Dissertations / Theses on the topic 'Singular Perturbations'
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Dyachenko, Evgueniya, and Nikolai Tarkhanov. "Singular perturbations of elliptic operators." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/6950/.
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We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.
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Hashemi, Seyed Naser. "Singular perturbations in coupled stochastic differential equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ65244.pdf.
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Ashino, Ryuichi. "On Nagumo's Hs-stability in singular perturbations." 京都大学 (Kyoto University), 1991. http://hdl.handle.net/2433/86434.
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Beauchamp, Gerson. "Algorithms for singular systems." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/15368.
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Neuner, Christoph. "Generalized Titchmarsh-Weyl functions and super singular perturbations." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-113389.
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In this thesis we study certain singular Sturm-Liouville differential expressions from an operator theoretic point of view.In particular we are interested in expressions that involve strongly singular potentials as introduced by Gesztesy and Zinchenko.On the ODE side, analyzing these expressions involves the so-called $m$-functions, often generalized Nevanlinna functions, who encapsulate spectral information of the underlying problem.The aim of the two papers in this thesis is to further understanding on the operator theory side.In the first paper, we use a model for super singular perturbations to describe a family of induced self-adjoint realizations of a perturbed Schr\"o\-din\-ger operator, i.e., with a potential of the form $c/x^2 + q$ where $q$ is a perturbation.Following the unperturbed example of Kurasov and Luger, we find that the so-called $Q$-function appearing in this approach is in good agreement with the above named $m$-function.Furthermore, we show that the operator model can be chosen such that $Q \equiv m$.In the second paper, we present a negative result in this area, namely that the supersingular perturbations model cannot be used for all strongly singular potentials.For a potential with a stronger singularity at the origin, namely $1/x^4$, we discuss the asymptotic behaviour of the Weyl solution at zero.It turns out that this function cannot be regularized appropriately and the operator model breaks down.
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Elago, David. "Robust computational methods for two-parameter singular perturbation problems." Thesis, University of the Western Cape, 2010. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_1693_1308039217.
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This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.
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Lu, Nan. "Normally elliptic singular perturbation problems: local invariant manifolds and applications." Diss., Georgia Institute of Technology, 2011. http://hdl.handle.net/1853/41090.
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In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small
singular parameters. We also use our results on local invariant manifolds to study
the persistence of homoclinic solutions under weakly dissipative and conservative per-
turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some
careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly
dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external
forcing and other potential fields.
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Li, Yongfeng. "Nonlinear oscillation and control in the BZ chemical reaction." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/26565.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.Committee Chair: Yi, Yingfei; Committee Member: Chow, Shui-Nee; Committee Member: Dieci, Luca; Committee Member: Verriest, Erik; Committee Member: Weiss, Howie. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Mason, Colin Stuart. "Boundary perturbations and ultracontractivity of singular second order elliptic operators." Thesis, King's College London (University of London), 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.395943.
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Shankar, Uday J. "Singular-perturbation analysis of climb-cruise-dash optimization." Thesis, Virginia Tech, 1985. http://hdl.handle.net/10919/45736.
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The method of singular-perturbation analysis is applied to the determination of range-fuel-time optimal aircraft trajectories.
The problem is shown to break down into three sub-problems which are studied separately. In particular, the inner layer containing the altitude path-angle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work.
As a step forward in solving the ensuing nonlinear two-point boundary-value problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundary-layer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angle-of-attack from the linear analysis is discussed.
Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method.Master of Science
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Agostiniani, Virginia. "Variational results for nematic elastomers and singular perturbations of evolution problems." Doctoral thesis, SISSA, 2012. http://hdl.handle.net/20.500.11767/4687.
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Sultanov, Oskar, Leonid Kalyakin, and Nikolai Tarkhanov. "Elliptic perturbations of dynamical systems with a proper node." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7046/.
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The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain.
We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.
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Garrett, Frederick Earl. "Fast half-loop maneuvers for the F/A-18 fighter aircraft using a singular pertubation feedback control law." Thesis, This resource online, 1988. http://scholar.lib.vt.edu/theses/available/etd-04122010-083818/.
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Tidefelt, Henrik. "Structural algorithms and perturbations in differential-algebraic equations." Licentiate thesis, Linköping : Department of Electrical Engineering, Linköpings universitet, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-9011.
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Gorius, Thomas [Verfasser]. "Approximate Model Inversion of Flexible Multibody Systems Based on Singular Perturbations / Thomas Gorius." Aachen : Shaker, 2014. http://d-nb.info/1058315234/34.
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Coiculescu, Ion. "Dynamics, Thermodynamic formalism and Perturbations of Transcendental Entire Functions of Finite Singular Type." Thesis, University of North Texas, 2005. https://digital.library.unt.edu/ark:/67531/metadc4783/.
