Observability and Controllability for the Schrödinger Equation on Quotients of Groups of Heisenberg Type

Clotilde Fermanian Kammerer (1) , Cyril Letrouit (2, 3, 4)

1 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 CaGE - Control And GEometry
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

We give necessary and sufficient conditions for the controllability of a Schrödinger equation involving a subelliptic operator on a compact manifold. This subelliptic operator is the sub-Laplacian of the manifold that is obtained by taking the quotient of a group of Heisenberg type by one of its discrete subgroups. This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrödinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schrödinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.

Domaines: Equations aux dérivées partielles [math.AP]; Théorie des représentations [math.RT]; Théorie spectrale [math.SP]

Clotilde Fermanian Kammerer, Cyril Letrouit. Observability and Controllability for the Schrödinger Equation on Quotients of Groups of Heisenberg Type. 2020. ⟨hal-02951662v1⟩