Skip to main content
SpringerLink
Log in

Approximate controllability of a class of semilinear systems with boundary degeneracy

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper we consider the approximate controllability of a class of degenerate semilinear systems. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary. We prove that the control systems are approximately controllable and the controls can be taken to be of quasi bang-bang form.

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. Alabau-Boussouira F., Cannarsa P., Fragnelli G.: Carleman estimates for degenerate parabolic operators with applications to null controllability. J. Evol. Equ. 6(2), 161–204 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barbu V.: Controllability of parabolic and Navier–Stokes equations. Sci. Math. Jpn. 56(1), 143–211 (2002)

    MATH  MathSciNet  Google Scholar 

  3. Cannarsa P., Fragnelli G.: Null controllability of semilinear degenerate parabolic equations in bounded domains. Electron. J. Differential Equations 2006(136), 1–20 (2006)

    MathSciNet  Google Scholar 

  4. Cannarsa P., Martinez P., Vancostenoble J.: Persistent regional null controllability for a class of degenerate parabolic equations. Commun. Pure Appl. Anal. 3(4), 607–635 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cannarsa P., Martinez P., Vancostenoble J.: Null controllability of degenerate heat equations. Adv. Differential Equations 10(2), 153–190 (2005)

    MATH  MathSciNet  Google Scholar 

  6. Cannarsa, P., Fragnelli, G. and Vancostenoble, J., Linear degenerate parabolic equations in bounded domains: controllability and observability, Systems, control, modeling and optimization, 163–173, IFIP Int. Fed. Inf. Process., 202, Springer, New York, 2006.

  7. Cannarsa P., Fragnelli G., Vancostenoble J.: Regional controllability of semilinear degenerate parabolic equations in bounded domains. J. Math. Anal. Appl. 320(2), 804–818 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cannarsa P., Martinez P., Vancostenoble J.: Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47(1), 1–19 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Escauriaza L., Fernández F.J.: Unique continuation for parabolic operators. Ark. Mat. 41(1), 35–60 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fabre C., Puel J., Zuazua E.: Approximate controllability of a semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125(1), 31–61 (1995)

    MATH  MathSciNet  Google Scholar 

  11. Fernández-Cara E., Zuazua E.: The cost of approximate controllability for heat equations: The linear case. Adv. Differential Equations 5, 465–514 (2000)

    MATH  MathSciNet  Google Scholar 

  12. Fernández-Cara E., Zuazua E.: Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Lineaire 17(5), 583–616 (2000)

    Article  MATH  Google Scholar 

  13. Fursikov, A. V., Imanuvilov, O. Y., Controllability of evolution equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.

  14. Gao H., Hou X., Pavel N.H.: Optimal control and controllability problems for a class of nonlinear degenerate diffusion equations. Panamer. Math. J. 13(1), 103–126 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order, Second edition. Springer-Verlag, Berlin (1983)

    Google Scholar 

  16. Kakutani S.: A generalization of Brouwer’s fixed point theorem. Duke Math J 8, 457–459 (1941)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin P., Gao H., Liu X.: Some results on controllability of a nonlinear degenerate parabolic system by bilinear control. J. Math. Anal. Appl. 326(2), 1149–1160 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lions, J. L., Remarques sur la contrôlabilité approchée, Proceedings of “Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos”, University of Málaga, Spain, October, 1990.

  19. Lions J.L.: Remarks on approximate controllability. J. Anal. Math. 59, 103–116 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu X., Gao H.: Controllability of a class of Newtonian filtration equations with control and state constraints. SIAM J. Control Optim. 46(6), 2256–2279 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Martinez P., Raymond J.P., Vancostenoble J.: Regional null controllability of a linearized Crocco-type equation. SIAM J. Control Optim. 42(2), 709–728 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Martinez P., Vancostenoble J.: Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ. 6(2), 325–362 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Oleinik O.A., Radkevic E.V., Second order differential equations with nonnegative characteristic form, American Mathematical Society, Rhode Island and Plenum Press, New York, 1973.

  24. Wang C.: Approximate controllability of a class of degenerate systems. Appl. Math. Comput. 203(1), 447–456 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wang C., Wang L., Yin J., Zhou S.: Hölder continuity of weak solutions of a class of linear equations with boundary degeneracy. J. Differential Equations 239(1), 99–131 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yin J., Wang C.: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differential Equations 237(2), 421–445 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Wu, Z., Zhao, J., Yin, J., Li, H., Nonlinear diffusion equations, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Jilin University, 130012, Changchun, People’s Republic of China

    Chunpeng Wang

Authors
  1. Chunpeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chunpeng Wang.

Additional information

Supported by the NNSF, FANEDD, NCET (985 program) and SRPDYS (Jilin Province) of China.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, C. Approximate controllability of a class of semilinear systems with boundary degeneracy. J. Evol. Equ. 10, 163–193 (2010). https://doi.org/10.1007/s00028-009-0044-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-009-0044-4

Mathematics Subject Classification (2000)

Keywords

Search

Navigation