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On the lack of null-controllability of the heat equation on the half space. (English) Zbl 0991.35010

The authors study the null-controllability property of the linear heat equation on the half-space with a \(L^2\) Dirichlet boundary control. Using similarity variables and weighted Sobolev spaces and developing solutions in Fourier series, they reduce the control problem to a sequence of one-dimensional controlled systems. For the null-controllability properties of this type of systems it was proved that no initial datum \(u_0\) belonging to any Sobolev space of negative order may be driven to zero in finite time. But this negative result was complemented by showing that there exist initial data with exponentially growing Fourier coefficients for which null-controllability holds in finite time with \(L^2\)-controls.
Reviewer: J.Y.Park

MSC:

35B37 PDE in connection with control problems (MSC2000)
93B05 Controllability
35K05 Heat equation
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