Null-controllability of hypoelliptic quadratic differential equations

By Karine Beauchard, K. Pravda-Starov. 2016

76 citations recorded by the source corpus.

We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dea9252e80791ec421fb7c78dd4cf77b77979629

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Spectral estimates for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations (2018, methodology): “We refer the reader to the works [4, 38] for detailed discussions of various physics models whose evolution turns out to be ruled by accretive quadratic operators with zero singular space and to which therefore apply the above result of null-controllability.”