Observability and controllability for the Schrödinger equation on quotients of groups of Heisenberg type. (Observabilité et contrôlabilité de l’équation de Schrödinger sur des quotients de groupes de type Heisenberg.) (English. French summary) Zbl 1475.35377
Summary: We give necessary and sufficient conditions for the controllability of a Schrödinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups. 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, contrary 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.
MSC:
| 35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
| 35H20 | Subelliptic equations |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 93B07 | Observability |
| 35Q93 | PDEs in connection with control and optimization |
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