WIAS Preprint No. 2006, (2014)

Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials


  • Colli, Pierluigi
    ORCID: 0000-0002-7921-5041
  • Farshbaf Shaker, Mohammad Hassan
    ORCID: 0000-0003-0543-5938
  • Gilardi, Gianni
    ORCID: 0000-0002-0651-4307
  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604

2010 Mathematics Subject Classification

  • 74M15 49K20 35K61


  • Optimal control, parabolic obstacle problems, MPECs, dynamic boundary conditions, optimality conditions




In this paper, we investigate optimal boundary control problems for Cahn--Hilliard variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace--Beltrami operator. The cost functional is of standard tracking type, and box constraints for the controls are prescribed. We prove existence of optimal controls and derive first-order necessary conditions of optimality. The general strategy, which follows the lines of the recent approach by Colli, Farshbaf-Shaker, Sprekels (see Appl. Math. Optim., 2014) to the (simpler) Allen--Cahn case, is the following: we use the results that were recently established by Colli, Gilardi, Sprekels in the preprint arXiv:1407.3916 [math.AP] for the case of (differentiable) logarithmic potentials and perform a so-called ``deep quench limit''. Using compactness and monotonicity arguments, it is shown that this strategy leads to the desired first-order necessary optimality conditions for the case of (non-differentiable) double obstacle potentials.

Appeared in

  • SIAM J. Control Optim., 53 (2015) pp. 2696--2721.

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