23 research outputs found
Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
This paper is concerned with a distributed optimal control problem for a
nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version
of a model for two-species phase segregation on an atomic lattice under the
presence of diffusion. The local model has been investigated in a series of
papers by P. Podio-Guidugli and the present authors; the nonlocal model studied
here consists of a highly nonlinear parabolic equation coupled to an ordinary
differential inclusion of subdifferential type. The inclusion originates from a
free energy containing the indicator function of the interval in which the
order parameter of the phase segregation attains its values. It also contains a
nonlocal term modeling long-range interactions. Due to the strong nonlinear
couplings between the state variables (which even involve products with time
derivatives), the analysis of the state system is difficult. In addition, the
presence of the differential inclusion is the reason that standard arguments of
optimal control theory cannot be applied to guarantee the existence of Lagrange
multipliers. In this paper, we employ recent results proved for smooth
logarithmic potentials and perform a so-called `deep quench' approximation to
establish existence and first-order necessary optimality conditions for the
nonsmooth case of the double obstacle potential.Comment: Key words: distributed optimal control, phase field systems, double
obstacle potentials, nonlocal operators, first-order necessary optimality
conditions. The interested reader can also see the related preprints
arXiv:1511.04361 and arXiv:1605.07801 whose results are recalled and used for
the analysis carried out in this pape
Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach
In this paper, we investigate a distributed optimal control problem for a
convective viscous Cahn-Hilliard 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 a difficulty for the
analysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by the
same authors, the bulk and surface free energies are of double obstacle type,
which renders the state constraint nondifferentiable. It is well known that for
such cases standard constraint qualifications are not satisfied so that
standard methods do not apply to yield the existence of Lagrange multipliers.
In this paper, we overcome this difficulty by taking advantage of results
established in the quoted paper for logarithmic nonlinearities, using a
so-called `deep quench approximation'. We derive results concerning the
existence of optimal controls and the first-order necessary optimality
conditions in terms of a variational inequality and the associated adjoint
system.Comment: Key words: Cahn-Hilliard system, convection term, dynamic boundary
conditions, double obstacle potentials, optimal velocity control, optimality
conditions. arXiv admin note: text overlap with arXiv:1702.0190
Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials
The paper arXiv:1804.11290 contains well-posedness and regularity results for
a system of evolutionary operator equations having the structure of a
Cahn-Hilliard system. The operators appearing in the system equations were
fractional versions in the spectral sense of general linear operators A and B
having compact resolvents and are densely defined, unbounded, selfadjoint, and
monotone in a Hilbert space of functions defined in a smooth domain. The
associated double-well potentials driving the phase separation process modeled
by the Cahn-Hilliard system could be of a very general type that includes
standard physically meaningful cases such as polynomial, logarithmic, and
double obstacle nonlinearities. In the subsequent paper arXiv:1807.03218, an
analysis of distributed optimal control problems was performed for such
evolutionary systems, where only the differentiable case of certain polynomial
and logarithmic double-well potentials could be admitted. Results concerning
existence of optimizers and first-order necessary optimality conditions were
derived. In the present paper, we complement these results by studying a
distributed control problem for such evolutionary systems in the case of
nondifferentiable nonlinearities of double obstacle type. For such
nonlinearities, it is well known that the standard constraint qualifications
cannot be applied to construct appropriate Lagrange multipliers. To overcome
this difficulty, we follow here the so-called "deep quench" method. We first
give a general convergence analysis of the deep quench approximation that
includes an error estimate and then demonstrate that its use leads in the
double obstacle case to appropriate first-order necessary optimality conditions
in terms of a variational inequality and the associated adjoint state system.Comment: Key words: Fractional operators, Cahn-Hilliard systems, optimal
control, double obstacles, necessary optimality condition
Optimal Distributed Control of a Cahn-Hilliard-Darcy System with Mass Sources
In this paper, we study an optimal control problem for a two-dimensional
Cahn-Hilliard-Darcy system with mass sources that arises in the modeling of
tumor growth. The aim is to monitor the tumor fraction in a finite time
interval in such a way that both the tumor fraction, measured in terms of a
tracking type cost functional, is kept under control and minimal harm is
inflicted to the patient by administering the control, which could either be a
drug or nutrition. We first prove that the optimal control problem admits a
solution. Then we show that the control-to-state operator is Fr\'echet
differentiable between suitable Banach spaces and derive the first-order
necessary optimality conditions in terms of the adjoint variables and the usual
variational inequality
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Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: A deep quench approach
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 a
difficulty for the analysis. In contrast to the previous paper Optimal
velocity control of a viscous CahnHilliard system with convection and dynamic
boundary conditions by the same authors, the bulk and surface free energies
are of double obstacle type, which renders the state constraint
nondifferentiable. It is well known that for such cases standard constraint
qualifications are not satisfied so that standard methods do not apply to
yield the existence of Lagrange multipliers. In this paper, we overcome this
difficulty by taking advantage of results established in the quoted paper for
logarithmic nonlinearities, using a so-called deep quench approximation. We
derive results concerning the existence of optimal controls and the
first-order necessary optimality conditions in terms of a variational
inequality and the associated adjoint system
Optimal distributed control of a generalized fractional Cahn-Hilliard system
In the recent paper `Well-posedness and regularity for a generalized
fractional Cahn-Hilliard system' (arXiv:1804.11290) by the same authors,
general well-posedness results have been established for a a class of
evolutionary systems of two equations having the structure of a viscous
Cahn-Hilliard system, in which nonlinearities of double-well type occur. The
operators appearing in the system equations are fractional versions in the
spectral sense of general linear operators A,B having compact resolvents, which
are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of
functions defined in a smooth domain. In this work we complement the results
given in arXiv:1804.11290 by studying a distributed control problem for this
evolutionary system. The main difficulty in the analysis is to establish a
rigorous Frechet differentiability result for the associated control-to-state
mapping. This seems only to be possible if the state stays bounded, which, in
turn, makes it necessary to postulate an additional global boundedness
assumption. One typical situation, in which this assumption is satisfied,
arises when B is the negative Laplacian with zero Dirichlet boundary conditions
and the nonlinearity is smooth with polynomial growth of at most order four.
Also a case with logarithmic nonlinearity can be handled. Under the global
boundedness assumption, we establish existence and first-order necessary
optimality conditions for the optimal control problem in terms of a variational
inequality and the associated adjoint state system.Comment: Key words: fractional operators, Cahn-Hilliard systems, optimal
control, necessary optimality condition
Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property
Optimal velocity control of a viscous Cahn--Hilliard system with convection and dynamic boundary conditions
In this paper, we investigate a distributed optimal control problem for a convective viscous Cahn--Hilliard 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 Fr'echet 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