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Null-controllability of hypoelliptic quadratic differential equations. (English. French summary) Zbl 1403.93041

Summary: We study the null-controllability of parabolic equations associated with 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 with 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 with 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 with 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.

MSC:

93B05 Controllability
35H10 Hypoelliptic equations
35K65 Degenerate parabolic equations
93B07 Observability
93C20 Control/observation systems governed by partial differential equations
93B28 Operator-theoretic methods
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References:

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