Skip to main content
SpringerLink
Log in

Boundary Exponential Stabilization for a Class of Coupled Reaction-Diffusion Equations with State Delay

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

Abstract

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results.

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. de Oliveira LAF. 1998. On reaction-diffusion systems. Electron J Diff Equ, No. 24, pp.1–10.

  2. Boutayeb S, Abta A. Boundary control for a class of coupled reaction-diffusion equations with state delay. Manuscript submitted for publication.

  3. Baccoli A, Orlov Y, Pisano A. On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters. Proc. CDC. Los Angeles, California, USA; 2014. p. 5222–8.

  4. Baccoli A, Pisano A, Orlov Y. Boundary control of coupled reaction diffusion processes with constant parameters. Automatica 2015;54:80–90.

    Article  MathSciNet  Google Scholar 

  5. Badraoui S. Analytic semigroups generated by an operator matrix in L2(Ω) × L2(Ω). Electron J Diff Equ 2012;2012(166):1–7.

    MathSciNet  Google Scholar 

  6. Chaplain MAJ. Reaction-diffusion pre-patterning and its potential role in tumour invasion. J Biol Syst 1995;03(04):929–36.

    Article  Google Scholar 

  7. Deutscher J, Kerschbaum S. Backstepping control of coupled linear parabolic PIDEs with spatially-varying coefficients. IEEE Trans Autom Control 2017; 63:4218–33.

    Article  MathSciNet  Google Scholar 

  8. Driver DR. Ordinary and delay differential equations. New York: Springer-Verlag; 1977.

    Book  Google Scholar 

  9. Engel K-J, Nagel R, Vol. 194. One-parameter semigroups for linear evolution equations. New York: Springer-Verlag; 1999. Graduate Texts in Mathematics.

    Google Scholar 

  10. Gallier J. Geometric methods and applications: for computer science and engineering, 2nd ed. New York: Springer; 2011.

    Book  Google Scholar 

  11. Hale JK, Vol. 3. Functional differential equations, Math Appl Sci. New York: Springer-Verlag; 1971.

    Book  Google Scholar 

  12. Hale JK, Verduyn Lunel SM, Vol. 99. Introduction to functional-differential equations Appl Math Sci. New York: Springer-Verlag; 1993.

    Book  Google Scholar 

  13. Hashimoto T, Krstic M. Stabilization of reaction diffusion equations with state delay using boundary control input. IEEE Trans Autom Control 2016;61 (12):4041–7.

    Article  MathSciNet  Google Scholar 

  14. Holmes EE, Lewis MA, Banks JE, Veit RR. Partial differential equations in ecology: spatial interactions and population dynamics. Ecol Wiley 1994; 75(1):17–29.

    Article  Google Scholar 

  15. Kato T. Perturbation theory for linear operators. Berlin: Springer-Verlag; 1995. Reprint of the 1980 edition.

    Book  Google Scholar 

  16. Kolmanovskii VB, Shaikhet LE. 1996. Control of systems with aftereffet. American Mathematical Society.

  17. Liu W. Boundary feedback stabilization of an unstable heat equation. SIAM J Control Optim 2003;42:1033–42.

    Article  MathSciNet  Google Scholar 

  18. Logemann H, Townley S. The effect of small delays in the feedback loop on the stability of neutral systems. Syst Control Lett 1996;27:267–74.

    Article  MathSciNet  Google Scholar 

  19. Li C-K., Zhang F. Eigenvalue continuity and Gersgorin’s theorem. Electron J Linear Algebra 2019;35:619–25.

    Article  MathSciNet  Google Scholar 

  20. Lorenzi L, Bertoldi M. 2006. Analytical methods for Markov semigroups. Chapman & Hall/CRC monographs and research notes in mathematics.

  21. Meurer T, Kugi A. Tracking control for boundary controlled parabolic PDEs with varying parameters Combining backstepping and differential flatness. Automatica 2009;45:1182–94.

    Article  MathSciNet  Google Scholar 

  22. Mondie S, Kharitonov VL. Exponential estimates for retarded time-delay systems : an LMI approach. IEEE Trans Autom Control 2005;50(2):268–73.

    Article  MathSciNet  Google Scholar 

  23. Murray JD, Stanley EA, Brown DL. On the spatial spread of rabies among foxes. Proc R Soc Lond B Biol Sci 1986;229(1255):111–50.

    Article  Google Scholar 

  24. Mugnolo D. Asymptotics of semigroups generated by operator matrices. Arab J Math 2014;3:419–35.

    Article  MathSciNet  Google Scholar 

  25. Pazy A. Semigroups of linear operators and applications to partial differential equations. New York: Springer-Verlag; 1983.

    Book  Google Scholar 

  26. Orlov Y, Dochain D. Discontinuous feedbackstabilization of minimum- phase semilinear infinite-dimensional systems with application to chemical tubularreactor. IEEE Trans Autom Control 2002;47(8):1293–304.

    Article  Google Scholar 

  27. Radde- N. 2009. The impact of time delays on the robustness of biological oscillators and the effect of bifurcations on the inverse problem. URASIP J Bioinform Syst Biol.

  28. Smyshlyaev A, Krstic M. Closed-Form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans Aut Contr 2004;49 (12):2185–202.

    Article  MathSciNet  Google Scholar 

  29. Smyshlyaev A, Krstic M. On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica 2005;41:1601–8.

    Article  MathSciNet  Google Scholar 

  30. Vazquez R. M Krstic Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients. IEEE Trans Autom Control 2016;62:2023–33.

    Google Scholar 

  31. Webb GF. Autonomous nonlinear functional differential equations and nonlinear semigroups. J Math Anal Appl 1974;46:1–12.

    Article  MathSciNet  Google Scholar 

  32. Wu J. Theory and applications of partial functional differential equations. New York: Springer-Verlag; 1996.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Poly-disciplinary Faculty, Cadi Ayyad University, P.O. Box 4162, Safi, Morocco

    Abdelhadi Abta & Salahaddine Boutayeb

Authors
  1. Abdelhadi Abta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salahaddine Boutayeb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Salahaddine Boutayeb.

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

Abta, A., Boutayeb, S. Boundary Exponential Stabilization for a Class of Coupled Reaction-Diffusion Equations with State Delay. J Dyn Control Syst 28, 829–850 (2022). https://doi.org/10.1007/s10883-021-09551-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-021-09551-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation