141 research outputs found
Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D
In this paper we study a distributed optimal control problem for a nonlocal
convective Cahn--Hilliard equation with degenerate mobility and singular
potential in three dimensions of space. While the cost functional is of
standard tracking type, the control problem under investigation cannot easily
be treated via standard techniques for two reasons: the state system is a
highly nonlinear system of PDEs containing singular and degenerating terms, and
the control variable, which is given by the velocity of the motion occurring in
the convective term, is nonlinearly coupled to the state variable. The latter
fact makes it necessary to state rather special regularity assumptions for the
admissible controls, which, while looking a bit nonstandard, are however quite
natural in the corresponding analytical framework. In fact, they are
indispensable prerequisites to guarantee the well-posedness of the associated
state system. In this contribution, we employ recently proved existence,
uniqueness and regularity results for the solution to the associated state
system in order to establish the existence of optimal controls and appropriate
first-order necessary optimality conditions for the optimal control problem
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
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
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
On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems
The phase separation of an isothermal incompressible binary fluid in a porous
medium can be described by the so-called Brinkman equation coupled with a
convective Cahn-Hilliard (CH) equation. The former governs the average fluid
velocity , while the latter rules evolution of , the
difference of the (relative) concentrations of the two phases. The two
equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular,
the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg
force) which is proportional to , where is the chemical
potential. When the viscosity vanishes, then the system becomes the
Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the
theoretical and the numerical viewpoints. However, theoretical results on the
CHHS system are still rather incomplete. For instance, uniqueness of weak
solutions is unknown even in 2D. Here we replace the usual CH equation with its
physically more relevant nonlocal version. This choice allows us to prove more
about the corresponding nonlocal CHHS system. More precisely, we first study
well-posedness for the CHB system, endowed with no-slip and no-flux boundary
conditions. Then, existence of a weak solution to the CHHS system is obtained
as a limit of solutions to the CHB system. Stronger assumptions on the initial
datum allow us to prove uniqueness for the CHHS system. Further regularity
properties are obtained by assuming additional, though reasonable, assumptions
on the interaction kernel. By exploiting these properties, we provide an
estimate for the difference between the solution to the CHB system and the one
to the CHHS system with respect to viscosity
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
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
Regularity results and optimal velocity control of the convective nonlocal Cahn-Hilliard equation in 3D
In this contribution, we study an optimal control problem for the celebrated
nonlocal Cahn-Hilliard equation endowed with the singular Flory-Huggins
potential in the three-dimensional setting. The control enters the governing
state system in a nonlinear fashion in the form of a prescribed solenoidal,
that is a divergence-free, vector field, whereas the cost functional to be
minimized is of tracking-type. The novelties of the present paper are twofold:
in addition to the control application, the intrinsic difficulties of the
optimization problem forced us to first establish new regularity results on the
nonlocal Cahn-Hilliard equation that were unknown even without the coupling
with a velocity field and are therefore of independent interest. This happens
to be shown using the recently proved separation property along with ad hoc
H\"older regularities and a bootstrap method. For the control problem, the
existence of an optimal strategy as well as first-order necessary conditions
are then established
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Optimal distributed control of two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular potential
In this paper, we consider a two-dimensional diffuse interface model
for the phase separation of an incompressible and isothermal binary fluid
mixture with matched densities. This model consists of the NavierStokes
equations, nonlinearly coupled with a convective nonlocal CahnHilliard
equation. The system rules the evolution of the (volume-averaged) velocity u
of the mixture and the (relative) concentration difference ' of the two
phases. The aim of this work is to study an optimal control problem for such
a system, the control being a time-dependent external force v acting on the
fluid. We first prove the existence of an optimal control for a given
tracking type cost functional. Then we study the differentiability properties
of the control-to-state map v 7! [u; '], and we establish first-order
necessary optimality conditions. These results generalize the ones obtained
by the first and the third authors jointly with E. Rocca in [19]. There the
authors assumed a constant mobility and a regular potential with polynomially
controlled growth. Here, we analyze the physically more relevant case of a
degenerate mobility and a singular (e.g., logarithmic) potential. This is
made possible by the existence of a unique strong solution which was recently
proved by the authors and C. G. Gal in [14]
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