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In this dissertation, we study the dynamics, fractal geometry and the topology of the Julia set of functions in the family H which is a set in the class S, the Speiser class of entire transcendental functions which have only finitely many singular values. One can think of a function from H as a generalized expanding function from the cosh family. We shall build a version of thermodynamic formalism for functions in H and we shall show among others, the existence and uniqueness of a conformal measure. Then we prove a Bowen's type formula, i.e. we show that the Hausdorff dimension of the set of returning points, is the unique zero of the pressure function. We shall also study conjugacies in the family H, perturbation of functions in the family and related dynamical properties. We define Perron-Frobenius operators for some functions naturally associated with functions in the family H and then, using fundamental properties of these operators, we shall prove the important result that the Hausdorff dimension of the subset of returning points depends analytically on the parameter taken from a small open subset of the n-dimensional parameter space.
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Bennett, James Cameron, and james bennett@student rmit edu au. "Mathematical Analysis of Film Blowing." RMIT University. Mathematical and Geospatial Sciences, 2008. http://adt.lib.rmit.edu.au/adt/public/adt-VIT20081128.115021.
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Film blowing is a highly complex industrial process used to manufacture thin plastic films for uses in a wide range of applications; for example, plastic bags. The mathematical modelling of this process involves the analysis of highly nonlinear differential equations describing the complex phenomena arising in the film blowing process, and requires a sophisticated mathematical approach. This dissertation applies an innovative combination of tools, namely analytic, numerical and heuristic mathematical techniques to the analysis of the film blowing process. The research undertaken examines, in particular, a two-point boundary value problem arising from the modelling of the radial profile of the polymer film. For even the simplest modelling of this process, namely the isothermal Newtonian model, the resulting differential equation is a highly nonlinear, second order one, with an extra degree of difficulty due to the presence of a small parameter multiplying the highest derivative. Thus, the problem falls into the category of a nonlinear singular perturbation problem. Analytic techniques are applied to the isothermal Newtonian blown film model to obtain a closed form explicit approximation to the film bubble radius. This is then used as a base approximation for an iterative numerical scheme to obtain an improved numerical solution of the problem. The process is extended to include temperature variations, varying viscosity (Power law model) and viscoelastic effects (Maxwell model). As before, closed form approximations are constructed for these models which are used to launch numerical schemes, whose solutions display good accuracy. The results compare well with results obtained by purely numerical solutions in the literature.
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Pham, Hoang. "A perturbation solution for forced response of systems displaying eigenvalue veering and mode localization." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/19120.
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Kass, Guy. "Perturbations singulières de problèmes d'évolution linéaires." Nancy 1, 1988. http://www.theses.fr/1988NAN10038.
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La solution du problème perturbé d'ordre 1 : mepsilon uepsilon + u'epsilon = fepsilon , uepsilon (O) = uepsilon ::(O), uepsilon appartient à Vepsilon, vérifie l'estimation à priori. |uepsilon ||vepsilon + ||u'epsilon ||v'epsilon
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Flandoli, Franco, and Michael Högele. "A solution selection problem with small stable perturbations." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7120/.
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The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic
function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.
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DeLashmutt, Timothy E. "Modeling a proton exchange membrane fuel cell stack." Ohio : Ohio University, 2008. http://www.ohiolink.edu/etd/view.cgi?ohiou1227224687.
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Munyakazi, Justin Bazimaziki. "Higher Order Numerical Methods for Singular Perturbation Problems." Thesis, Online Access, 2009. http://etd.uwc.ac.za/usrfiles/modules/etd/docs/etd_gen8Srv25Nme4_6335_1277251056.pdf.
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Hamouda, Makram. "Perturbations singulières pour des EDP linéaires et non linéaires en presence de discontinuités." Phd thesis, Université Paris Sud - Paris XI, 2001. http://tel.archives-ouvertes.fr/tel-00001931.
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Ma thèse porte sur l'étude des couches limites et de perturbations singulières (\textit{i.e.} des problèmes caractérisés par la présence d'un petit paramètre qui tend vers zéro) dans des conditions plus délicates que d'habitude, à savoir lorsque la solution limite n'est pas régulière. Je considère ainsi deux classes de problèmes réguliers associes à un laplacien et à un bilaplacien, et un problème non linéaire dérivé du problème de Plateau (surfaces minimas), pour lequels la fonction limite possède une singularité (discontinuité simple pour les premiers problèmes, dérivée normale infinie sur certaines parties de la frontière pour le second).\\ La première partie de cette thèse est consacrée à l'étude de deux modèles linéaires singuliers associés à des perturbations singulières pour des EDPs ayant une fonction source singulière. Ce type d'équations fait l'objet de plusieurs applications, par exemple les problèmes de flambement en élasticité, les tourbillons singuliers en mécanique des fluides, le problème de la charge critique pour une poutre ou une plaque élastoplastique, le problème du contrôle automatique de la trajectoire d'un mobile et le problème du bord arrière pour l'écoulement autour d'une aile. De manière classique, la présence d'un petit paramètre dans des équations aux dérivées partielles entraîne, dans certains cas, l'apparition d'une couche limite classique près du bord du domaine pour la solution dite régularisée. Cependant, si on considère en plus une fonction source discontinue (voire une distribution), on constate que de nouvelles couches limites apparaissent à l'intérieur du domaine; l'étude de celles-ci constitue le principal but de cette première partie. Dans la deuxième partie, on s'intéresse à l'étude du problème des surfaces minimales sur une couronne. Pour certaines classes de données au bord, ce problème n'admet pas de solution et sa solution faible dite ``généralisée'' admet une dérivée infinie. On introduit alors une méthode de régularisation elliptique qui entraîne une couche limite près du bord. Le résultat fondamental de cette partie consiste à donner explicitement une approximation pour cette solution régularisée.
