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Optimal Boundary Control of a Simplified Ericksen–Leslie System for Nematic Liquid Crystal Flows in 2D

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Abstract

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen–Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg–Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Cecilia Cavaterra

  2. Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, Italy

    Elisabetta Rocca

  3. School of Mathematical Sciences and Shanghai, Key Laboratory for Contemporary Applied Mathematics, Fudan University, 200433, Shanghai, China

    Hao Wu

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  1. Cecilia Cavaterra
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  2. Elisabetta Rocca
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  3. Hao Wu
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Correspondence to Hao Wu.

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Communicated by F. Lin

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Cavaterra, C., Rocca, E. & Wu, H. Optimal Boundary Control of a Simplified Ericksen–Leslie System for Nematic Liquid Crystal Flows in 2D . Arch Rational Mech Anal 224, 1037–1086 (2017). https://doi.org/10.1007/s00205-017-1095-2

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