Coron, Jean-Michel; Gagnon, Ludovick; Morancey, Morgan Rapid stabilization of a linearized bilinear 1-D Schrödinger equation. (English. French summary) Zbl 1392.35281 J. Math. Pures Appl. (9) 115, 24-73 (2018). Summary: 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. The continuity and invertibility of the transformation follows from exact controllability of the linearized system. Cited in 18 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 93D15 Stabilization of systems by feedback 93B52 Feedback control 93B05 Controllability Keywords:Schrödinger equation; rapid stabilization; integral transform PDFBibTeX XMLCite \textit{J.-M. Coron} et al., J. Math. Pures Appl. (9) 115, 24--73 (2018; Zbl 1392.35281) Full Text: DOI arXiv References: [1] Balogh, A.; Krstić, M., Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability, Eur. J. Control, 8, 3, 165-175, (2002) · Zbl 1293.93404 [2] Beauchard, K., Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9), 84, 7, 851-956, (2005) · Zbl 1124.93009 [3] Beauchard, K.; Coron, J.-M., Controllability of a quantum particle in a moving potential well, J. Funct. 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