Abstract
We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.
Highlights
We study parabolic equations posed on the whole Euclidean space Rd,(∂t + P ) f (t, x) = 1ω(t)(x)u(t, x), x ∈ Rd, t > 0, f |t=0 = f0 ∈ L2(Rd ), (1)and controlled by a source term u locally distributed in a time-dependent control subset ω(t ) ⊂ Rd
We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability
The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are given to ensure null-controllability. 2020 Mathematics Subject Classification. 93B05, 35H10
Summary
Controlled by a source term u locally distributed in a time-dependent control subset ω(t ) ⊂ Rd. We consider in this work two specific classes of hypoelliptic quadratic parabolic equations, and we aim at pointing out necessary or sufficient geometric conditions on the moving control subsets (ω(t ))t∈I to ensure null-controllability. The geometric condition (2) was not expected to be sharp to ensure null-controllability, and a new breakthrough was made by Veselicand the second author in [10], who established that the following notion of thickness is a necessary and sufficient condition on fixed control subsets to ensure the null-controllability of the heat equation posed on the whole Euclidean space Rd in some positive time, as well as in any positive time: Definition 3 ((δ, α)-thick set). For this second class of hypoelliptic quadratic parabolic equations, we provide only a sufficient condition for null-controllability. The transport phenomena at play for this second class of equations are far more complex and this topic is not studied in this work
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