Abstract
We consider a problem of stabilization of a one-dimensional anti-stable linearized Schrödinger equation subject to boundary control. The controller is designed through the estimated state and is designed in the case that only displacement is available. The method of “backstepping” is adopted in the investigation. We then combine the control and observer designs into an output-feedback compensator and prove exponential stability of the closed-loop system.
Similar content being viewed by others
References
Chen G, Delfour MC, Krall AM, Payre G. Modeling, stabilization and control of serially connected beam. SIAM J Control Optim. 1987;25:526–546.
Chen G. Energy decay estimates and exact boundary value controllability for the wave equation in a boundary domain. J Math Pure Appl. 1979;58:249–273.
Guo BZ, Xu CZ. The stabilization of a one-dimensional wave equation by boundary feedback with noncollocated observation. IEEE Trans Autom Control. 2007;52:371–377.
Guo BZ, Shao ZC. Regularity of a Schrödinger equation with Dirichlet control and colocated observation. Syst Control Lett. 2005;54:1135–1142.
Guo BZ. Riesz basis approach to the stabilization of a flexible beam with a tip mass. SIAM J Control Optim. 2001;39:1736–1747.
Krstic M, Smyshlyaev A. Boundary control of PDEs: a course on backstepping designs. Philadelphia: SIAM; 2008.
Krstic M, Guo BZ, Balogh A, Smyshlyaev A. Output-feedback stabilization of an unstable wave equation. Automatica. 2008;44:63–74.
Kobayashi T. Stabilization of infinite-dimensional undamped second order system by using a parallel compensator. IMA J Math Control Inform. 2004;21:85–94.
Komornik V, Loreti P. Fourier series in control theory. New York: Springer; 2005.
Lasiecka I, Triggiani R. Optimal regularity, exact controllability and uniform stabilization of Schrödinger equation with Dirichlet control. Differ Integral Equ. 1992;5:521–535.
Lasiecka I, Triggiani R. Control theory for partial differential equations: continuous and approxiamation theories-II: abstract hyperbolic-like systems over a finite time horizon. Cambridge: Cambridge University Press; 2000.
Luo ZH, Guo BZ, Morgul O. Stability and stabilization of infinite dimensional systems with applications. London: Springer; 1999.
Luo ZH, Guo BZ, Morgul O. Stability and stabilization of infinite dimensional systems with applications. New York: Springer; 1999.
Machtyngier E, Zuazua E. Stabilization of the Schrödinger equation. Port Math. 1994;51:243–256.
Machtyngier E. Exact controllability for the Schrödinger equation. SIAM J Control Optim. 1994;32:24–34.
Phung KD. Observability and control of Schrödinger equation. SIAM J Control Optim. 2001;40:211–230.
Smyshlyaev A, Krstic M. Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Syst Control Lett. 2009;58:617–623.
Slemrod M. Stabilization of boundary control systems. J Differ Equ. 1976;22:402–415.
Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans Autom Control. 2004;49:2185–2202.
Smyshlyaev A, Krstic M. Backstepping observers for a class of parabolic PDEs. Syst Control Lett. 2005;54:613–625.
Weiss G. Admissible observation operators for linear semigroups. Israel J Math. 1989;65:17–43.
Weiss G. Admissibility of unbounded control operators. SIAM J Control Optim. 1989;27:527–545.
Acknowledgements
This work was supported by the National Natural Science Foundation of China.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, JJ., Wang, JM. Output-Feedback Stabilization of an Anti-stable Schrödinger Equation by Boundary Feedback with Only Displacement Observation. J Dyn Control Syst 19, 471–482 (2013). https://doi.org/10.1007/s10883-013-9189-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-013-9189-0