The aim of this paper is to study the null controllability of a large class of quasilinear parabolic equations. In a first step we prove that the associated linear parabolic equations with non-constant diffusion coefficients are approximately null controllable by the means of regular controls and that these controls depend continuously to the diffusion coefficient. A fixed-point strategy is employed in order to prove the null approximate controllability for the considered quasilinear parabolic equations. We also show the exact null controllability in arbitrary small time for a class of parabolic equations including the parabolic $p$-Laplacian with $p \in (\frac{3}{2}, 2)$. The theoretical results are numerically illustrated combining a fixed point algorithm and a reformulation of the controllability problem for linear parabolic equation as a mixed-formulation which is numerically solved using a finite elements method.
Regional citrate anticoagulation use in intermittent hemodialysis is limited by the increased risk of metabolic complications due to faster solute exchanges than with continuous renal replacement therapies. Several simplifications have been proposed. The objective of this study was to validate a mathematical model of hemodialysis anticoagulated with citrate that was then used to evaluate different prescription scenarios on anticoagulant effectiveness (free calcium concentration in dialysis filter) and calcium balance. A study was conducted in hemodialyzed patients with a citrate infusion into the arterial line and a 1.25 mmol/L calcium dialysate. Calcium and citrate concentrations were measured upstream and downstream of the citrate infusion site and in the venous line. The values measured in the venous lines were compared with those predicted by the model using Bland and Altman diagrams. The model was then used with 22 patients to make simulations. The model can predict the concentration of free calcium, bound to citrate or albumin, accurately. Irrespective of the prescription scenario a decrease in free calcium below 0.4 mmol/L was obtained only in a fraction of the dialysis filter. A zero or slightly negative calcium balance was observed, and should be taken into account in case of prolonged use.
The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald–DeWaele law) in dimension $N \in \{2, 3\}$. We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald–DeWaele, Carreau–Yasuda, Herschel–Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a $p$-Laplacian for the symmetrized gradient for $p \in [1, 2)$.
We analyze a method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russells "stabilizability implies controllability" principle and a finite elements method of order $\theta$ with vanishing numerical viscosity. We show that the algorithm is convergent for any initial data in the energy space and that the error is of order $\theta$ for sufficiently smooth initial data. Both results are consequences of the uniform exponential decay of the discrete solutions guaranteed by the added viscosity and improve previous estimates obtained in the literature. Several numerical examples for the wave and the beam equations are presented to illustrate the method analyzed in this article.
The aim of this paper is to study the boundary controllability
of the linear elasticity system as a first-order system in
both space and time. Using the observability inequality known
for the usual second-order elasticity system, we deduce an
equivalent observability inequality for the associated
first-order system. Then, the control of minimal $L^2$-norm
can be found as the solution to a spacetime mixed formulation.
This first-order framework is particularly interesting from a
numerical perspective since it is possible to solve the
space-time mixed formulation using only piecewise linear
$C^0$-finite elements. Numerical simulations illustrate the
theoretical results.
This work is concerned with the distributed controllability of
the one-dimensional wave equation over non-cylindrical
domains. The controllability in that case has been obtained in
[Castro-Cindea-Munch, Controllability of the linear
one-dimensional wave equation with inner moving forces, SIAM
J. Control Optim 2014] for domains satisfying the usual
geometric optics condition. In the present work, we first show
that the corresponding observability property holds true
uniformly in a precise class of non-cylindrical domains.
Within this class, we then consider, for a given initial
datum, the problem of the optimization of the control support
and prove its well-posedness. Numerical experiments are then
discussed and highlight the influence of the initial condition
on the optimal domain.
We consider a contact model with power-law friction in the
antiplane context. Our study focuses on the boundary optimal
control, paying special attention to optimality conditions and
computational methods. Depending on the form of the power-law
friction, we are able to deduce an optimality condition for the
original problem or for a regularized version of it.
Furthermore, we introduce and analyze a computational technique
based on linearization, saddle point theory and a fixed point
method.
The aim of this paper is to propose a method to model
and numerically simulate the inertial migration of
particles in three-dimensional channels. The initial
problem, coupling Navier-Stokes equations to the
equations modeling the displacement of a spherical
particle immersed in the fluid, is replaced by a first
order expansion with respect to a small Reynolds
number. We reduce the computation of the velocity of a
spherical particle situated at a given position in a
channel to the numerical solutions of several Stokes
elementary problems. The proposed method is employed
to numerically approach the steady solutions for
different domain configurations.
Calcium has two important roles in haemodialysis. It
participates in the activation of blood coagulation and calcium
intakes have a major impact on patient mineral and bone metabolism.