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Branco, Meireles Joao. "Singular Perturbations and Ergodic Problems for degenerate parabolic Bellman PDEs in R^m with Unbounded Data." Doctoral thesis, Università degli studi di Padova, 2015. http://hdl.handle.net/11577/3424194.
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In this thesis we treat the first singular perturbation problem of a stochastic model with unbounded and controlled fast variables with success. Our methods are based on the theory of viscosity solutions, homogenisation of fully nonlinear PDEs and a careful analysis of the associated ergodic stochastic control problem in the whole space R^m. The text is divided in two parts.
In the first chapter, we investigate the existence and uniqueness as well as a suitable stability of the solution to the associated ergodic problem that are crucial to characterize the effective Hamiltonian of the limit (effective) Cauchy problem in Chapter II of this thesis. The main achievement obtained in this part is a purely analytical proof for the uniqueness of solution to such ergodic problem. Since the state space of the problem is not compact, in general there are infinitely many solutions to the ergodic problem. However, if one restrict the class of solutions to the set of bounded-below functions, then it is known that uniqueness holds up to an additive constant. The existing proof relies on some probabilistic techniques employing the invariant probability measure for the associated stochastic process. Here we give a new proof, purely analytic, based on the strong maximum principle. We believe that our results can be interesting and useful for researchers in the PDE community.
In the second chapter, we introduce our singular perturbation model of a stochastic control problem and we prove our main result: the convergence of the value function $V^\epsilon$ associated to the problem to the solution of the limiting equation. More precisely, we prove that the functions
\underline{V} (t,x):=\liminf_{(\epsilon,t',x') \to (0,t,x)} \inf_{y \in \mathbb{R}^m} V^\epsilon (t',x',y)
and
\bar{V} (t,x) :=(\sup_{R} \bar{V}_R)^* (t,x) \text{ (upper semi-continuous envelope of $\sup_{R} \bar{V}_R$ )}
where $\bar{V}_{R} (t,x):=\limsup_{(\epsilon, t',x') \to (0,t,x)} \sup_{y \in B_R (0)} V^\epsilon (t',x',y)$,
are, respectively, a super and a subsolution of the effective Cauchy problem.
As a corollary of this result, $V^\epsilon$ converges to the unique solution $V$ of the effective equation provided the equation admits the comparison principle for discontinuous viscosity solutions. The justification of this convergence is not trivial at all. It especially involves some regularity issues and a careful treatment of viscosity techniques and stochastic analysis. This result has never been obtained before.In questa tesi viene trattato con successo il primo problema di perturbazione singolare di un modello stocastico con variabili veloci controllate e non limitate. I metodi si basano sulla teoria delle soluzioni di viscosità, omogeinizzazione dei PDE completamente non lineari, e su un'attenta analisi del problema stocastico ergodico associato, valido nell'intero spazio R^m. Il testo è diviso in due parti. Nel primo capitolo, saranno studiate l'esistenza, l'unicità e alcune proprietà di stabilità della soluzione del problema ergodico, riferito sopra, che sono essenziali per caratterizzare il Hamiltoniano effettivo che appare in un Problema di Cauchy "limite", che sarà descritto nel capitolo II di questa tesi. Il principale contributo, presentato in questa parte, è una prova puramente analitica dell'unicità della soluzione di questo problema ergodico. Siccome lo stato dello spazio del problema non è compatto, in generale ci sono un numero infinito di soluzioni a questo problema. Tuttavia, se uno limitasse la classe di soluzioni all'insieme di funzioni limitate inferiormente, allora è noto che l'unicità sarà mantenuta a meno di una costante. La prova esistente si basa su alcune tecniche probabilistiche che impiegano la misura di probabilità invariante per l'associato processo stocastico. Qua verrà data una nuova prova, puramente analitica, basata sul principio del massimo. Si ritiene che il risultato potrà essere interessante ed utile per i ricercatori che lavorano all'interno della comunità di ricerca delle Equazioni Differenziali alle derivate Parziali (PDE). Nel secondo capitolo, sarà introdotto un modello di perturbazione singolare di un problema di controllo stocastico, e provato il risultato principale: la convergenza della funzione valore $V^\epsilon$, associata al nostro problema, per soluzione dell'equazione limite. Più precisamente, sarà provato che le funzioni: \underline{V} (t,x):=\liminf_{(\epsilon,t',x') \to (0,t,x)} \inf_{y \in \mathbb{R}^m} V^\epsilon (t',x',y) e \bar{V} (t,x) :=(\sup_{R} \bar{V}_R)^* (t,x) \text{ (upper semi-continuous envelope of $\sup_{R} \bar{V}_R$ )} dove $\bar{V}_{R} (t,x):=\limsup_{(\epsilon, t',x') \to (0,t,x)} \sup_{y \in B_R (0)} V^\epsilon (t',x',y)$, sono, rispettivamente, una super soluzione e una sottosoluzione del problema effettivo di Cauchy. Come corollario di questo risultato, $V^\epsilon$ converge all'unica soluzione V della equazione effettiva se l'equazione limite permette il principio di comparazione per le soluzioni di viscosità discontinue. La motivazione di questa convergenza non è ovvia del tutto. Coinvolge specialmente alcuni problemi di regolarità e un trattamento attento delle tecniche di viscosità e di analisi stocastica. Questo risultato è nuovo e non è mai stato ottenuto, prima d'ora, nella letteratura Matematica.
