Rapid Stabilization of a Linearized Bilinear 1-D Schrödinger Equation

Jean-Michel Coron (1, 2) , Ludovick Gagnon (3) , Morgan Morancey (4)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 I2M - Institut de Mathématiques de Marseille

We consider the one dimensional Schrödinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.

Domaines: Optimisation et contrôle [math.OC]

Jean-Michel Coron, Ludovick Gagnon, Morgan Morancey. Rapid Stabilization of a Linearized Bilinear 1-D Schrödinger Equation. Journal de Mathématiques Pures et Appliquées, 2018, 115, ⟨10.1016/j.matpur.2017.10.006⟩. ⟨hal-01408179⟩