Optimal boundary control of a viscous Cahn--Hilliard system with dynamic boundary condition and double obstacle potentials
- Colli, Pierluigi
- Farshbaf Shaker, Mohammad Hassan
- Gilardi, Gianni
- Sprekels, Jürgen
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.
- SIAM J. Control Optim., 53 (2015) pp. 2696--2721.