Local Exact Controllability to the Trajectories of the Cahn–Hilliard Equation

Abstract

In this paper we prove the local exact controllability to the trajectories of the Cahn–Hilliard equation, which is a nonlinear fourth-order parabolic equation, by means of a control supported on an interior open interval. To prove this result we derive a Carleman estimate that allows us to conclude, thanks to a duality argument, the null controllability of the linearized equation around a given solution. Then, we apply a local inversion theorem to extend the control result to the nonlinear equation.

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References

  1. 1.

    Alekseev, V., Tikhomirov, V., Fomin, S.: Optimal control. In: Contemporary Soviet Mathematics, Springer Science & Business Media (1987)

  2. 2.

    Baudouin, L., Cerpa, E., Crépeau, E., Mercado, A.: Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation. Appl. Anal. 92(10), 2084–2102 (2013)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Cahn, J.: On spinodal decomposition. Acta Metall. 9(9), 795–801 (1961)

    Google Scholar 

  4. 4.

    Cahn, J., Hilliard, J.: Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    MATH  Google Scholar 

  5. 5.

    Cahn, J., Hilliard, J.: Spinodal decomposition. Reprise Acta Metall. 19(2), 151–161 (1971)

    Google Scholar 

  6. 6.

    Carreño, N.: Local controllability of the N-dimensional Boussinesq system with N-1 scalar controls in an arbitrary control domain. Math. Control Relat. Fields 2(4), 361–382 (2012)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Carreño, N., Cerpa, E.: Local controllability of the stabilized Kuramoto–Sivashinsky system by a single control acting on the heat equation. J. Math. Pures Appl. 106(4), 670–694 (2016)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Carreño, N., Guerrero, S.: Local null controllability of the N-dimensional Navier–Stokes system with N-1 scalar controls in an arbitrary control domain. J. Math. Fluid Mech. 14(1), 139–152 (2013)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Carreño, N., Guzmán, P.: On the cost of null controllability of a fourth-order parabolic equation. J. Differ. Equat. 261(11), 6485–6520 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Cerpa, E., Mercado, A.: Local exact controllability to the trajectories of the 1-D Kuramoto–Sivashinsky equation. J. Differ. Equat. 250(4), 2024–2044 (2011)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cerpa, E., Mercado, A., Pazoto, A.: On the boundary control of a parabolic system coupling KS-KdV and heat equations. Sci. Ser. A 22, 55–74 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Cerpa, E., Mercado, A., Pazoto, A.: Null controllability of the stabilized Kuramoto–Sivashinsky system with one distributed control. SIAM J. Control Optim. 53(3), 1543–1568 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Cerpa, E., Guzmán, P., Mercado, A.: On the control of the linear Kuramoto–Sivashinsky equation. ESAIM Control Optim. Calc. Var. 23(1), 165–194 (2017)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology. Functional and Variational Methods, vol. 2. Springer, Berlin (2000)

    MATH  Google Scholar 

  15. 15.

    Díaz, J., Ramos, Á.: On the approximate controllability for higher order parabolic nonlinear equations of the Cahn–Hilliard type, In: Control and Estimation of Distributed Parameters Systems, International Series of Numerical Mathematics, vol. 126, Birkhauser (1998)

  16. 16.

    Elliott, C.: The Cahn–Hilliard model for the kinetics of phase separation, In: Mathematical Models for Phase Change Problems, International Series on Numerical Mathematics, vol. 88, Birkhauser (1989)

  17. 17.

    Elliott, C., Zheng, S.: On the Cahn–Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)

    MATH  Google Scholar 

  18. 18.

    Fernández-Cara, E., Guerrero, S., Imanuvilov, O., Puel, J.-P.: Local exact controllability of the Navier–Stokes system. J. Math. Pures Appl. 83(12), 1501–1542 (2004)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Fernández-Cara, E., Guerrero, S., Imanuvilov, O., Puel, J.-P.: Some controllability results for the N-dimensional Navier–Stokes system and Boussinesq systems with N-1 scalar controls. SIAM J. Control Optim. 45(1), 146–173 (2006)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Fursikov, A., Imanuvilov, O.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Seoul (1996)

    MATH  Google Scholar 

  21. 21.

    Gao, P.: : Insensitizing controls for the Cahn–Hilliard type equation. Electron. J. Qual. Theory Differ. Equt. 35, 1–22 (2014)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Gao, P.: A new global Carleman estimate for the one-dimensional Kuramoto–Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem. Nonlinear Anal. 117, 133–147 (2015)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Gao, P.: A new global Carleman estimate for Cahn–Hilliard type equation and its applications. J. Differ. Equat. 260(1), 427–444 (2016)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Guerrero, S.: Local exact controllability to the trajectories of the Boussinesq system. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(1), 29–61 (2006)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Guzmán, Patricio: Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation. J. Math. Anal. Appl. 408(1), 275–290 (2013)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Imanuvilov, Oleg: Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim. Calc. Var. 6, 39–72 (2001)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Novick-Cohen, A.: The Cahn–Hilliard equation. In: Dafermos, C.M., Pokorny, M. (eds.) Handbook of Differential Equations: Evolutionary Equations, pp. 201–228. Elsevier, Amsterdam (2008)

    Google Scholar 

  28. 28.

    Novick-Cohen, A., Segel, L.: Nonlinear aspects of the Cahn–Hilliard equation. Phys. D 10(3), 277–298 (1984)

    MathSciNet  Google Scholar 

  29. 29.

    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, vol. 68, Springer, New York (1997)

  30. 30.

    Yong, J., Zheng, S.: Feedback stabilization and optimal control for the Cahn–Hilliard equation. Nonlinear Anal. 17(5), 431–444 (1991)

    MathSciNet  MATH  Google Scholar 

  31. 31.

    Zheng, J.: Time optimal controls of the Cahn–Hilliard equation with internal control. Optim. Control Appl. Methods 36(4), 566–582 (2015)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Zheng, S.: Asymptotic behavior of solution to the Cahn–Hillard equation. Appl. Anal. 23(3), 165–184 (1986)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Zhou, Z.: Observability estimate and null controllability for one-dimensional fourth order parabolic equation. Taiwan. J. Math. 16(6), 1991–2017 (2012)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author dedicates this work to his parents, Mario Guzmán and Mariela Meléndez, for their unlimited love and support. This study has been partially supported by Basal Project FB0008, FONDECYT 1140741 and PIIC UTFSM 2015.

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Correspondence to Patricio Guzmán.

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Guzmán, P. Local Exact Controllability to the Trajectories of the Cahn–Hilliard Equation. Appl Math Optim 82, 279–306 (2020). https://doi.org/10.1007/s00245-018-9500-2

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Keywords

  • Cahn–Hilliard equation
  • Parabolic equation
  • Internal control
  • Null controllability
  • Carleman estimates

Mathematics Subject Classification

  • 35K35
  • 93B05
  • 93B07