Lyapunov control of a quantum particle in a decaying potential. (English) Zbl 1176.35169
Summary: A Lyapunov-based approach for the trajectory generation of an \(N\)-dimensional Schrödinger equation in whole \(\mathbb R^N\) 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.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B35 | Stability in context of PDEs |
| 35P05 | General topics in linear spectral theory for PDEs |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 93C20 | Control/observation systems governed by partial differential equations |
Keywords:
nonlinear control of PDEs; approximate stabilization; Lyapunov techniques; dispersive estimates; pre-compactnessReferences:
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