The aim of this article is to propose a mathematical model for
calcium ions concentration in a dialyzer during haemodialysis using
a citrate dialysate. The model is composed of two elements. The
first describes the flows of blood and dialysate in a dialyzer
fibre. It was obtained by asymptotic analysis and takes into account
the anisotropy of the fibres forming a dialyzer. Newtonian and
non-Newtonian blood rheologies were tested. The second part of the
model predicts the evolution of the concentration of five chemical
species present in these fluids. The fluid velocity field drives the
convective part of a convection–reaction–diffusion system that
models the exchange of free and complexed calcium. We performed
several numerical experiments to calculate the free calcium
concentration in the blood in a dialyzer using dialysates with or
without citrate. The choice of blood rheology had little effect on
the fluid velocity field. Our model predicts that only a citrate
based dialysate without calcium can decrease free calcium
concentration at the blood membrane interface low enough to inhibit
blood coagulation. Moreover for a given calcium dialysate
concentration, adding citrate to the dialysate decreases total
calcium concentration in the blood at the dialyzer outlet. This
decrease of the calcium concentration can be compensated by infusing
in the dialyzed blood a quantity of calcium computed from the model.
This article deals with the boundary observability
and controllability properties of a space finite-differences
semidiscretization of the clamped beam equation. We make a
detailed spectral analysis of the system and, by combining
numerical estimates with asymptotic expansions, we localize all
the eigenvalues of the corresponding discrete operator depending
on the mesh size $h$. Then, an Ingham's type inequality and a
discrete multiplier method allow us to deduce that the uniform
(with respect to $h$) observability property holds if and only
if the eigenfrequencies are filtered out in the range ${\cal
O}\left(1/h^4\right)$.
We introduce a direct method that makes it possible to solve
numerically inverse type problems for linear hyperbolic
equations posed in ${\rm{\Omega }}\times (0,T)$ − $\rm{\Omega}$,
a bounded subset of ${{\mathbb{R}}}^{N}$. We consider the
simultaneous reconstruction of both the state and the source
term from a partial boundary observation. We employ a
least-squares technique and minimize the $L^2$-norm of the
distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the
optimality conditions are reduced to a mixed formulation
involving both the state to reconstruct and a Lagrange
multiplier. Under usual geometric conditions, we show the
well-posedness of this mixed formulation (in particular the
inf–sup condition) and then introduce a numerical approximation
based on space-time finite element discretization. We prove the
strong convergence of the approximation and then discuss several
examples in the one- and two-dimensional cases.
We introduce in this document
a direct method allowing to solve nu-merically inverse type
problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed
in $\Omega \times (0, T)$ - $\Omega$ a bounded subset of
$\mathbb{R}^n$ - from a partial distributed observation. We
employ a least-squares technic and minimize the $L^2$-norm of
the distance from the observa-tion to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the
optimality conditions are reduced to a mixed formulation
involving both the state to reconstruct and a Lagrange
multiplier. Under usual geometric optic conditions, we show the
well-posedness of this mixed formulation (in particular the
inf-sup condition) and then introduce a numerical approximation
based on space-time finite elements discretization. We prove the
strong convergence of the approximation and then discussed
several example for $N = 1$ and $N = 2$. The problem of the
reconstruction of both the state and the source term is also
addressed.
In this paper we study, from a control theoretic
view point, a 1D model of fluid-particle interaction. More
precisely, we consider a point mass moving in a pipe filled with
a fluid. The fluid is modelled by the viscous Burgers equation
whereas the point mass obeys Newton's second law. The control
variable is a force acting on the mass point. The main result of
the paper asserts that for any initial data there exist a time
$T>0$ and a control such that, at the end of the control
process, the particle reaches a point arbitrarily close to a
given target, whereas the velocities of the fluid and of the
point mass are driven exactly to zero. Therefore, within this
simplified model, we can control simultaneously the fluid and
the particle, by using inputs acting on the moving point only.
Moreover, the main result holds without any smallness
assumptions on the initial data. Alternatively, we can see our
results as yielding controllability of the viscous Burgers
equation by a moving internal boundary.
This paper deals with the numerical computation of
distributed null controls for the 1D wave equation. We consider
supports of the controls that may vary with respect to the time
variable. The goal is to compute approximations of such controls
that drive the solution from a prescribed initial state to zero
at a large enough controllability time. Assuming a geometric
optic condition on the support of the controls, we first prove a
generalized observability inequality for the homogeneous wave
equation. We then introduce and prove the well-posedness of a
mixed formulation that characterizes the controls of minimal
square-integrable norm. Such mixed formulation, introduced in
[Cindea and Münch, A mixed formulation for the direct
approximation of the control of minimal $L^2$-norm for linear
type wave equations], and solved in the framework of the
(space-time) finite element method, is particularly well-adapted
to address the case of time dependent support. Several numerical
experiments are discussed.