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Mudavanhu, Blessing. "A new renormalization method for the asymptotic solution of multiple scale singular perturbation problems /." Thesis, Connect to this title online; UW restricted, 2002. http://hdl.handle.net/1773/6794.
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Mallwitz, Enno. "Nearly Gaussian Curvature Perturbations in Ekpyrotic Cosmologies." Doctoral thesis, Humboldt-Universität zu Berlin, 2019. http://dx.doi.org/10.18452/19805.
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In dieser Arbeit studieren wir das ekpyrotische Szenario, welches ein kosmologisches Modell des frühen Universums ist. Dieses Modell erklärt mit Hilfe einer kontrahierenden ekpyrotischen Phase die "Anfangsbedingungen" des Universums. Das bedeutet, dass der konventionelle "Urknall" durch einem Rückprall ersetzt wird. In dieser Arbeit versuchen wir Unstimmigkeiten zwischen den Vorhersagen der ekpyrotischen Modelle und den Messungen der Kosmologischen Hintergrundstrahlung des Planck Satelliten zu lösen.
Den Planck Messungen zufolge sind die ursprünglichen adiabatischen Fluktuationen fast skaleninvariant und gaußverteilt. Während der ekpyrotischen Phase werden typischer Weise Flutuationen mit nicht-Gaußschen Korrekturen erzeugt. Wir schlagen zwei Ansätze vor, um diese Unstimmigkeit zu beheben.
In dem nicht-minimalen entropischen Mechanismus werden fast skaleninvariante entropische Fluktuationen mit Hilfe einer nicht-minimalen kinetischen Kopplung zwischen zwei Skalarfeldern erzeugt. Wir werden zeigen, dass die nicht-Gaußschen Korrekturen während der ekpyrotischen Phase genau Null sind. Dies führt zu insgesamt kleinen nicht-Gaußschen Korrekturen nach der Umwandlung von entropischen zu adiabatischen Fluktuationen.
Im Folgendem werden wir eine kinetische Umwandlung untersuchen, die nach einem nicht-singulären Rückprall stattfindet.
Das Wachstum der entropischen Fluktuationen während des Rückpralls hat zur Folge, dass die möglichen nicht-Gaußschen Korrekturen, die zur Zeit der ekpyrotischen Phase erzeugt wurden, während des Rückpralls unterdrückt werden.
Im letzten Teil der Arbeit gehen wir ein gravierendes Problem des inflationären Paradigmas an, welches "slow-roll eternal inflation" genannt wird.
Wir schlagen ein Modell vor, das Ideen von Inflation und Ekpyrosis verbindet. Während der Konflation expandiert das Universum beschleunigt. Die adiabatischen Fluktuationen verhalten sich jedoch wie bei ekpyrotischen Modellen und wird "slow-roll eternal inflation" verhindert.In this thesis, we study the ekpyrotic scenario, which is a cosmological model of the early universe. In this model the ``initial conditions'' of the universe are determined by a contracting ekpyrotic phase, which means that the conventional ``Big Bang'' is replaced by a bounce. The following thesis addresses the tension between ekpyrotic predictions and the observations of the Cosmic Microwave Background radiation by the Planck team. According to the Planck data, the primordial curvature fluctuations are nearly scale-invariant and Gaussian. However, during ekpyrosis, the fluctuations have typically sizable non-Gaussian signatures. In this thesis, we propose two approaches in order to resolve the tension with observations. In the non-minimal entropic mechanism, nearly scale-invariant entropy perturbations are created due to a non-minimal kinetic coupling between two scalar fields. We will show that the non-Gaussian corrections during ekpyrosis are precisely zero leading to overall small non-Gaussian signatures after the conversion process from entropy perturbations to curvature perturbations. In the following, we will consider a kinetic conversion phase, which takes place after a non-singular bounce. Due to the growth of entropy perturbations during the bounce phase, the possibly large non-Gaussian corrections created during the ekpyrotic phase become suppressed during the bounce. The last part of this thesis addresses a major problem of the inflationary paradigm: Due to large adiabatic fluctuations, slow-roll eternal inflation creates infinitely many physically distinct pocket universes. We propose a model in the framework of scalar-tensor theories, which conflated ideas of both inflation and ekpyrosis. During conflation, the universe undergoes accelerated expansion, but there are no large adiabatic fluctuations like during ekpyrosis resulting in the absence of slow-roll eternal inflation.
