Null controllability for a class of degenerate parabolic equations with the gradient terms

Abstract

This paper concerns a class of control systems governed by degenerate parabolic equations with the gradient terms, which are independent of the diffusion terms. The Carleman estimates and the observability inequalities for the equations are established when the degeneracy is relatively weak. Subsequently, it is proved that the control systems are null controllable. Moreover, the result can be generalized to the semilinear equations by using the fixed point theorem.

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References

  1. 1.

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

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

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

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Black, F. and Scholes M., The pricing of options and corporate liabilities, J. Political Economy, 81 (1973), 637–659.

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

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

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cannarsa, P., Fragnelli, G. and Rocchetti, D., Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2(4)(2007), 695–715.

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Cannarsa, P., Fragnelli, G. and Rocchetti, D., Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8(4)(2008), 583–616.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

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

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Du, R., Approximate controllability of a class of semilinear degenerate systems with boundary control, J.Differential Equations, 256(9)(2014), 3141–3165.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Du, R., Wang, C. and Zhou, Q., Approximate Controllability of a Semilinear System Involving a Fully Nonlinear Gradient Term, Appl Math Optim, 70(2014), 165–183.

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Du, R. and Wang, C., Null controllability of a class of systems governed by coupled degenerate equations, Applied Mathematics Letters , 26(2013) 113–119.

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

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

    MathSciNet  MATH  Google Scholar 

  15. 15.

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

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Flores, C. and Teresa, L., Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Math. Acad. Sci. Paris, 348(2010), 391–396.

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

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

  18. 18.

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

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Imanuvilov, Oleg Yu. and Yamamoto, Masahiro,Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for Semilinear Parabolic Equations, Publ. RIMS, Kyoto Univ., 39(2003), 227–274.

  20. 20.

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

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

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

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

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

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

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

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    North, G. R., Howard, L., Pollard D. and Wielicki B., Variational formulation of Budyko-Sellers climate models, J. Atmos. Sci., 36(1979), 255–259.

    Article  Google Scholar 

  25. 25.

    Sakthivel, R., Ganesh, R., Ren, Y. and Anthoni, S.M., Approximate controllability of nonlinear fractional dynamical systems, Commun Nonlinear Sci Number Simulat, 18(12)(2013), 3498–3508.

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Sakthivel, R., Ren, Y. and Mahmudov, N. I., Approximate controllability of second-order stochastic differential equations with impulsive effects, Modern Physics Letters B , 24(14)(2010), 1559.

  27. 27.

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

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Wang, C., Approximate controllability of a class of semilinear systems with boundary degeneracy, J. Evol. Equ., 10(1)(2010),163–193.

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Wang, C., Du, R., Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, SIAM J. Control Optim., 52(3)(2014), 1457–1480.

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Wang, C., Zhou, Y. , Du, R. and Liu, Q., Carleman estimates and null controllability for a class of degenerate parabolic equations with convection terms, Discrete Contin. Dyn. Syst. Ser. B., 23(10)(2018), 4207–4222.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Runmei Du.

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Supported by the Natural Science Foundation for Young Scientists of Jilin Province, China (No. 20170520048JH) and National Natural Science Foundation of China (No. 11571137).

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Du, R. Null controllability for a class of degenerate parabolic equations with the gradient terms. J. Evol. Equ. 19, 585–613 (2019). https://doi.org/10.1007/s00028-019-00487-8

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Keywords

  • Carleman estimate
  • Null controllability
  • Degeneracy

Mathematics Subject Classification

  • 93B05
  • 93C20
  • 35K65