Skip to main content
SpringerLink
Log in

Extendability and Optimal Control After Quenching for Some Ordinary Differential Equations

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

Abstract

This paper first studies the extendability after quenching for some ordinary differential equations. With respect to a solution, which quenches at finite time, we give a criterion of its non-extendability after quenching; it implies a necessary condition of extendability after quenching for this solution. Then, we consider an optimal control problem after quenching. We establish the Pontryagin’s maximum principle to this problem.

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. Kawarada, H.: On solutions of initial-boundary problem for \(u_t=u_{xx}+1/(1-u)\). Publ. Res. Inst. Math. Sci. 10, 729–736 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chan, C.Y., Kaper, H.G.: Quenching for semilinear singular parabolic problems. SIAM J. Math. Anal. 20(3), 558–566 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, J.S., Hu, B.: The profile near quenching time for the solution of a singular semilinear heat equation. Proc. Edinb. Math. Soc. 40, 437–456 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bandle, C., Brunner, H.: Blowup in diffusion equations: a survey. J. Comput. Appl. Math. 97, 3–22 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barron, E.N., Liu, W.: Optimal control of the blowup time. SIAM J. Control Optim. 34(1), 102–123 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lin, P., Wang, G.: Blowup time optimal control for ordinary differential equations. SIAM J. Control Optim. 49, 73–105 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lin, P., Wang, G.: Some properties for blowup parabolic equations and their application. Journal de Mathematiques Pures et Appliques 101, 223–255 (2014)

    Article  MATH  Google Scholar 

  8. Lou, H., Wang, W.: Optimal blowup time for controlled ordinary differential equations. ESAIM: COCV 21(3), 815–834 (2015)

    Article  MathSciNet  Google Scholar 

  9. Lin, P.: Quenching time optimal control for some ordinary differential equations. J. Appl. Math. 2014, 127809 (2014). doi:10.1155/2014/127809

    Google Scholar 

  10. Lou, H., Wen, J., Xu, Y.: Time optimal control problems for some non-smooth systems. Math. Control Relat. Fields 4, 289–314 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lou, H., Wang, W.: Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Math. Control Relat. Fields 5(3), 517–527 (2015)

    Article  MathSciNet  Google Scholar 

  12. Li, X., Yong, J.: Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser, Boston (1995)

    Book  Google Scholar 

  13. Wang, G., Wang, L.: State constrained optimal control governed by non-well-posed parabolic differential equations. SIAM J. Control Optim. 40(5), 1517–1539 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author gratefully acknowledges Professor Jiongmin Yong and Professor Hongwei Lou for their useful suggestions. This work was partially supported by the National Natural Science Foundation of China under Grant 11471070 and the National Basis Research Program of China (973 Program) under Grant 2011CB808002.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Ping Lin

Authors
  1. Ping Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Lin.

Additional information

Communicated by Boris Vexler.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, P. Extendability and Optimal Control After Quenching for Some Ordinary Differential Equations. J Optim Theory Appl 168, 769–784 (2016). https://doi.org/10.1007/s10957-015-0858-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0858-x

Keywords

Mathematics Subject Classification

Search

Navigation