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Waldron, William Michael. "Optimal vertical plane booster guidance including pitch dynamics." Diss., This resource online, 1996. http://scholar.lib.vt.edu/theses/available/etd-10042006-143908/.
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López, Zazueta Claudia. "Réduction dynamique de réseaux métaboliques par la théorie des perturbations singulières : application aux microalgues." Thesis, Université Côte d'Azur (ComUE), 2018. http://www.theses.fr/2018AZUR4108/document.
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Les lipides des microalgues et les glucides de cyanobactéries peuvent être transformés en biodiesel et en bioéthanol, respectivement. L'amélioration de la production de ces molécules doit prendre en compte les entrées périodiques (principalement la lumière) forçant le réseau métabolique de ces organismes photosynthétiques. Il est donc nécessaire de tenir compte de la dynamique du réseau métabolique en réduisant sa dimension pour assurer la maniabilité mathématique. Le but de ce travail est de concevoir une approche originale pour réduire les réseaux métaboliques dynamiques tout en conservant la dynamique de base. Cette méthode est basée sur une séparation en échelles de temps. Pour une classe de modèles de réseaux métaboliques décrits par des ODE, la dynamique des systèmes réduits est calculée à l'aide du théorème de Tikhonov pour les systèmes singulièrement perturbés. Cette approximation quasi-stationnaire coïncide avec la dynamique du réseau d'origine, avec une erreur bornée. L'approche est d'abord développée pour les systèmes de réaction pouvant être linéarisés autour d'un point de travail et forcés par des entrées continues. Ensuite, une généralisation de cette méthode est donnée pour les réseaux à réactions rapides de cinétiques de Michaelis-Menten et tout type de cinétiques lentes, prenant également en compte un nombre fini d'entrées continues externes. La méthode de réduction met en évidence une relation entre la grandeur de la concentration des métabolites et la gamme des vitesses de réaction : les métabolites consommés par les réactions rapides ont une concentration inférieure d'un ordre de grandeur à celle des métabolites consommés à faible vitesse. Cette propriété est satisfaite pour les métabolites à dynamique rapide ne se trouvant pas dans un piège de flux, concept introduit dans ce travail. Le système réduit peut être calibré avec des données expérimentales à l'aide d'une procédure d'identification dédiée basée sur la minimisation. L'approche est illustrée par un réseau métabolique de microalgues autotrophes, comprenant le métabolisme central et représentant la dynamique des glucides et des lipides. Cette approche permet de bien ajuster les données expérimentales de Lacour et al. (2012) avec la microalgue Tisochrysis lutea. Enfin, un schéma visant à optimiser la production de molécules cibles est proposé en utilisant le système réduitLipids from microalgae and carbohydrates from cyanobacteria can be transformed into biodiesel and bioethanol, respectively. Enhancing the production of these molecules must account for the periodic inputs (mainly light) forcing the metabolic network of these photosynthetic organisms. It is therefore necessary to account for the dynamics of the metabolic network, while reducing its dimension to ensure mathematical tractability. The aim of this work is to design an original approach to reduce dynamic metabolic networks while keeping the core dynamics. This method is based on time-scale separation. For a class of metabolic network models described by ODE, the dynamics of the reduced systems are computed using the theorem of Tikhonov for singularly perturbed systems. This Quasi Steady State Approximation accurately coincides with the original network dynamics, with a bounded error. The approach is first developed for reaction systems that can be linearized around a working point and that are forced by external continuous inputs. Then, a generalization of this method is given for networks with fast reactions of Michaelis-Menten kinetics and any type of slow kinetics, also considering a finite number of external continuous inputs. The reduction method highlights a relation between the concentration magnitude of the metabolites and the range of the reaction rates: the metabolites that are consumed by fast reactions have concentration one order of magnitude lower than metabolites consumed at slow rates. This property is satisfied for metabolites with fast dynamics that are not in a flux trap, a concept introduced in this work. The reduced system can be calibrated with experimental data using a dedicated identification procedure based on minimization. The approach is illustrated with an autotrophic microalgae metabolic network, including the core metabolism and representing the carbohydrates and lipids dynamics. The approach efficiently fits the experimental data from Lacour et al. (2012) with the microalgae Tisochrysis lutea. Finally, a scheme to optimize the production of target molecules is proposed using the reduced system
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Kunert, Gerd, Zoubida Mghazli, and Serge Nicaise. "A posteriori error estimation for a finite volume discretization on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601352.
