Skip to main content
SpringerLink
Log in

Null Controllability and Finite Time Stabilization for the Heat Equations with Variable Coefficients in Space in One Dimension via Backstepping Approach

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bardos C., Tartar L.: Sur l’unicité rétrograde des équations paraboliques et quelques questions voisines. Arch. Rational Mech. Anal. 50, 10–25 (1973)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bekiaris-Liberis N., Krstic M.: Compensating the distributed effect of a wave PDE in the actuation or sensing path of multi-input and MIMO LTI systems. Systems Control Lett. 59, 713–719 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bekiaris-Liberis N., Krstic M.: Compensating the distributed effect of diffusion and counter-convection in multi-input and multi-output LTI systems. IEEE Trans. Automat. Control 56, 637–643 (2011)

    Article  MathSciNet  Google Scholar 

  4. Cerpa E., Coron J.-M.: Rapid stabilization for a Korteweg–de Vries equation from the left Dirichlet boundary condition. IEEE Trans. Automat. Control 58, 1688–1695 (2013)

    Article  MathSciNet  Google Scholar 

  5. Coron, J.-M.: Control and nonlinearity, vol. 136 of Mathematical Surveys and Monographs, American Mathematical Soc., Providence, RI, 2007

  6. Coron, J.-M., d’Andréa-Novel, B.: Stabilization of a rotating body beam without damping. IEEE Trans. Automat. Control, 43, 608–618 (1998)

  7. Coron J.-M., Lü Q.: Local rapid stabilization for a Korteweg–de Vries equation with a Neumann boundary control on the right. J. Math. Pures Appl. 102, 1080–1120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Coron J.-M., Lü Q.: Fredholm transform and local rapid stabilization for a Kuramoto–Sivashinsky equation. J. Diff. Equations 259, 3683–3729 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Coron J.-M., Vazquez R., Krstic M., Bastin G.: Local Exponential H 2 Stabilization of a 2 × 2 Quasilinear Hyperbolic System Using Backstepping. SIAM J. Control Optim. 51, 2005–2035 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Di Meglio, F., Vazquez, R., Krstic, M.: Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans. Automat. Control, 58, 3097–3111 (2013)

  11. Elharfi A.: Explicit construction of a boundary feedback law to stabilize a class of parabolic equations. Differential and Integral Equations 21, 351–362 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Escauriaza L., Alessandrini G.: Null-controllability of one-dimensional parabolic equations. ESAIM Control Optim. Calc. Var. 14, 284–293 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fattorini O. H., Russell D. L.: Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43, 272–292 (1971)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Friedman, A.: Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964

  15. Fursikov, A. V., Imanuvilov, O. Y.:Controllability of evolution equations, vol. 34, Seoul National University, Seoul, 1996

  16. Guo Y.-J. L., Littman W.: Null boundary controllability for semilinear heat equations. Appl. Math. Optim. 32, 281–316 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu, L., Di Meglio, F.: Finite-time backstepping stabilization of 3 × 3 hyperbolic systems, IEEE European Control Conference, under review, 2015

  18. Hu, L., Di Meglio, F., Vazquez, R., Krstic, M.: Boundary control design of homodirectional and general heterodirectional linear hyperbolic PDEs, Preprint, 2015

  19. Jones B. F.: A fundamental solution for the heat equation which is supported in a strip. J. Math. Anal. Appl. 60, 314–324 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Krstic M., Guo B. Z., Balogh A., Smyshlyaev A.: Output-feedback stabilization of an unstable wave equation. Automatica 44, 63–74 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Krstic, M., Kanellakopoulos, I., Kokotovic, P. V.: Nonlinear and adaptive control design, John Wiley & Sons, Inc., New York, 1995

  22. Krstic M., Smyshlyaev A.: Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems Control Lett. 57, 750–758 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Krstic, M., Smyshlyaev, A.: Boundary control of PDEs: A course on backstepping designs, vol. 16, SIAM, Philadelphia, 2008

  24. Lebeau G., Robbiano L.: Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations 20, 335–356 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Littman W.: Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 567–580 (1978)

    MathSciNet  MATH  Google Scholar 

  26. Liu W.-J.: Boundary feedback stabilization of an unstable heat equation. SIAM J. Control Optim. 42, 1033–1043 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu W.-J., Krstic M.: Backstepping boundary control of Burgers’s equation with actuator dynamics. Systems Control Lett. 41, 291–303 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Liu, W.-J., Krstic, M.: Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation. Nonlinear Anal. Ser. A: Theory Methods, 43, 485–507 (2001)

  29. Martin P., Rosier L., Rouchon P.: Null controllability of the heat equation using flatness. Automatica 50, 3067–3076 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Miller L.: The control transmutation method and the cost of fast controls. SIAM J. Control Optim. 45, 762–772 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Reissig M.: Hyperbolic equations with non-Lipschitz coefficients. Rend. Sem. Mat. Univ. Politec. Torino 61, 135–181 (2003)

    MathSciNet  MATH  Google Scholar 

  32. Smyshlyaev A., Cerpa E., Krstic M.: Boundary stabilization of a 1-D wave equation with in-domain antidamping. SIAM J. Control Optim. 48, 4014–4031 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Smyshlyaev A., Krstic M.: Closed-form boundary state feedbacks for a class of 1−D partial integro-differential equations. IEEE Trans. Automat. Control 49, 2185–2202 (2004)

    Article  MathSciNet  Google Scholar 

  34. Smyshlyaev A., Krstic M.: On control design for PDEs with space-dependent diffusivity or time-dependent reactivity. Automatica 41, 1601–1608 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Smyshlyaev A., Krstic M.: Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary. Systems Control Lett. 58, 617–623 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vazquez R., Krstic M.: Control of 1-D parabolic PDEs with Volterra nonlinearities, part I: design. Automatica 44, 2778–2790 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, UMR 7598, UPMC Univ Paris 06, Sorbonne Universités, 4 place Jussieu, 75252, Paris, France

    Jean-Michel Coron

  2. EPFL SB MATHAA CAMA, Station 8, 1015, Lausanne, Switzerland

    Hoai-Minh Nguyen

Authors
  1. Jean-Michel Coron
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hoai-Minh Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hoai-Minh Nguyen.

Additional information

Communicated by A. Bressan

JMC was supported by ERC advanced Grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7).

HMN was supported by ERC advanced Grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Coron, JM., Nguyen, HM. Null Controllability and Finite Time Stabilization for the Heat Equations with Variable Coefficients in Space in One Dimension via Backstepping Approach. Arch Rational Mech Anal 225, 993–1023 (2017). https://doi.org/10.1007/s00205-017-1119-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1119-y

Mathematics Subject Classification

Search

Navigation