On the exact controllability of Kirchhoff plates (Q1177289)

The Kirchoff plate is the system \[ \begin{aligned} y''-h^ 2\Delta y''+\Delta^ 2 y=0 &\text{ in }\Omega\times(0,T) \\ y(T)=y'(T)=0 &\text{ in }\Omega \\ y=\Delta y=0 &\text{ on }\Gamma_ -\times(0,T) \\ y=0\text{ and }\Delta y=v &\text{ on }\Gamma_ +\times(0,T) \end{aligned} \] \(\Omega\) is a bounded domain in \(\mathbb{R}^ n\) with boundary \(\Gamma\) of class \(C^ 4\). At \(x\in \Gamma\), the outward unit normal is \(\nu(x)\) and \(x\in \Gamma_ +\) if \(x\cdot\nu(x)>0\) else \(x\in \Gamma_ -\). (\(k\), \(T>0\) are constants.) It is proven that all initial states \((y^ 0,y^ 1)\in (H^ 2(\Omega)\cap H_ 0^ 1(\Omega))\times H_ 0^ 1(\Omega)\) are exactly null-controllable if \(T>k\cdot\sup\{\| x\|: x\in \Omega\}\). The proof is based on the Hilbert uniqueness method of J.-L. Lions, together with some ideas of G. Lebeau, A. Haraux, which are combined in an abstract theorem (\S 2).