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Null controllability of the heat equation with boundary Fourier conditions: the linear case. (English) Zbl 1106.93009
Summary: In this paper, we prove the global null controllability of the linear heat equation completed with linear Fourier boundary conditions of the form $\frac{\partial y}{\partial n}+\beta \phantom{\rule{0.166667em}{0ex}}y=0$. We consider distributed controls with support in a small set and nonregular coefficients $\beta =\beta \left(x,t\right)$. For the proof of null controllability, a crucial tool will be a new Carleman estimate for the weak solutions of the classical heat equation with nonhomogeneous Neumann boundary conditions.
##### MSC:
 93B05 Controllability 35K20 Second order parabolic equations, initial boundary value problems 49J20 Optimal control problems with PDE (existence)
##### Keywords:
controllability; heat equation; Fourier conditions