Quadratic differential equations : partial Gelfand-Shilov smoothing effect and null-controllability

Abstract

We study the partial Gelfand-Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated to a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated to this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the nullcontrollability of a large class of quadratic differential operators.

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

“…On one hand, one can indeed consider more degenerate cases of evolution equations associated to accretive non-selfadjoint quadratic operators with possibly non zero singular spaces. This question was recently addressed by Alphonse in [1] (Theorem 1.12), who obtained some null-controllability results from fixed thick subsets for evolution equations associated to certain classes of quadratic operators with non zero singular spaces that enjoy only partial Gelfand-Shilov smoothing effects. Theorem 1.12 in [1] is actually derived from the abstract observability result (Theorem 3.2) established in the present work and used with frequency cutoff projections.…”

“…This question was recently addressed by Alphonse in [1] (Theorem 1.12), who obtained some null-controllability results from fixed thick subsets for evolution equations associated to certain classes of quadratic operators with non zero singular spaces that enjoy only partial Gelfand-Shilov smoothing effects. Theorem 1.12 in [1] is actually derived from the abstract observability result (Theorem 3.2) established in the present work and used with frequency cutoff projections. On the other hand, the other approach based on the Gelfand-Shilov smoothing effects and projections onto the first Hermite modes which applies only for evolution equations associated to quadratic operators with zero singular spaces can be push further by taking advantage of the up to now unused exponential decay of the solutions in order to weaken the thickness assumption of the control support.…”

Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

“…As an application of the splitting formula given by Theorem 1.1 and the estimate given by Theorem 1.2, we investigate the regularizing properties of the evolution operators e −tq w for all t ≥ 0. As pointed out in the works [2,14,17,18,32], the understanding of this smoothing effect is closely related to the structure of the singular space S. Indeed, the notion of singular space allows to study the propagation of Gabor singularities for the solutions of the quadratic differential equations…”

“…In the case when the singular space S is not necessary trivial nor symplectic but satisfies the condition S ⊂ Ker(Im F ), with F the Hamilton map of the quadratic form q, some partial Gelfand-Shilov smoothing effects in any positive time t > 0 for the semigroup (e −tq w ) t≥0 were obtained by the first author in [2] (Theorem 1.4), with some control of the associated seminorms as t → 0 + . Moreover, we mention that under an algebraic condition on the matrices Q and B, the regularizing effects of the Ornstein-Uhlenbeck operator (1.1), whose singular space is not symplectic nor satisfies the condition S ⊂ Ker(Im F ), see (1.34) with R = 0, were studied by the two authors in [3] (Theorem 1.2).…”

“…The operator P = P + 1 2 Tr(B) is therefore a quadratic operator and it follows from a straightforward computation, see e.g. [2] (Section 5), that the Hamilton map F and the singular space S of P are respectively given by (1.32)…”