Optimal Control of a Semidiscrete Cahn--Hilliard--Navier--Stokes System

In this paper, the optimal boundary control of a time-discrete Cahn--Hilliard--Navier--Stokes system is studied. A general class of free energy potentials is considered which, in particular, includes the double-obstacle potential. The latter homogeneous free energy density yields an optimal control...

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Bibliographic Details
Published in:SIAM journal on control and optimization Vol. 52; no. 1; pp. 747 - 772
Main Authors: Hintermüller, M., Wegner, D.
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Applied mathematics
Approximation
Decomposition
Energy
Fluid mechanics
Mathematical programming
Polymers
Reynolds number
ISSN:0363-0129, 1095-7138
Online Access:Get full text
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Summary:In this paper, the optimal boundary control of a time-discrete Cahn--Hilliard--Navier--Stokes system is studied. A general class of free energy potentials is considered which, in particular, includes the double-obstacle potential. The latter homogeneous free energy density yields an optimal control problem for a family of coupled systems, which result from a time discretization of a variational inequality of fourth order and the Navier--Stokes equation. The existence of an optimal solution to the time-discrete control problem as well as an approximate version is established. The latter approximation is obtained by mollifying the Moreau--Yosida approximation of the double-obstacle potential. First order optimality conditions for the mollified problems are given, and in addition to the convergence of optimal controls of the mollified problems to an optimal control of the original problem, first order optimality conditions for the original problem are derived through a limit process. The newly derived stationarity system is related to a function space version of C-stationarity. [PUBLICATION ABSTRACT]
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ISSN:0363-0129
1095-7138
DOI:10.1137/120865628