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Title: Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions
Authors: Colli, PierluigiGilardi, GianniSprekels, Jürgen
Publishers Version: https://doi.org/10.20347/WIAS.PREPRINT.2427
Issue Date: 2017
Published in: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik , Volume 2427, ISSN 2198-5855
Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Abstract: In this paper, we investigate a distributed optimal control problem for a convective viscous CahnHilliard system with dynamic boundary conditions. Such systems govern phase separation processes between two phases taking place in an incompressible fluid in a container and, at the same time, on the container boundary. The cost functional is of standard tracking type, while the control is exerted by the velocity of the fluid in the bulk. In this way, the coupling between the state (given by the associated order parameter and chemical potential) and control variables in the governing system of nonlinear partial differential equations is bilinear, which presents an additional difficulty for the analysis. The nonlinearities in the bulk and surface free energies are of logarithmic type, which entails that the thermodynamic forces driving the phase separation process may become singular. We show existence for the optimal control problem under investigation, prove the Fréchet differentiability of the associated control-to-state mapping in suitable Banach spaces, and derive the first-order necessary optimality conditions in terms of a variational inequality and the associated adjoint system. Due to the strong nonlinear couplings between state variables and control, the corresponding proofs require a considerable analytical effort.
Keywords: Cahn–Hilliard system; convection term; dynamic boundary conditions; optimal velocity control; optimality conditions; adjoint state system
DDC: 510
License: This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.
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