Skip to main content
SpringerLink
Log in

Stabilization for Schrödinger Equation with Internal Damping and Boundary Disturbance

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this paper, the stabilization for Schrödinger equation subject to internal damping and boundary disturbance at the control end is investigated. Due to its immeasurability, the nonlinear observer system is designed to obtain the state information, and the existence of weak solution and its convergence for the nonlinear observer system are proved. The feedback control is realized by the backstepping transformation. In addition, the adaptive disturbance rejection control is applied to estimate the disturbance. According to the observer and disturbance estimation, the feedback control is finally designed to stabilize the system asymptotically.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Peirce AP, Dahleh MA, Rabitz H. Optimal control of quantum-mechanical systems: existence, numerical approximation, and applications. Phys Rev A 1988;37(12):4950–64.

    Article  MathSciNet  Google Scholar 

  2. Brif C, Chakrabarti R, Rabitz H. Control of quantum phenomena. Adv Chem Phys 2012;148:1–76.

    MATH  Google Scholar 

  3. Dahleh MA, Peirce AP, Rabitz H, Ramakrishna V. Control of molecular motion. Proc IEEE 1996;84(1):7–15.

    Article  Google Scholar 

  4. Machtyngier E. Exact controllability for the schrödinger equation. SIAM J Control Optim 1994;32(1):24–34.

    Article  MathSciNet  Google Scholar 

  5. Machtyngier E, Zuazua E. Stabilization of the schrödinger equation. Portugal Math 1994;51(2):243–56.

    MathSciNet  MATH  Google Scholar 

  6. Ito K, Kunisch K. Optimal bilinear control of an abstract schrödinger equation. SIAM J Control Optim 2007;46(1):274–287.

    Article  MathSciNet  Google Scholar 

  7. Mirrahimi M, Rouchon P, Turinici G. Lyapunov control of bilinear schrödinger equations. Automatica 2005;41:1987–94.

    Article  MathSciNet  Google Scholar 

  8. Friesecke G, Henneke F, Kunisch K, Radon J. Frequency-sparse optimal quantum control. Math Control Related Fields 2018;8(1):155–76.

    Article  MathSciNet  Google Scholar 

  9. Sprengel M, Ciaramella G, Borzì A. Investigation of optimal control problems governed by a time-dependent Kohn-Sham model. Berlin: Springer; 2017.

    MATH  Google Scholar 

  10. Sprengel M, Ciaramella G, Borzì A. A COKOSNUT code for the control of the time-dependent Kohn-Sham model. Comput Phys Commun 2017;214: 231–8.

    Article  Google Scholar 

  11. Hintermuller M, Marahrens D, Markowich PA, Sparber C. Optimal bilinear control of gross-pitaevskii equations. SIAM J Control Optim 2013; 51(3):2509–43.

    Article  MathSciNet  Google Scholar 

  12. Cazenave T. Semilinear schrödinger equations. New York University Courant Institute of Mathematical Sciences; 2003.

  13. Ciaramella G, Sprengel M, Borzì A. A theoretical investigation of time-dependent Kohn-Sham equations: new proofs. Applicable Analysis. 2019. https://doi.org/10.1080/00036811.2019.1679792.

  14. Jerome JW. Time dependent closed quantum systems: nonlinear Kohn-Sham potential operators and weak solutions. J Math Anal Appl 2015;429:995–1006.

    Article  MathSciNet  Google Scholar 

  15. Borzì A, Salomon J, Volkwein S. Formulation and numerical solution of finite-level quantum optimal control problems. J Comput Appl Math;216: 170–97.

  16. Cong S. Control of quantum systems: Theory and Methods. Wiley; 2014.

  17. Krstic M, Guo BZ, Smyshlyaev A. Boundary controllers and observers for the linearized schrödinger equation. SIAM J Control Optim 2011;49(4): 1479–97.

    Article  MathSciNet  Google Scholar 

  18. Han J. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009;56(3):900–6.

    Article  Google Scholar 

  19. Han J. The ”Extended State Observer” of a class of uncertain systems. Control Decis 1995;1:85–8.

    Google Scholar 

  20. Orlov YV, Utkin VI. Use of sliding modes in distributed system control problems. Autom Remote Control 1983;43(9):1127–35.

    MathSciNet  MATH  Google Scholar 

  21. Orlov YV. Discontinuous System: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. Springer; 2009.

  22. Krstic M, Smyshlyaev A. 2008. Boundary Control of PDEs: A Course on Backstepping Designs Society for Industrial and Applied Mathematics.

  23. Liu DY, Zhang LP, Xu GQ. Stabilisation of Timoshenko beam system with a tip payload under the unknown boundary external disturbances. Int J Control 2015;88(9):1830–40.

    Article  MathSciNet  Google Scholar 

  24. Guo BZ, Jin FF. Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance. IEEE Trans Autom Control 2014;60(3): 824–30.

    Article  MathSciNet  Google Scholar 

  25. Liu JJ, Wang JM. Stabilization of one-dimensional wave equation with nonlinear boundary condition subject to boundary control matched disturbance. IEEE Trans Autom Control 2019;64(7):3068–73.

    Article  MathSciNet  Google Scholar 

  26. Kang W, Guo BZ. Stabilisation of unstable cascaded heat partial differential equation system subject to boundary disturbance. IET Control Theory Appl. 2016;10(9):1027–39.

    Article  MathSciNet  Google Scholar 

  27. Kang W, Fridman E. Boundary constrained control of delayed nonlinear schrödinger equation. IEEE Trans Autom Control 2018;63(11):3873–80.

    Article  Google Scholar 

  28. Guo BZ, Liu JJ. Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional schrödinger equation subject to boundary control matched disturbance. Int J Robust Nonlinear Control 2014;24(16):2194–212.

    Article  Google Scholar 

  29. Cui HY, Xu GQ, Han ZJ. Distributed tracking control of the schrödinger equation with internal disturbance. IMA J Math Control Inf 2017;34(3): 831–50.

    MATH  Google Scholar 

  30. Cui HY, Han ZJ, Xu GQ. Controller design to stabilization of schrödinger equation with boundary input disturbance. Appl Anal 2020;99(5):796–813.

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research was supported by the Science and Technology Development Fund of Tianjin Education Commission for Higher Education, China (Grant No. 2018KJ231).

Author information

Authors and Affiliations

  1. Department of Mathematics, Tianjin University of Commerce, Tianjin, 300134, People’s Republic of China

    Haoyue Cui

  2. School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, People’s Republic of China

    Yining Chen

  3. School of Mathematics, Tianjin University, Tianjin, 300350, People’s Republic of China

    Genqi Xu

Authors
  1. Haoyue Cui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yining Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Genqi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haoyue Cui.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cui, H., Chen, Y. & Xu, G. Stabilization for Schrödinger Equation with Internal Damping and Boundary Disturbance. J Dyn Control Syst 28, 971–987 (2022). https://doi.org/10.1007/s10883-021-09564-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-021-09564-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation