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A Note on the Feedback Stabilization of a Cahn–Hilliard Type System with a Singular Logarithmic Potential

  • Gabriela MarinoschiEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 22)

Abstract

This article deals with the internal feedback stabilization of a phase field system of Cahn–Hilliard type involving a logarithmic potential F, and extends the recent results provided in Barbu et al. (J Differ Equ 262:2286–2334, 2017) for the double-well potential. The stabilization is searched around a stationary solution, by a feedback controller with support in a subset ω of the domain. The controller stabilizing the linearized system is constructed as a finite combination of the unstable modes of the operator acting in the linear system and it is further provided in a feedback form by solving a certain minimization problem. Finally, it is proved that this feedback form stabilizes the nonlinear system too, if the stationary solution has not large variations. All these results are provided in the three-dimensional case for a regularization of the singular potential F, and allow the same conclusion for the singular logarithmic potential in the one-dimensional case.

Keywords

Cahn–Hilliard system Closed loop system Feedback control Logarithmic potential Stabilization 

AMS (MOS) Subject Classification

93D15 35K52 35Q79 35Q93 93C20 

Notes

Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th anniversary. The author would like to thank Professor Pierluigi Colli and the anonymous reviewer for the observations made on the earlier version of the paper.

References

  1. 1.
    Barbu, V.: Stabilization of Navier-Stokes Flows. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barbu, V., Triggiani, R.: Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J. 53, 1443–1494 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barbu, V., Wang, G.: Internal stabilization of semilinear parabolic systems. J. Math. Anal. Appl. 285, 387–407 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Barbu, V., Lasiecka, I., Triggiani, R.: Tangential boundary stabilization of Navier-Stokes equations. Mem. Am. Math. Soc. 181(852), 1–128 (2006)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Barbu, V., Colli, P., Gilardi, G., Marinoschi, G.: Feedback stabilization of the Cahn–Hilliard type system for phase separation. J. Differ. Equ. 262, 2286–2334 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  7. 7.
    Caginalp, G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2, 258–267 (1958)Google Scholar
  9. 9.
    Cherfils, L., Miranville, A., Zelik, S.: The Cahn–Hilliard equation with logarithmic potentials. Milan J. Math. 79, 561–596 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Colli, P., Gilardi, G., Marinoschi, G.: A boundary control problem for a possibly singular phase field system with dynamic boundary conditions. J. Math. Anal. Appl. 434, 432–463 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Triggiani, R.: Boundary feedback stabilizability of parabolic equations. Appl. Math. Optim. 6, 201–220 (1980)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Mathematical Statistics and Applied Mathematics of the Romanian AcademyBucharestRomania

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