On the Lack of Null-Controllability of the Heat Equation on the Half Space

Sorin Micu and Enrique Zuazua

Abstract: We study the null-controllability property of the linear heat equation on the half-space with a $L^2$ Dirichlet boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. By separation of variables the multi-dimensional control problem is reduced to an infinite family of one-dimensional controlled systems. Next, the results for this type of systems proved in [18] are used in order to show that, roughly speaking, controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.

Keywords: Heat equation; Similarity variables; Control; Moments.

Classification (MSC2000): 35B37, 35K05.

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