Null-controllability of hypoelliptic quadratic differential equations

Karine Beauchard,Karel Pravda-Starov

doi:10.5802/jep.62

Abstract

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.

Highlights

Abstract. — We study the null-controllability of parabolic equations associated with a general class of hypoelliptic quadratic differential operators
As a first result in this work (Theorem 1.3), we prove that condition (1.6) is sufficient for the null-controllability of all hypoelliptic Ornstein-Uhlenbeck equations
In the second part of this work, we prove that the parabolic equations (1.1) associated with a general class of hypoelliptic quadratic operators are null-controllable from open sets satisfying condition (1.6) in any positive time

Summary

Miscellaneous facts about quadratic differential operators

Let qw(x, Dx) be a quadratic operator defined by the Weyl quantization (1.2) of a complex-valued quadratic form q on the phase space R2n. According to the above description of the singular space, these quadratic operators are exactly those whose Weyl symbols have a non-negative real part Re q 0, becoming positive definite (1.16). 0, after averaging by the linear flow of the Hamilton vector field associated with its imaginary part These quadratic operators are known [29, Th. 1.2.1] to generate contraction semigroups (e−tqw )t 0 on L2(Rn), which are smoothing in the Schwartz space for any positive time. According to [53, Th. 1.2.1], this integer k0 ∈ [0, 2n − 1] is directly related to the loss of derivatives 0 δ = 2k0/(2k0 + 1) < 1 in the global subelliptic estimate (1.10) satisfied by any quadratic operator whose Weyl symbol has a non-negative real part and a zero singular space.

Statements of the main results

Adapted Lebeau-Robbiano method for observability

Application