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A singularly perturbed reaction diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using \emph{anisotropic meshes} which can improve the accuracy of the discretization considerably. The main focus is on \emph{a posteriori} error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient \emph{a posteriori} error estimation is achieved for the finite volume method on anisotropic meshes.
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Visser, Hendrikus. "Energy management of three-dimensional minimum-time intercept." Diss., Virginia Polytechnic Institute and State University, 1985. http://hdl.handle.net/10919/49954.
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A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission.
The proposed scheme has roots in two well-known techniques: singular perturbations and neighboring-optimal guidance. Use of singular-perturbation ideas is made in terms of the assumed trajectory-family structure. A heading/energy family of prestored point-mass-model state-Euler solutions is used as the baseline in this scheme. The next step is to generate a near-optimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot consist of the reference controls plus correction terms which are linear combinations of the altitude and path-angle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboring-optimal guidance. However, in contrast to the neighboring-optimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach.
A detailed description of the feedback laws and of some of the mathematical tools used to construct the controller is presented. The construction of the feedback laws requires a substantial preflight computational effort, but the computation times for on-board execution of the feedback laws are very modest. Other issues relating to practical implementation are addressed as well.
Numerical examples, comparing open-loop optimal and approximate feedback solutions for a sample high-performance fighter, illustrate the attractiveness of the guidance scheme. Optimal three-dimensional flight in the presence of a terrain limit is studied in some detail.Ph. D.
incomplete_metadata
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Hachemi, Fouad El. "Analyse de stabilité des systèmes à commutations singulièrement perturbés." Thesis, Université de Lorraine, 2012. http://www.theses.fr/2012LORR0229/document.
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Un grand nombre de phénomènes nous entourant peuvent être décrit par des modèles hybrides, c'est-à-dire, mettant en jeu simultanément une dynamique continu et une dynamique discrète. Également, il n'est pas rare que ces dynamiques puissent évoluer dans des échelles de temps différentes. Dans cette thèse, nous nous intéressons à l'analyse de stabilité des systèmes à commutations singulièrement perturbés à temps continu. En présence de commutations, l'analyse de stabilité des systèmes singulièrement perturbés dite "classique" (séparation des échelles de temps) n'est plus valable. En nous plaçant en dimension deux et en considérant deux modes, nous donnons une caractérisation complète du comportement asymptotique de tels systèmes lorsque le paramètre de perturbation tend vers zéro. Ensuite, nous étudions la discrétisation des systèmes à commutations singulièrement perturbés, en portant un intérêt particulier aux méthodes de discrétisation permettant de préserver la stabilité et les fonctions de Lyapunov quadratiques communesMany phenomena we encounter can be described by hybrid models, namely, consisting of one continuous dynamic and one discret dynamic at the same time. Moreover, these dynamics often evolves in different time scales. In this thesis, we deal with the stability analysis of singularly perturbed switched systems in continuous time. When we consider switchings, the "classical" approach (decoupling fast and slow dynamics) allowing to analyse stability of singularly perturbed systems doesn't hold anymore. Considering second order singularly perturbed switched systems woth two modes, we completely characterize de stability behavior of such systems when the perturbation parameter goes to zero. Then, we study the discretization of singularly perturbed switched systems. In particular, we focus on methods allowing to preserve stability and common quadratic Lyapunov functions
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Badaoui, Noad. "Dynamique et estimation paramétrique pour les gyroscopes laser à milieu amplificateur gazeux." Thesis, Paris Sciences et Lettres (ComUE), 2016. http://www.theses.fr/2016PSLEM058/document.
