310 research outputs found
Optimal control of an Allen-Cahn equation with singular potentials and dynamic boundary condition
In this paper, we investigate optimal control problems for Allen-Cahn
equations with singular nonlinearities and a dynamic boundary condition
involving singular nonlinearities 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. Parabolic problems with nonlinear dynamic
boundary conditions involving the Laplace-Beltrami operation have recently
drawn increasing attention due to their importance in applications, while their
optimal control was apparently never studied before. In this paper, we first
extend known well-posedness and regularity results for the state equation and
then show the existence of optimal controls and that the control-to-state
mapping is twice continuously Fr\'echet differentiable between appropriate
function spaces. Based on these results, we establish the first-order necessary
optimality conditions in terms of a variational inequality and the adjoint
state equation, and we prove second-order sufficient optimality conditions.Comment: Key words: optimal control; parabolic problems; dynamic boundary
conditions; optimality condition
A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation
In this paper, we consider a rather general linear evolution equation of
fractional type, namely a diffusion type problem in which the diffusion
operator is the th power of a positive definite operator having a discrete
spectrum in . We prove existence, uniqueness and differentiability
properties with respect to the fractional parameter . These results are then
employed to derive existence as well as first-order necessary and second-order
sufficient optimality conditions for a minimization problem, which is inspired
by considerations in mathematical biology.
In this problem, the fractional parameter serves as the "control
parameter" that needs to be chosen in such a way as to minimize a given cost
functional. This problem constitutes a new class of identification problems:
while usually in identification problems the type of the differential operator
is prescribed and one or several of its coefficient functions need to be
identified, in the present case one has to determine the type of the
differential operator itself.
This problem exhibits the inherent analytical difficulty that with changing
fractional parameter also the domain of definition, and thus the underlying
function space, of the fractional operator changes
A boundary control problem for the viscous Cahn-Hilliard equation with dynamic boundary conditions
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.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditions,
adjoint state syste
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
Optimal control of a phase field system of Caginalp type with fractional operators
In their recent work `Well-posedness, regularity and asymptotic analyses for
a fractional phase field system' (Asymptot. Anal. 114 (2019), 93-128; see also
the preprint arXiv:1806.04625), two of the present authors have studied phase
field systems of Caginalp type, which model nonconserved, nonisothermal phase
transitions and in which the occurring diffusional operators are given by
fractional versions in the spectral sense of unbounded, monotone, selfadjoint,
linear operators having compact resolvents. In this paper, we complement this
analysis by investigating distributed optimal control problems for such
systems. It is shown that the associated control-to-state operator is Fr\'echet
differentiable between suitable Banach spaces, and meaningful first-order
necessary optimality conditions are derived in terms of a variational
inequality and the associated adjoint state variables.Comment: 38 pages. Key words: fractional operators, phase field system,
nonconserved phase transition, optimal control, first-order necessary
optimality condition
Pavel Krejčí turns sixty and receives the Bernard Bolzano Honorary Medal for Merit in Mathematical Sciences
summary:The recent development of mathematical methods of investigation of problems with hysteresis has shown that the structure of the hysteresis memory plays a substantial role. In this paper we characterize the hysteresis operators which exhibit a memory effect of the Preisach type (memory preserving operators). We investigate their properties (continuity, invertibility) and we establish some relations between special classes of such operators (Preisach, Ishlinskii and Nemytskii operators). For a general memory preserving operator we derive an energy inequality
On a Cahn-Hilliard system with convection and dynamic boundary conditions
This paper deals with an initial and boundary value problem for a system
coupling equation and boundary condition both of Cahn-Hilliard type; an
additional convective term with a forced velocity field, which could act as a
control on the system, is also present in the equation. Either regular or
singular potentials are admitted in the bulk and on the boundary. Both the
viscous and pure Cahn-Hilliard cases are investigated, and a number of results
is proven about existence of solutions, uniqueness, regularity, continuous
dependence, uniform boundedness of solutions, strict separation property. A
complete approximation of the problem, based on the regularization of maximal
monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and
rigorously discussed.Comment: Key words: Cahn-Hilliard system, convection, dynamic boundary
condition, initial-boundary value problem, well-posedness, regularity of
solution
On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation
This paper investigates a nonlocal version of a model for phase separation on
an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55
(2006) 105-118. The model consists of an initial-boundary value problem for a
nonlinearly coupled system of two partial differential equations governing the
evolution of an order parameter and the chemical potential. Singular
contributions to the local free energy in the form of logarithmic or
double-obstacle potentials are admitted. In contrast to the local model, which
was studied by P. Podio-Guidugli and the present authors in a series of recent
publications, in the nonlocal case the equation governing the evolution of the
order parameter contains in place of the Laplacian a nonlocal expression that
originates from nonlocal contributions to the free energy and accounts for
possible long-range interactions between the atoms. It is shown that just as in
the local case the model equations are well posed, where the technique of
proving existence is entirely different: it is based on an application of
Tikhonov's fixed point theorem in a rather unusual separable and reflexive
Banach space.Comment: The paper is dedicated to our friend Paolo Podio-Guidugli on the
occasion of his 75th birthday with best wishe
On the longtime behavior of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions
In this paper, we study the longtime asymptotic behavior of a phase
separation process occurring in a three-dimensional domain containing a fluid
flow of given velocity. This process is modeled by a viscous convective
Cahn-Hilliard system, which consists of two nonlinearly coupled second-order
partial differential equations for the unknown quantities, the chemical
potential and an order parameter representing the scaled density of one of the
phases. In contrast to other contributions, in which zero Neumann boundary
conditions were are assumed for both the chemical potential and the order
parameter, we consider the case of dynamic boundary conditions, which model the
situation when another phase transition takes place on the boundary. The phase
transition processes in the bulk and on the boundary are driven by free
energies functionals that may be nondifferentiable and have derivatives only in
the sense of (possibly set-valued) subdifferentials. For the resulting
initial-boundary value system of Cahn-Hilliard type, general well-posedness
results have been established in a recent contribution by the same authors. In
the present paper, we investigate the asymptotic behavior of the solutions as
times approaches infinity. More precisely, we study the -limit (in a
suitable topology) of every solution trajectory. Under the assumptions that the
viscosity coefficients are strictly positive and that at least one of the
underlying free energies is differentiable, we prove that the -limit is
meaningful and that all of its elements are solutions to the corresponding
stationary system, where the component representing the chemical potential is a
constant.Comment: Key words: Cahn-Hilliard systems, convection, dynamic boundary
conditions, well-posedness, asymptotic behavior, omega-limit. arXiv admin
note: text overlap with arXiv:1704.0533
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
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