Approximate controllability of the Stokes system on cylinders by external unidirectional forces
The paper studies the approximate controllability of the Stokes system in \((0,T) \times \Omega\) \[ {\mathbf y}_t -\Delta {\mathbf y} +\nabla p=u \chi_\omega {\mathbf e}, \quad \text{div} {\mathbf y} =0, \quad {\mathbf y} (0,\cdot)=\mathbf{0}, \quad {\mathbf y} |_{\partial \Omega}=\text\textbf{0}. \] Here, \({\mathbf y} (t,x)= (y_1, y_2, y_3)\) and \(p\) denote the flow velocity and the pressure of the flow, respectively; \(\Omega\) is a cylinder \(G\times R^1 \subset {\mathbf R}^3\) with \(G\) a bounded domain in \({\mathbf R}^2\); \(\chi_\omega\) the characteristic function of a subdomain \(\omega \subset \Omega\); \({\mathbf e}\) a given vector in \({\mathbf R}^3\); and \(u=u(t,x)\) a control function. An uncontrollability result is first shown when \(\omega\) is the periodic extension (with respect to the variable \(x_3)\) of an open set of \(\Omega \times(0,L_3)\); \({\mathbf e} =(0,0,1)\); and the solution \({\mathbf y}\) is sought within the \(L_3\)-periodic functions with respect to \(x_3\) with the controls \(u\) \(L_3\)-periodic too. Second, approximate controllability is shown in the space \({\mathcal H}^0\) when \(G\) is a rectangle; and \({\mathbf e}= {\mathbf e}_1= (1,0,0)\), where \({\mathcal H}^0\) is the closure of the set of functions \(w\in C^\infty_0 (\Omega)\) with \(\text{div} {\mathbf w} =0\) in the space \((L^2 (\Omega))^3\).
scientific article; zbMATH DE number 1022295