We propose a sequential data assimilation scheme
using Luenberger type observers when only some space restricted
time under-sampled measurements are available. More precisely,
we consider a wave-like equation for which we assume known the
restriction of the solution to an open non-empty subset of the
spatial domain and for some time samples (typically the sampling
step in time is much larger than the time discretization step).
To assimilate the available data, two strategies are proposed
and analyzed. The first strategy consists in assimilating data
only if they are available and the second one in assimilating
interpolation of the available data at all the discretization
times. In order to tackle the spurious high frequencies which
appear when we discretize the wave equation, for both
strategies, we introduce a numerical viscous term. In this case,
we prove some error estimates between the exact solution and our
observers. Numerical simulations illustrate the theoretical
results in the case of the one dimensional wave equation.
This paper deals with the numerical computation of
null controls for the wave equation with a potential. The goal
is to compute approximations of controls that drive the solution
from a prescribed initial state to zero at a large enough
controllability time. In [Cindea, Fernandez-Cara & Münch,
Numerical controllability of the wave equation through primal
methods and Carleman estimates, 2013], a so called primal
method is described leading to a strongly convergent
approximation of boundary controls : the controls minimize
quadratic weighted functionals involving both the control and
the state and are obtained by solving the corresponding
optimality condition. In this work, we adapt the method to
approximate the control of minimal square-integrable norm. The
optimality conditions of the problem are reformulated as a mixed
formulation involving both the state and his adjoint. We prove
the well-posedeness of the mixed formulation (in particular the
inf-sup condition) then discuss several numerical experiments.
The approach covers both the boundary and the inner
controllability. For simplicity, we present the approach in the
one dimensional case.
This paper deals with the numerical computation of
boundary null controls for the 1D wave equation with a
potential. The goal is to compute an approximation of controls
that drive the solution from a prescribed initial state to zero
at a large enough controllability time. We do not use in this
work duality arguments but explore instead a direct approach in
the framework of global Carleman estimates. More precisely, we
consider the control that minimizes over the class of admissible
null controls a functional involving weighted integrals of the
state and of the control. The optimality conditions show that
both the optimal control and the associated state are expressed
in terms of a new variable, the solution of a fourth-order
elliptic problem defined in the space-time domain. We first
prove that, for some specific weights determined by the global
Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method,
we introduce a family of finite-dimensional approximate control
problems and we prove a strong convergence result. Numerical
experiments confirm the analysis. We complete our study with
several comments.
This paper studies the numerical approximation of periodic
solutions for an exponentially stable linear hyperbolic equation
in the presence of a periodic external force f . These
approximations are obtained by combining a fixed point algorithm
with the Galerkin method. It is known that the energy of the
usual discrete models does not decay uniformly with respect to
the mesh size. Our aim is to analyze this phenomenon’s
consequences on the convergence of the approximation method and
its error estimates. We prove that, under appropriate regularity
assumptions on f , the approximation method is always
convergent. However, our error estimates show that the
convergence’s properties are improved if a numerically vanishing
viscosity is addaed to the system. The same is true if the
nonhomogeneous term f is monochromatic. To illustrate our
theoretical results we present several numerical simulations
with finite element approximations of the wave equation in one
or two dimensional domains and with different forcing terms.
We analyze an observer strategy based on partial --i.e. in a
subdomain -- measurements of the solution of a wave equation, in
order to compensate for unknown initial conditions. We prove the
exponential convergence of this observer under a non-standard
observability condition, whereas using measurements of the
time-derivative of the solution would lead to a standard
observability condition arising in stabilization and exact
controlability. Nevertheless, we directly relate our specific
condition to the classical geometric control condition. Finally,
we provide some numerical illustrations of the effectiveness of
the approach.
We propose an observer-based approach to circumvent the issue of
unbounded approximation errors -- with respect to the length of
the time window considered -- in the discretization of wave-like
equations in bounded domains, which covers the cases of the wave
equation per se and of linear elasticity as well as beam, plate
and shell formulations, and so on. Namely, taking advantage of
some measurements available on the system over time, we adopt a
strategy inspired from sequential data assimilation and by which
the discrete system is dynamically corrected using the
discrepancy between the solution and the measurements. In
addaition to the classical cornerstones of numerical analysis
made up by stability and consistency, we are thus led to
incorporating a third crucial requirement pertaining to
observability -- to be preserved through discretization. The
latter property warrants exponential stability for the corrected
dynamics, hence provides bounded approximation errors over time.
