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

Paul Alphonse (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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 null-controllability of a large class of quadratic differential operators.

Mots clés: Quadratic operators; Gelfand-Shilov regularity; Gevrey regularity; Null-controllability

Domaines: Equations aux dérivées partielles [math.AP]

Paul Alphonse. Quadratic differential equations : partial Gelfand-Shilov smoothing effect and null-controllability. 2019. ⟨hal-02011642v2⟩