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Author: Sprekels, Jürgen × Start date: 2010 × Type: Text × Type: report × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: dynamic boundary conditions ×

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Now showing 1 - 10 of 14
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    A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved.
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    Second-order analysis of a boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary condition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Farshbaf Shaker, Mohammad Hassan; Gilardi, Gianni; Sprekels, Jürgen
    In this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn-Hilliard equation with possibly singular potentials and dynamic boundary conditions.
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    Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fréchet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.
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    Optimal boundary control of a viscous Cahn-Hilliard system with dynamic boundary condition and double obstacle potentials
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Farshbaf Shaker, Mohammad Hassan; Gilardi, Gianni; Sprekels, Jürgen
    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.
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    Global existence for a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    In this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.
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    A deep quench approach to the optimal control of an Allen-Cahn equation with dynamic boundary conditions and double obstacles
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Colli, Pierluigi; Farshbaf-Shaker, M. Hassan; Sprekels, Jürgen
    In this paper, we investigate optimal control problems for Allen-Cahn variational inequalities with a dynamic boundary condition involving double obstacle potentials and the Laplace-Beltrami operator. The approach covers both the cases of distributed controls and of boundary controls. 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 is the following: we use the results that were recently established by two of the authors in the paper [5] 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.
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    A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    A boundary control problem for the pure Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first-order necessary conditions for optimality are proved.
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    Optimal boundary control of a nonstandard Cahn-Hilliard system with dynamic boundary condition and double obstacle inclusions : dedicated to our friend Prof. Dr. Gianni Gilardi on the occasion of his 70th birthday
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Sprekels, Jürgen
    In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105–118. The model consists of a strongly coupled system of nonlinear parabolic differential inclusions, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically; the system is complemented by initial and boundary conditions. For the order parameter of the phase separation process, a dynamic boundary condition involving the Laplace–Beltrami operator is assumed, which models an additional nonconserving phase transition occurring on the surface of the domain. We complement in this paper results that were established in the recent contribution appeared in Evol. Equ. Control Theory 6 (2017), pp. 35–58, by the two authors and Gianni Gilardi. In contrast to that paper, in which differentiable potentials of logarithmic type were considered, we investigate here the (more difficult) case of nondifferentiable potentials of double obstacle type. For such nonlinearities, the standard techniques of optimal control theory to establish the existence of Lagrange multipliers for the state constraints are known to fail. To overcome these difficulties, we employ the following line of approach: we use the results contained in the preprint arXiv:1609.07046 [math.AP] (2016), pp. 1–30, 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 (nondifferentiable) double obstacle potentials.
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    On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    The Cahn-Hilliard and viscous Cahn-Hilliard equations with singular and possibly nonsmooth potentials and dynamic boundary condition are considered and some well-posedness and regularity results are proved.
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    Optimal velocity control of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Gilardi, Gianni; Sprekels, Jürgen
    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.