Special care is needed to establish the required observability
at the discrete level, in particular due to the fact that we
focus on an original observer method adapted to measurements of
the main variable, whereas measurements of the time-derivative
-- admissible, of course, albeit less frequent in practical
systems -- lead to a stability analysis in which existing
results can be more directly applied. We also provide some
detailed application examples with several such wave-like
equations, and the corresponding numerical assessments
illustrate the performance of our approach.
We consider the motion of a stretched string coupled with a
rigid body at one end and we study the existence of periodic
solution when a periodic force $f$ acts on the body. The main
difficulty of the study is related to the weak dissipation that
characterizes this hybrid system, which does not ensure a
uniform decay rate of the energy. Under addaitional regularity
conditions on $f$, we use a perturbation argument in order to
prove the existence of a periodic solution. In the last part of
the paper we present some numerical simulations based on the
theoretical results.
We propose a new method for the approximation of exact controls
of a second order infinite dimensional system with bounded input
operator. The algorithm combines Russell's ``stabilizability
implies controllability'' principle with the Galerkin's method.
The main new feature brought in by this work consists in giving
precise error estimates. In order to test the efficiency of the
method, we consider two illustrative examples (with the finite
element approximations of the wave and the beam equations) and
we describe the corresponding simulations.
In this paper we study a controllability problem for a
simplified 1-d nonlinear system which models the self-propelled
motion of a rigid body in a fluid located on the real axis. The
control variable is the difference of the velocities of the
fluid and the solid and depends only on time. The main result of
the paper asserts that any final position and velocity of the
rigid body can be reached by a suitable input function.
In this work we prove that the exact internal observability for
the Euler-Bernoulli equation is robust with respect to a class
of linear perturbations. Our results yield, in particular, that
for rectangular do- mains we have the exact observability in an
arbitrarily small time and with an arbitrarily small observation
region. The usual method of tack- ling lower order terms, using
Carleman estimates, cannot be applied in this context. More
precisely, it is not known if Carleman estimates hold for the
evolution Euler-Bernoulli equation with arbitrarily small
observation region. Therefore we use a method combining
frequency domain techniques, a compactness-uniqueness argument
and a Carleman estimate for elliptic problems.
We study the exact controllability of a nonlinear plate equation
by the means of a control which acts on an internal region of
the plate. The main result asserts that this system is locally
exactly controllable if the associated linear Euler-Bernoulli
system is exactly controllable. In particular, for rectangular
domains, we obtain that the Berger system is locally exactly
controllable in arbitrarily small time and for every open and
nonempty control region.
The aim of this paper is to prove a uniform observability
inequality for a finite differences semi-discretization of a
clamped beam equation. A discrete multiplier method is employed
in order to obtain the uniform observability of the eigenvectors
of the matrix driving the semi-discrete system, corresponding to
eigenfrequencies smaller than a precise filtering threshold.
This result can be generalized to the uniform observability of
every filtered solution. Numerical simulations, concerning the
dual controllability problem, illustrate the theoretical
results.
The aim of this work is to study of the numerical approximation
of the controls for the hinged beam equation. A consequence of
the numerical spurious high frequencies is the lack of the
uniform controllability property of the semi-discrete model for
the beam equation, in the classical setting. We solve this
deficiency by adding a vanishing numerical viscosity term, which
will damp out these high frequencies. An approximation algorithm
based on the conjugate gradient method and some numerical
experiments are presented.
In this paper we summarize some
recent results from cite{Hib2013} concerning the controllability
of a one dimensional fluid-structure model. These results are
confirmed by numerical experiments in some particular cases.
More precisely, we consider a simplified model for a point
swimmer moving, according to Newton's law, in an one dimensional
fluid which is modeled by the viscous Burgers equation. The
control variable is the relative velocity of the swimmer with
respect to the fluid. The main result of the paper gives the
conditions in which we can drive the fluid and the velocity of
the point mass to zero. A set of reachable positions of the
point mass is also obtained. From the numerical point of view,
we compute the $L^2$-minimal norm control for a linearized and
simplified model. The method we used combine a finite elements
discretization in space, a finite-difference centered scheme in
time and the conjugate gradient method.
Electroporation consists in increasing the permeability of a
tissue by applying high voltage pulses. In this paper we discuss
the question of optimal placement and optimal loading of
electrodes such that electroporation holds only in a given open
set of the domain. The electroporated set of the domain is where
the norm of the electric field is above a given threshold value.
We use a standard gradient algorithm to optimize the loading of
the electrodes and shape sensitivity analysis and a gradient
algorithm in order to move the electrodes. We also discuss the
choice of objective functions to be chosen in the gradient
algorithm.
