Semi-global weak stabilization of bilinear Schrödinger equations

Karine Beauchard (1) , Vahagn Nersesyan (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 LMV - Laboratoire de Mathématiques de Versailles

We consider a linear Schrödinger equation, on a bounded domain, with bilinear control, representing a quantum particle in an electric field (the control). Recently, Nersesyan proposed explicit feedback laws and proved the existence of a sequence of times $(t_n)_{n \in \mathbb{N}}$ for which the values of the solution of the closed loop system converge weakly in $H^2$ to the ground state. Here, we prove the convergence of the whole solution, as $t \rightarrow + \infty$. The proof relies on control Lyapunov functions and an adaptation of the LaSalle invariance principle to PDEs.

Domaines: Equations aux dérivées partielles [math.AP]

Karine Beauchard, Vahagn Nersesyan. Semi-global weak stabilization of bilinear Schrödinger equations. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2010, 348 (19-20), pp.1073-1078. ⟨10.1016/j.crma.2010.09.002⟩. ⟨hal-00485045v2⟩