Skip to main content
SpringerLink
Log in

Stabilization of Series Heat Equations with One Boundary Control

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

Abstract

In this paper, we consider the exponential stabilization for a series system of N heat equations, where one boundary control is only applied to the leading (first) heat equation, and the boundary temperature of the previous heat is fed into the boundary heat flux of the next one. With N invertible transformations, we present a boundary feedback control so that the closed-loop system is equivalent to an exponentially stable target system. Some numerical simulation results are presented to validate the theoretical conclusions.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Gu JJ, Wang JM. Sliding mode control for N-coupled reaction-diffusion PDEs with boundary input disturbances. Int J Robust Nonlinear Control. 2019;29:1437–61.

    Article  MathSciNet  MATH  Google Scholar 

  2. Gu JJ, Wang JM. Backstepping state feedback regulator design for an unstable reaction-diffusion PDE with long time delay. J Dyn Control Syst. 2018;24:563–76.

    Article  MathSciNet  MATH  Google Scholar 

  3. Jia YN, Liu JJ, Li SJ. Output feedback stabilization for a cascaded heat PDE-ODE system subject to uncertain disturbance. International Journal of Robust and Nonlinear Control. 2018;28:5173–90.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Li T, Rao BP. Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls. Chin Ann Math. 2013;34B(1):139–60.

    Article  MathSciNet  MATH  Google Scholar 

  6. Liu JJ, Wang JM. Output-feedback stabilization of an anti-stable Schrödinger equation by boundary feedback with only displacement observation. J Dyn Control Syst. 2013;19:471–82.

    Article  MathSciNet  MATH  Google Scholar 

  7. Liu JJ, Wang JM. Boundary stabilization of a cascade of ODE-wave systems subject to boundary control matched disturbance. International Journal of Robust and Nonlinear Control. 2017;27:252–80.

    Article  MathSciNet  MATH  Google Scholar 

  8. Mahamud R, Park C. Reciprocating air flow for Li-ion battery thermal management to improve temperature uniformity. J Power Sources. 2011;196:5685–96.

    Article  Google Scholar 

  9. Meglio FD, Vazquez R, Krstic M. Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans on Automatic Control. 2013;58(12):3097–111.

    Article  MathSciNet  MATH  Google Scholar 

  10. Ni W, Cheng DZ. Leader-following consensus of multi-agent systems under fixed and switching topologies. Systems Control Letters. 2010;59:209–17.

    Article  MathSciNet  MATH  Google Scholar 

  11. Olfati-Saber R, Murray M. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans on Automatic Control. 2004;49(9):1520–33.

    Article  MathSciNet  MATH  Google Scholar 

  12. Pazy A. Semigroup of linear operators and applications to partial differential equations. New York, NY, USA: Springer-Verlag; 1983.

    Book  MATH  Google Scholar 

  13. Pisano A, Baccoli A, Orlov Y, Usai E. Boundary control of coupled reaction-advection-diffusion equations having the same diffusivity parameter. IFAC-Papers OnLine. 2016;49(8):086–91.

    Article  MathSciNet  Google Scholar 

  14. Popovski P, Yomo H. Wireless network coding by amplify-and-forward for bi-directional traffic flows. IEEE communications letters. 2007;11:16–8.

    Article  Google Scholar 

  15. Ren BB, Wang JM, Krstic M. Stabilization of an ODE-Schröinger cascade. System Control Letters. 2013;62:503–10.

    Article  MATH  Google Scholar 

  16. Smets IY, Claes JE, November EJ, Bastin GP, Van Impe JF. Optimal adaptive control of (bio)chemical reactors: past, present and future. Journal of Process Control. 2004;14:795–805.

    Article  Google Scholar 

  17. Ge SS, He W, How BVE, Choo YS. Boundary control of a coupled nonlinear flexible marine riser. IEEE Trans Automatic Control 2010;18(5):1080-1091.

  18. Smyshlyaev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans on Automatic Control. 2004;49(12):2185–202.

    Article  MathSciNet  MATH  Google Scholar 

  19. Smyshlyaev A, Krstic M. Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems Control Letters. 2008;57:750–8.

    Article  MathSciNet  MATH  Google Scholar 

  20. Vazquez R, Krstic M. Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients. IEEE Trans on Automatic Control. 2017;62:2026–33.

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang JM, Su LL, Li HX. Stabilization of an unstable reaction-diffusion PDE cascaded with a heat equation. Systems Control Letters. 2015;76:8–18.

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu KN, Tian T, Wang LM. Synchronization for a class of coupled linear partial differential systems via boundary control. Journal of Franklin Institute. 2016;353:4062–73.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is supported by the National Nature Science Foundation of China with grant number 62073037.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Lina Pan & Jun-Min Wang

Authors
  1. Lina Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jun-Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lina Pan.

Additional information

Publisher’s Note

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

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, L., Wang, JM. Stabilization of Series Heat Equations with One Boundary Control. J Dyn Control Syst 29, 969–987 (2023). https://doi.org/10.1007/s10883-022-09618-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-022-09618-w

Keywords

MSC Classification

Search

Navigation