Skip to main content
SpringerLink
Log in

Time Optimal Controls of Semilinear Heat Equation with Switching Control

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper is concerned with the bang-bang property of time optimal controls, governed by a semilinear heat equation in a bounded domain with switching controls acting locally into two open subsets. The proofs rely on an observability estimate from a positive measurable set in time for the linear heat equation, and a Kakutani fixed point argument.

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. Fattorini, H.O.: Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems. North-Holland Mathematics Studies 201, Elsevier, Amsterdam (2005)

  2. Kunisch, K., Wang, L.: Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints. J. Math. Anal. Appl. 395, 114–130 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kunisch, K., Wang, L.: Time optimal control of the heat equation with pointwise control constraints. ESAIM: COCV 19, 460–485 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  5. Mizel, V.J., Seidman, T.I.: An abstract bang-bang principle and time optimal boundary control of the heat equation. SIAM J. Control Optim. 35, 1204–1216 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Phung, K.D., Wang, G.: An observability estimate for parabolic equations from a measurable set in time and its applications. J. Eur. Math. Soc. 15, 681–703 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wang, G.: \(L^\infty \)-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47, 1701–1720 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993)

    MATH  Google Scholar 

  9. Barbu, V.: The time optimal control of Navier–Stokes equations. Syst. Control Lett. 30, 93–100 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kunisch, K., Wang, L.: Bang-bang property of time optimal controls of Burgers equation. Discrete Contin. Dyn. Sys. Ser. A 34, 3611–3637 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Phung, K.D., Wang, L., Zhang, C.: Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 477–499 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wang, L., Wang, G.: The optimal time control of a phase-field system. SIAM J. Control Optim. 42, 1483–1508 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lü, Q., Zuazua, E.: Robust null controllability for heat equations with unknown switching control mode. Discrete Contin. Dyn. Sys. Ser. A 34, 4183–4210 (2014)

    Article  MATH  Google Scholar 

  14. Shorten, R., Wirth, F., Mason, O., Wulff, K., King, C.: Stability criteria for switched and hybrid systems. SIAM Rev. 49, 545–592 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zuazua, E.: Switching control. J. Eur. Math. Soc. 13, 85–117 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Arada, N., Raymond, J.P.: Time optimal problems with Dirichlet boundary controls. Discrete Contin. Dyn. Sys. Ser. A 9, 1549–1570 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Raymond, J.P., Zidani, H.: Pontryagin’s principle for time-optimal problems. J. Optim. Theory Appl. 101, 375–402 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fernández-Cara, E., Zuazua, E.: The cost of approximate controllability for heat equations: the linear case. Adv. Differen. Equ. 5, 465–514 (2000)

    MATH  Google Scholar 

  19. Duyckaerts, E., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1–41 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their extremely careful reading of a previous version of the paper and for their very valuable suggestions. This work was partially supported by the National Science Foundation of China under Grants 11371285 and 91130022.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Lijuan Wang & Qishu Yan

Authors
  1. Lijuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qishu Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qishu Yan.

Additional information

Communicated by Viorel Barbu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Yan, Q. Time Optimal Controls of Semilinear Heat Equation with Switching Control. J Optim Theory Appl 165, 263–278 (2015). https://doi.org/10.1007/s10957-014-0606-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0606-7

Keywords

Mathematics Subject Classifications

Search

Navigation