We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on L 2 (R n ) satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using the interpolation theory, we get global L 2 subelliptic estimates for the these operators.
We study fractional hypoelliptic Ornstein-Uhlenbeck operators acting on L 2 (R n) satisfying the Kalman rank condition. We prove that the semigroups generated by these operators enjoy Gevrey regularizing effects. Two byproducts are derived from this smoothing property. On the one hand, we prove the null-controllability in any positive time from thick control subsets of the associated parabolic equations posed on the whole space. On the other hand, by using interpolation theory, we get global L 2 subelliptic estimates for the these operators.
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.
We prove that the thickness is a necessary and sufficient geometric condition that ensures the (rapid) stabilization or the approximate null-controllability with uniform cost of a large class of evolution equations posed on the whole space R n and associated with operators of the form F (|Dx|), the function F : [0, +∞) → R being bounded below and continuous. We also provide explicit feedbacks and constants associated with these stabilization properties. Our results apply in particular for the half heat equation associated with the function F (t) = t, for which null-controllability is known to fail from thick control supports. More generally, the notion of thickness was known to be a necessary and sufficient condition for the null-controllability of the fractional heat equations associated with the functions F (t) = t 2s in the case s > 1, and that this null-controllability property from thick control supports does not hold when 0 < s ≤ 1.
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