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Les gyroscopes laser à gaz constituent une solution technique de haute performances dans les problématiques de navigation inertielle. Néanmoins, pour de très faibles vitesses de rotation, les petites imperfections des miroirs de la cavité optique font que les deux faisceaux contra-propageant sont verrouillés en phase. En conséquence, les mesures en quadrature de leur différence de phase ne permettent plus de remonter directement aux vitesses de rotation à l'intérieur d'une zone autour de zéro, dite zone aveugle statique, ou, si l'on utilise une procédure d'activation mécanique, dite zone aveugle dynamique. Ce travail montre qu'il est néanmoins possible, en utilisant des méthodes issues du filtrage et de l'estimation, de remonter aux vitesses de rotation mêmes si ces dernières sont en zone aveugle. Pour cela, on part d'une modélisation physique de la dynamique que l'on simplifie par des techniques de perturbations singulières pour en déduire une généralisation des équations de Lamb. Il s'agit de quatre équations différentielles non-linéaires qui décrivent la dynamique des intensités et des phases des deux faisceaux contra-propageant. Une étude qualitative par perturbations régulières, stabilité exponentielle des points d'équilibre et applications de Poincaré permet de caractériser les zones aveugles statiques et dynamiques en fonction des imperfections dues aux miroirs. Il est alors possible d'estimer en ligne avec un observateur asymptotique fondé sur les moindre carrés récursifs ces imperfections en rajoutant aux deux mesures en quadrature celles des deux intensités. La connaissance précise de ces imperfections permet alors de les compenser dans la dynamique de la phase relative, et ainsi d'estimer les rotations en zone aveugle. Des simulations numériques détaillées illustrent l'intérêt de ces observateurs pour augmenter la précision des gyroscopes à gazGaz ring laser gyroscopes provide a high performance technical solution for inertial navigation. However, for very low rotational speeds, the mirrors imperfections of the optical cavity induce a locking phenomena between the phases of the two counter-propagating Laser beams. Hence, the measurements of the phase difference can no longer be used when the speed is within an area around zero, called lock-in zone, or,if a procedure of mechanical dithering is implemented, dithering lock-in zone. Nevertheless, this work shows that it is possible using filtering and estimation methods to measure the speed even within the lock-in zones. To achieve this result, we exploit a physical modeling of the dynamics that we simplify, using singular perturbation techniques, to obtain a generalization of Lamb's equations. There are four non-linear differential equations describing the dynamics of the intensities and phases of the two counter-propagating beams. A qualitative study by regular perturbation theory, exponential stability of the equilibrium points and Poincaré maps allows a characterisation of the lock-in zones according to the mirrors imperfections. It is then possible to estimate online, with an asymptotic observer based on recursive least squares, these imperfections by considering the additional measurements of the beam intensities. Accurate knowledge of these imperfections enables us to compensate them in the dynamic of the relative phase, and thus to estimate rotational speeds within the lock-in zones. Detailed numerical simulations illustrate the interest of those observers to increase the accuracy of gas ring laser gyroscopes
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Seydi, Ousmane. "Perturbations singulières des systèmes dynamiques en dimension infinie : théorie et applications." Phd thesis, Université Sciences et Technologies - Bordeaux I, 2013. http://tel.archives-ouvertes.fr/tel-00991857.
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L'objectif de cette thèse est d'étudier et de donner des outils pour la compréhension des problèmes de perturbations singulières pour des modèles épidémiques et des problèmes de dynamiques de populations. Les modèles considérés sont des équations structurées en âge qui peuvent dans certains cas se réécrire comme des équations à retard. L'étude de ces classes d'exemples s'est faite avec succès et a permis de comprendre et de mettre en évidence toute la complexité et l'étendue de ces problèmes. Comme on peut le remarquer dans la littérature, l'une des clés fondamentales à la compréhension de ces problèmes est l'étude des variétés normalement hyperboliques en dimension infinie que nous avons largement étudiées dans cette thèse. L'approche utilisée est la méthode de Lyapunov-Perron. Ce qui nous a amené à étudier les problèmes de persistance et d'existence de trichotomie (dichotomie) exponentielle qui sont des éléments fondamentaux dans l'utilisation de cette méthode.
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Grosman, Serguei. "The robustness of the hierarchical a posteriori error estimator for reaction-diffusion equation on anisotropic meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601418.
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Singularly perturbed reaction-diffusion problems
exhibit in general solutions with anisotropic
features, e.g. strong boundary and/or interior
layers. This anisotropy is reflected in the
discretization by using meshes with anisotropic
elements. The quality of the numerical solution
rests on the robustness of the a posteriori error
estimator with respect to both the perturbation
parameters of the problem and the anisotropy of the
mesh. The simplest local error estimator from the
implementation point of view is the so-called
hierarchical error estimator. The reliability
proof is usually based on two prerequisites:
the saturation assumption and the strengthened
Cauchy-Schwarz inequality. The proofs of these
facts are extended in the present work for the
case of the singularly perturbed reaction-diffusion
equation and of the meshes with anisotropic elements.
It is shown that the constants in the corresponding
estimates do neither depend on the aspect ratio
of the elements, nor on the perturbation parameters.
Utilizing the above arguments the concluding
reliability proof is provided as well as the
efficiency proof of the estimator, both
independent of the aspect ratio and perturbation
parameters.
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Grosman, Serguei. "Robust local problem error estimation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600475.
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Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the equilibrated residual method and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.
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Gatse, Franchel. "Spectre ordonné et branches analytiques d'une surface qui dégénère sur un graphe." Electronic Thesis or Diss., Orléans, 2020. http://www.theses.fr/2020ORLE3205.
