Skip to main content
SpringerLink
Log in

Optimal Control Problem for the Cahn–Hilliard/Allen–Cahn Equation with State Constraint

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

Abstract

In this paper, we consider a distributed optimal control problem for the Cahn–Hilliard/Allen–Cahn equation with state-constraint. The objective is to force the coverage y to have some specified properties or achieve a certain goal. Since the cost functional is discontinuous, together with state constraint, we employ a new penalty functional by the approximation of the cost functional, in this case, we derive the necessary optimality conditions for the approximating optimal control problem. Finally, by considering the limits of the necessary optimality conditions we have obtained, we solve the optimal control problem and derive the necessary optimality conditions.

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. Alt, H.W.: Lineare Funktionalanalysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  2. Antonopoulou, D.C., Karali, G., Millet, A.: Existence and regularity of solution for a stochastic Cahn–Hilliard/Allen–Cahn equation with unbounded noise diffusion. J. Differ. Equ. 260(3), 2383–2417 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barbu, V.: Optimal Control of Variational Inequalities, Pitman Research Notes in Mathematics. Pitman, London, Boston (1984)

    Google Scholar 

  4. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces, 4th edn. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  5. Benincasa, T., Donado Escobar, L.D., Morosanu, C.: Distributed and boundary optimal control of the Allen–Cahn equation with regular potential and dynamic boundary conditions. Int. J. Control 89(8), 1523–1532 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Colli, P., Sprekels, J.: Optimal control of an Allen–Cahn equation with singular potentials and dynamic boundary condition. SIAM J. Control Optim. 53(1), 213–234 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Colli, P., Farshbaf-Shaker, M.H., Gilardi, G., Sprekels, J.: Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials. SIAM J. Control Optim. 53(4), 2696–2721 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions. Appl. Math. Optim. 73(2), 195–225 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colli, P., Gilardi, G., Sprekels, J.: Optimal velocity control of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions. SIAM J. Control Optim. 56(3), 1665–1691 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gao, P.: Optimal distributed control of the Kuramoto—Sivashinsky equation with pointwise state and mixed control state constraints. IMA J. Math. Control Inf. 33(3), 791–811 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hintermüller, M., Wegner, D.: Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 50(1), 388–418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Israel, H.: Long time behavior of an Allen–Cahn type equation with a singular potential and dynamic boundary conditions. J. Appl. Anal. Comput. 2(1), 29–56 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Israel, H.: Well-posedness and long time behavior of an Allen–Cahn type equation. Commun. Pure Appl. Anal. 12(6), 2811–2827 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Israel, H., Miranville, A., Petcu, M.: Well-posedness and long time behavior of a perturbed Cahn–Hilliard system with regular potentials. Asymptot. Anal. 84(3/4), 147–179 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Karali, G., Katsoulakis, M.A.: The role of multiple microscopic mechanisms in cluster interface evolution. J. Differ. Equ. 235, 418–438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karali, G., Nagase, Y.: On the existence of solution for a Cahn–Hilliard/Allen–Cahn equation. Discret. Contin. Dyn. Syst. Ser. S 7(1), 127–137 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Karali, G., Ricciardi, T.: On the convergence of a fourth order evolution equation to the Allen–Cahn equation. Nonlinear Anal. 72, 4271–4281 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  19. Liu, C., Tang, H.: Existence of periodic solution for a Cahn–Hilliard/Allen–Cahn equation in two space dimensions. Evol. Equ. Control Theory 6(2), 219–237 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, C., Wang, Z.: Optimal control for a sixth order nonlinear parabolic equation. Math. Methods Appl. Sci. 38(2), 247–262 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Morosanu, C., Wang, G.: State-constrained optimal control for the phase-field transition system. Numer. Funct. Anal. Optim. 28(3/4), 379–403 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sideris, T.C.: Ordinary Differential Equations and Dynamical Systems, Atlantis Studies in Differential Equations. Atlantis Press, Paris (2013)

    Book  Google Scholar 

  23. Simon, J.: Compact sets in the space $L^{p}(0, T;B)$. Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Tachim Medjo, T.: Optimal control of the primitive equations of the ocean with state constraints. Nonlinear Anal. 73(3), 634–649 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)

    Book  MATH  Google Scholar 

  26. Tian, L., Shen, C., Ding, D.: Optimal control of the viscous Camassa–Holm equation. Nonlinear Anal. Real World Appl. 10(1), 519–530 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, G.: Optimal controls of 3-dimensional Navier–Stokes equations with state constraints. SIAM J. Control Optim. 41(2), 583–606 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, G., Zhao, Y., Li, W.: Some optimal control problems governed by elliptic variational inequalities with control and state constraint on the boundary. J. Optim. Theory Appl. 106(3), 627–655 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, X., Liu, C.: Optimal control problem for viscous Cahn–Hilliard equation. Nonlinear Anal. 74(17), 6348–6357 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhao, X., Liu, C.: Optimal control for the convective Cahn–Hilliard equation in 2D case. Appl. Math. Optim. 70(1), 61–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, X., Liu, C.: Existence of solutions to the Cahn–Hilliard/Allen–Cahn equation with degenerate mobility. Electron. J. Differ. Equ. 329, 22 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Zheng, J.: Optimal control problem for Lengyel–Epstein model with obstacles and state constraints. Nonlinear Anal. Model. Control 21(1), 18–39 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zheng, J., Wang, Y.: Optimal control problem for Cahn–Hilliard equations with state constraint. J. Dyn. Control Syst. 21(2), 257–272 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their deep thanks to the referee’s valuable suggestions for the revision and improvement of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Jilin University, Changchun, 130012, China

    Xiaoli Zhang, Huilai Li & Changchun Liu

Authors
  1. Xiaoli Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huilai Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Changchun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changchun Liu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the Jilin Scientific and Technological Development Program (No. 20170101143JC) and the National Natural Science Foundation of China (No. 11471164).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, X., Li, H. & Liu, C. Optimal Control Problem for the Cahn–Hilliard/Allen–Cahn Equation with State Constraint. Appl Math Optim 82, 721–754 (2020). https://doi.org/10.1007/s00245-018-9546-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-018-9546-1

Keywords

Mathematics Subject Classification

Search

Navigation