Smoothing Properties of Fractional Ornstein-Uhlenbeck Semigroups and Null-Controllability

Abstract

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.

Schauder theorems for a class of (pseudo-)differential operators on finite and infinite dimensional state spaces

“…Here we consider modified Ornstein-Uhlenbeck operators which are the object of very recent studies (e.g., [33,3,17]), heuristically given by…”

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

“…When the Ornstein-Uhlenbeck operator 1 2 Tr(Q∇ 2 x ) + Bx, ∇ x is hypoelliptic, the associated Markov semigroup (T (t)) t≥0 has the following explicit representation due to Kolmogorov [20]:…”

“…where ω ⊂ R n is a measurable subset with a positive Lebesgue measure and P is the generalized Ornstein-Uhlenbeck operator defined in (5.1). When R = 0 and P stands for a hypoelliptic Ornstein-Uhlenbeck operator, J. Bernier and the author proved in [1] (Theorem 1.3) that this equation is null-controllable in any positive time, once the control subset ω ⊂ R n is thick. When R = 0, we derive from Theorem 1.12 the following result: Theorem 5.2.…”

“…Therefore, the inclusion S ⊂ Ker(Im F ) does not hold when B = 0. However, by explicitly computing e −tq w u for all t ≥ 0 and u ∈ L 2 (R n ) and exploiting the Kalman rank condition, J. Bernier and the author proved in [1] (Theorem 1.2) that there exist some positive constants C > 1 and t 0 > 0 such that for all β ∈ N n , 0 < t ≤ t 0 and u ∈ L 2 (R n ),…”

“…No general result of null-controllability for the equation (1.25) is known up to now when the singular space of q is non-zero. However, when the quadratic form q is defined by (1.22), where B and Q are real n × n matrices, with Q symmetric positive semidefinite, B and Q 1 2 satisfying the Kalman rank condition, we recall that the singular space of q is S = R n × {0} (in particular, S is non-zero), and J. Bernier and the author proved in [1] (Theorem 1.8) that the parabolic equation (1.25) is null-controllable in any positive time from thick control subsets. Moreover, when B = 0 n and Q = 2I n , the quadratic form q is given by q(x, ξ) = |ξ| 2 and (1.25) is the heat equation posed on the whole space :…”