311 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 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
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
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
A boundary control problem for the pure Cahn-Hilliard equation with dynamic boundary conditions
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.
Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase
separation, singular potentials, optimal control, optimality conditionsComment: arXiv admin note: text overlap with arXiv:1407.391
Analysis and optimal boundary control of a nonstandard system of phase field equations
We investigate a nonstandard phase field model of Cahn-Hilliard type. The
model describes two-species phase segregation and consists of a system of two
highly nonlinearly coupled PDEs. It has been studied recently in the papers
arXiv:1103.4585 and arXiv:1109.3303 for the case of homogeneous Neumann
boundary conditions. In this paper, we investigate the case that the boundary
condition for one of the unknowns of the system is of third kind and
nonhomogeneous. For the resulting system, we show well-posedness, and we study
optimal boundary control problems. Existence of optimal controls is shown, and
the first-order necessary optimality conditions are derived. Owing to the
strong nonlinear couplings in the PDE system, standard arguments of optimal
control theory do not apply directly, although the control constraints and the
cost functional will be of standard type.Comment: Key words: nonlinear phase field systems, Cahn-Hilliard systems,
parabolic systems, optimal boundary control, first-order necessary optimality
conditions. The interested reader can also see the preprint arXiv:1106.3668
where a distributed optimal control problem is studied for a similar system.
arXiv admin note: significant text overlap with arXiv:1106.366
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