp. on System Theory Polytechnic Press, New York ( 1965), 321-337.
[3]	C. Fabre, J. P. Puel and E. Zuazua: Contrôlabilité approchée de l’équation de la chaleur semilinéaire, C.R.A.S. Paris, 315, ( 1992), 807-812. Zbl0770.35009 MR1184907 · Zbl 0770.35009
[4]	O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow, , Gordon and Breach Science Publishers, New York, 1987. Zbl0184.52603 MR254401 · Zbl 0184.52603
[5]	H. Lamb: Hydrodynamics, 6th ed., Cambridge Univ. Press, 1932. Zbl0828.01012 MR1317348 JFM58.1298.04 · Zbl 0828.01012
[6]	J.L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. Zbl0189.40603 MR259693 · Zbl 0189.40603
[7]	J.L. Lions: Remarks on approximate controllability, I. Analyse Math., 59, 1992, 103-116. Zbl0806.35101 MR1226954 · Zbl 0806.35101
[8]	J.L. Lions and E. Magenes: Problèmes aux limites non homogènes et applications, vol. 1, Dunod, Paris, 1968. Zbl0165.10801 MR247243 · Zbl 0165.10801
[9]	A.M. Micheletti: Perturbazione dello spettro dell operatore di Laplace, in relazione ad una variazione del campo, Ann. Scuola Norm. Sup. Pisa, 26(3), 1972, 151-169. Zbl0234.35073 MR367480 · Zbl 0234.35073
[10]	K. Uhlenbeck: Generic properties of eigenfunctions, American J. Math., 98(4),1 1976, 1059-1078. Zbl0355.58017 MR464332 · Zbl 0355.58017