The aim of this work is to study the exact observability of a
perturbed plate equation. A fast and strongly localized
observation result was proven using a perturbation argument of
an Euler-Bernoulli plate equation and a unique continuation
result for bi-Laplacian.
This paper describes a rigorous framework for reconstructing MR
images of the heart, acquired continuously over the cardiac and
respiratory cycle. The framework generalizes existing
techniques, commonly referred to as retrospective gating, and is
based on the properties of reproducing kernel Hilbert spaces.
The reconstruction problem is formulated as a moment problem in
a multidimensional reproducing kernel Hilbert spaces (a
two-dimensional space for cardiac and respiratory resolved
imaging). Several reproducing kernel Hilbert spaces were tested
and compared, including those corresponding to commonly used
interpolation techniques (sinc-based and splines kernels) and a
more specific kernel allowed by the framework (based on a
first-order Sobolev RKHS). The Sobolev reproducing kernel
Hilbert spaces was shown to allow improved reconstructions in
both simulated and real data from healthy volunteers, acquired
in free breathing.
This article describes a general framework for multiple coil MRI
reconstruction in the presence of elastic physiological motion.
On the assumption that motion is known or can be predicted, it
is shown that the reconstruction problem is equivalent to
solving an integral equation--known in the literature as a
Fredholm equation of the first kind--with a generalized kernel
comprising Fourier and coil sensitivity encoding, modified by
physiological motion information. Numerical solutions are found
using an iterative linear system solver. The different steps in
the numerical resolution are discussed, in particular it is
shown how over-determination can be used to improve the
conditioning of the generalized encoding operator. Practical
implementation requires prior knowledge of displacement fields,
so a model of patient motion is described which allows elastic
displacements to be predicted from various input signals (e.g.,
respiratory belts, ECG, navigator echoes), after a
free-breathing calibration scan. Practical implementation was
demonstrated with a moving phantom setup and in two
free-breathing healthy subjects, with images from the
thoracic-abdominal region. Results show that the method
effectively suppresses the motion blurring/ghosting artifacts,
and that scan repetitions can be used as a source of
over-determination to improve the reconstruction.
Le but de cette thèse est d'étudier, du point de vue théorique,
la contrôlabilité exacte de certaines équations aux dérivées
partielles qui modélisent les vibrations élastiques, et
d'appliquer les résultats ainsi obtenus à la résolution des
problèmes inverses provenant de l'imagerie par résonance
magnétique (IRM).
Cette thèse comporte deux parties. La première partie, intitulée
``Contrôlabilité et observabilité de quelques équations des
plaques'', discute la problématique de la contrôlabilité,
respectivement de l'observabilité, de l'équation des plaques
perturbées avec des termes linéaires ou non linéaires. Des
résultats récents ont prouvé que l'observabilité exacte d'un
système qui modélise les vibrations d'une structures élastique
(équation des ondes ou des plaques) implique l'existence d'une
solution du problème inverse de la récupération d'un terme
source dans l'équation à partir de l'observation. Ainsi, dans le
Chapitre 2 de cette thèse nous avons démontré l'observabilité
interne exacte de l'équation des plaques perturbées par des
termes linéaires d'ordre un et dans le Chapitre 3 la
contrôlabilité exacte locale d'une équation des plaques non
linéaire attribuée à Berger. Le Chapitre 4 introduit une méthode
numérique pour l'approximation des contrôles exactes dans des
systèmes d'ordre deux en temps.
La deuxième partie de la thèse est dédiée à l'imagerie par
résonance magnétique. Plus précisément, on s'intéresse aux
méthodes de reconstruction des images pour des objets en
mouvement, l'exemple typique étant l'imagerie cardiaque en
respiration libre. Dans le Chapitre 6, nous avons formulé la
reconstruction d'images cardiaques acquises en respiration libre
comme un problème des moments dans un espace de Hilbert à noyau
reproductif. L'existence d'une solution pour un tel problème des
moments est prouvée par des outils bien connus dans la théorie
du contrôle. Nous avons validé cette méthode en utilisant des
images simulées numériquement et les images de cinq volontaires
sains.
La connexion entre les deux parties de la thèse est réalisée par
le Chapitre 7 où l'on présente le problème inverse
d'identification d'un terme source dans l'équation des ondes à
partir d'une observation correspondante à un enregistrement IRM.
En conclusion, nous avons montré qu'on peut utiliser les outils
de la théorie de contrôle pour des problèmes inverses provenant de
l'IRM des objets en mouvement, à la condition de connaître
l'équation du mouvement.
