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Boundary controllability of a stationary Stokes system with linear convection observed on an interior curve. (English) Zbl 0958.93010

Let \(\Omega\) be an \(m\)-dimensional domain with boundary \(\Gamma,\) \(\Gamma_0 \subseteq \Gamma,\) \(L^2(\Gamma_0)^m_*\) the subspace of \(L^2(\Gamma_0)^m\) defined by the condition \(\int_{\Gamma_0} v \cdot \eta d\sigma = 0\) \((\eta\) the outer normal vector). The system for the state \((y, p) = (y(v), p(v))\) is \[ - \Delta y_j + {\partial \over \partial x_j} (a_j y_j) = f_j - {\partial p \over \partial x_j}, \quad \text{div} y = 0 \quad \text{in }\Omega, \]
\[ y= 0 \quad \text{on } \Gamma \setminus \Gamma_0, \quad y = v \quad \text{on } \Gamma_0, \] where \(f\) is given. The objective is to construct a control \(v \in L^2(\Gamma_0)^m_*\) such that \(\|y(v)|_S - \overline y\|_{L^2(S)} \leq \varepsilon,\) where \(S \subset \Omega\) is an \((m-1)\)-hypersurface and \(\overline y\) is a target in \(L^2(S).\) Under suitable conditions, the authors show that the problem has solutions for any \(\varepsilon\) and give a numerical method for their construction. The method deteriorates as \(\varepsilon \to 0,\) which has to do with the fact that the corresponding exact controllability problem may not have a solution.

MSC:

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35Q30 Navier-Stokes equations
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