Spectral inequality and optimal cost of controllability for the Stokes system (Q2835354)

The authors deal with a controllability for the Stokes system NEWLINE\[NEWLINEy_t-\Delta y +\nabla p=\mathbf{1}_\omega,\;\text{div}\,y=0\;\text{in}\,Q;\;y=0\;\text{on}\, \Sigma,\;y(0)=y_0\;\text{in}\,\varOmega,NEWLINE\]NEWLINE with \(\varOmega\subset \mathbb{R}^N\,(N\geq 2)\) a bounded connected open set, \(Q:=\varOmega \times (0,T),\;\Sigma:=\partial\varOmega \times (0,T),\;\nu\) the outward unit normal to \(\varOmega,\;\mathbf{H}=\{u\in L^2(\varOmega)^N;\;\text{div}\, u=0,\,u\cdot \nu =0\,\text{on}\,\partial\varOmega\}.\) The main result isNEWLINENEWLINETheorem. Let \(\omega\) be a nonempty subset of \(\varOmega.\) There exist constants \(C_1>0,\;C_2>0\) depending only on \(\varOmega,\;\omega\) such that for every \(T>0,\;y_0\in \mathbf{H}\) there exists a control \(f\in L^2(\omega\times (0,T))\) such that the associated solution of the Stokes system satisfies \(y(T)=0\) and the estimate \(\|f\|_{L^2(\omega\times (0,T))}\leq C_1e^{C_2/T}\|y_0\|_\mathbf{H}\) holds.