A Lyapunov-based approach for the trajectory generation of an N-dimensional Schrödinger equation in whole is proposed. For the case of a quantum particle in an N-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.
Cite this article
Mazyar Mirrahimi, Lyapunov control of a quantum particle in a decaying potential. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 5, pp. 1743–1765DOI 10.1016/J.ANIHPC.2008.09.006