Optimal Control Problem for Cahn–Hilliard Equations with State Constraint

1. Bai F., Elliott C.M., Gardiner A., Spence A., Stuart A.M. The viscous Cahn–Hilliard equation. Part I. Computations. Nonlinearity 1995;8:131–160.

   Article  MATH  MathSciNet  Google Scholar 

2. Barbu V. Local controllability of the phase field system. Nonlinear Anal. TMA. 2002;50:363–372.

   Article  MATH  MathSciNet  Google Scholar 

3. Barbu V. Optimal control of variational inequalities. In: Pitman research notes in mathematics, London, Boston.

4. Cahn J. W., Hilliard J. E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 1958;2:258–267.

   Article  Google Scholar 

5. Chen Z. Optimal boundary controls for a phase field model. IMA J. Math. Control and Information 1993;10:157–176.

   Article  MATH  MathSciNet  Google Scholar 

6. Colli P., Gilardi G., Podio-Guidugli P., Sprekels J. Distributed optimal control problem of a nonstandard system of phase field equations. Contin. Mech. Thermodyn. 2012;24:437–459.

   Article  MATH  MathSciNet  Google Scholar 

7. Elliott C.M., Kostin I.N. Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn–Hilliard equation. Nonlinearity 1996;9:687–702.

   Article  MATH  MathSciNet  Google Scholar 

8. Hinterm̈ller M., Wegner D. Distributed optimal control of the Chan–Hilliard system including the case of a double-obstacle homogeneous free energy density. SIAM J. Control Optim. 2012;50:388–418.

   Article  MathSciNet  Google Scholar 

9. Ito A., Kenmochi N., Niezgdka M. Phase separation model of Penrose–Fife type with Signorini boundary condition. Adv. Math. Sci. Appl. 2007;17:337–356.

   MATH  MathSciNet  Google Scholar 

10. Kenmochi N., Niezgódka M., Pawlow I. Subdifferential operator approach to the Cahn–Hilliard equation with constraint. J. Diff. Eqns. 1995;117:320–356.

    Article  MATH  Google Scholar 

11. Kubo M. The Cahn–Hilliard equation with time-dependent constraint. Nonlinear Anal. TMA. 2012;75:5672–5685.

    Article  MATH  MathSciNet  Google Scholar 

12. Lefter C., Sprekels J. Optimal boundary control of a phase field system modeling non-isothermal phase transitions. Adv. Math. Sci. Appl. 2007;17:181–194.

    MATH  MathSciNet  Google Scholar 

13. Li X., Yong J. Optimal control theory for infinite dimensional system. Boston: Birhauser; 1995.

    Book  Google Scholar 

14. Lions J.L. Optimal control of systems governed by partial differential equations. Berlin: Springer; 1971.

    Book  MATH  Google Scholar 

15. Miranville A. Generalized Cahn–Hilliard equations based on a microforce balance. J. Appl. Math. 2003;4:165–185.

    Article  MathSciNet  Google Scholar 

16. Moroşanu C., Wang G. State-constrained optimal control for the phase-field transition system. Numerical Numer. Funct. Anal. Optim. 2007;28:379–403.

    Article  MATH  Google Scholar 

17. Novick-Cohen A. The Cahn–Hilliard equation. In: Handbook of differential equations, Evolutionary Partial Differential Equations. Amsterdam: Elsevier, p. 2008.

18. Park J., Jeong J., Kang Y., Optimal control of parabolic variational inequalities with delays and state constraint. Nonlinear Anal. 2009;71:329–339.

    Article  MathSciNet  Google Scholar 

19. Ryu S.-U., Yagi A. Optimal control for an adsorbate-induced phase transition model. Appl. Math. Comput. 2005;171:420–432.

    Article  MATH  MathSciNet  Google Scholar 

20. Temam R. Infinite-dimensional dynamical systems in mechanics and physics, second ed. In: Applied mathematical sciences. New York: Springer, p. 1997.

21. Wang G. Optimal control of 3-dimensional Navier–Stokes equations with state constraints. SIAM J. Control Optim. 2002;41:583–606.

    Article  MATH  MathSciNet  Google Scholar 

22. Wang Q., Nakagiri S. Weak solutions of Cahn–Hilliard equations having forcing terms and optimal control problems. Res. Inst. Math. Sci. Kokyuüroku 2000;1128:172–180.

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

24. Zhao X., Liu C. Optimal control problem for viscous Cahn–Hilliard equation. Nonlinear Anal. TMA. 2011;74:6348–6357.

    Article  MATH  Google Scholar 

25. Zhao X., Liu C. 2011. Optimal control of the convective Cahn–Hilliard equation, Applicable Analysis.

Download references