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Local exact boundary controllability of the Boussinesq equation. (English) Zbl 0907.76020

Summary: We study the local exact boundary controllability problem for the Boussinesq equations that describe an incompressible fluid flow coupled to thermal dynamics. The result that we get in this paper is as follows: suppose that \(\widehat y(t,x)\) is a given solution of the Boussinesq equation where \(t \in(0,T)\), \(x \in \Omega\), \(\Omega\) is a bounded domain with \(C^\infty\)-boundary \(\partial \Omega\). Let \( y_0(x)\) be a given initial condition and \(\| \widehat y(0,\cdot) - y_0 \| < \varepsilon\) where \(\varepsilon=\varepsilon(\widehat y)\) is small enough. Then there exists boundary control \(u\) such that the solution \(y(t,x)\) of the Boussinesq equations satisfying \(y |_{(0,T) \times \partial \Omega} = u\), \(y |_{t=0} = y_0\) coincides with \(\widehat y(t,x)\) at the instant \(T\) : \(y(T,x) \equiv \widehat y(T,x)\).

MSC:

76D99 Incompressible viscous fluids
49J20 Existence theories for optimal control problems involving partial differential equations
93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
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