JournalsaihpcVol. 26, No. 5pp. 1793–1815

Some controllability results for the 2D Kolmogorov equation

  • K. Beauchard

    CMLA, ENS Cachan, CNRS, Universud, 61, avenue du Président Wilson, F-94230 Cachan, France
  • E. Zuazua

    Basque Center for Applied Mathematics (BCAM), Gran Via, 35-2, 48009 Bilbao, Spain
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Abstract

In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.

The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.

Cite this article

K. Beauchard, E. Zuazua, Some controllability results for the 2D Kolmogorov equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1793–1815

DOI 10.1016/J.ANIHPC.2008.12.005