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Dans ce travail, nous donnons un cadre général de surfaces riemanniennes qui dégénèrent sur des graphes métriques que nous appelons surfaces décomposables en cylindres et en jonctions. Les surfaces décomposables en cylindres et en jonctions dépendent d’un paramètre t qui traduit le mécanisme d’écrasement sur le graphe. Quand le paramètre t tend vers 0, les circonférences des cylindres tendent vers 0 et leurs longueurs restent fixes. On obtient ainsi les arêtes du graphe limite. Les jonctions, elles, sont écrasées dans toutes les directions et donc dégénèrent sur les sommets du graphe limite. Nous étudions alors le comportement asymptotique du spectre de ces variétés lors de cette déformation. Nous adoptons les points de vue de la convergence des valeurs propres ordonnées et de celle des branches analytiques. Ces deux approches sont fondamentalement différentes. Le cas des valeurs propres ordonnées est assez classique et nous retrouvons la convergence vers le spectre du graphe limite. L’étude des branches analytiques est plus original. Nous montrons la convergence et donnons une caractérisation des limites possibles. Ces résultats s’appliquent dans le cas des surfaces de translations qui possèdent une direction complètement périodiqueIn this work, we give a general framework of Riemannian surfaces that can degenerate on metric graphs and that we call surfaces made from cylinders and connecting pieces. The latter depend on a parameter t that describes the degeneration. When t goes to 0, the waists of the cylinders go to 0 but their lengths stay fixed. We thus obtain the edges of the limiting graph. The connecting pieces are squeezed in all directions and degenerate on the vertices of the limiting graph. We then study the asymptotic behaviour of the spectrum of these surfaces when t varies from two different points of view, considering the spectrum either as a sequence of ordered eigenvalues or as a collection of analytic eigenbranches. In the case of ordered eigenvalues, we recover a rather classical statement, and prove that the spectrum converges to the spectrum of the limiting object. The study of the analytic eigenbranches is more original. We prove that any such eigenbranch converges and we give a characterisation of the possible limits. These results apply to translation surfaces on which there is a completely periodic direction
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Negron, Luis G. "Initial-value technique for singularly perturbed two point boundary value problems via cubic spline." Master's thesis, University of Central Florida, 2010. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/4597.
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A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed.ID: 029051011; System requirements: World Wide Web browser and PDF reader.; Mode of access: World Wide Web.; Thesis (M.S.)--University of Central Florida, 2010.; Includes bibliographical references (p. 48-50).
M.S.
Masters
Department of Mathematics
Sciences
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Malloci, Ivan. "Sur les systèmes à commutation à deux échelles de temps : une application au contrôle de guidage de bande dans un laminoir à chaud." Phd thesis, Institut National Polytechnique de Lorraine - INPL, 2009. http://tel.archives-ouvertes.fr/tel-00439457.
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Dans cette thèse, on s'est attaché à résoudre un certain nombre de problèmes qui apparaissent lorsqu'on traite des problèmes concrets de contrôle: phénomènes à plusieurs échelles de temps, discontinuités de la commande lors du basculement d'un correcteur à un autre, nécessité de concevoir un nombre limité de correcteurs différents malgré une gamme très importante des produits traités. Pour illustrer concrètement les résultats obtenus, nous nous sommes appuyés sur un exemple industriel concret, le contrôle de guidage de bande durant le processus de laminage dans un laminoir à chaud. D'abord, nous proposons une solution convexe au problème de commande optimale linéaire quadratique pour les systèmes linéaires à deux échelles de temps en temps discret. Ensuite, nous établissons des conditions suffisantes, formulées sous la forme d'inégalités matricielles linéaires, qui permettent de vérifier la stabilité d'un système à commutation à deux échelles de temps et de synthétiser des correcteurs stabilisants. Nous proposons aussi dans ce travail une méthode pour minimiser les discontinuités sur la commande dans le cadre des systèmes à commutation. Dans le contexte du contrôle de guidage de bande pour un laminoir à chaud, nous ne pouvons pas négliger l'influence des paramètres incertains, qui sont dus principalement au fait que ce genre de système traite une gamme de produits très large. Donc, dans la synthèse du correcteur, nous prenons en compte ces variations en divisant l'ensemble des produits en plusieurs familles et en synthétisant un correcteur différent pour chaque famille.
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Azouit, Rémi. "Elimination adiabatique pour systèmes quantiques ouverts." Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLEM008/document.
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Cette thèse traite du problème de la réduction de modèle pour les systèmes quantiquesouverts possédant différentes échelles de temps, également connu sous le nom d’éliminationadiabatique. L’objectif est d’obtenir une méthode générale d’élimination adiabatiqueassurant la structure quantique du modèle réduit.On considère un système quantique ouvert, décrit par une équation maîtresse deLindblad possédant deux échelles de temps, la dynamique rapide faisant converger lesystème vers un état d’équilibre. Les systèmes associés à un état d’équilibre unique ouune variété d’états d’équilibre ("decoherence-free space") sont considérés. La dynamiquelente est traitée comme une perturbation. En utilisant la séparation des échelles de temps,on développe une nouvelle technique d’élimination adiabatique pour obtenir, à n’importequel ordre, le modèle réduit décrivant les variables lentes. Cette méthode, basée sur undéveloppement asymptotique et la théorie géométrique des perturbations singulières, assureune bonne interprétation physique du modèle réduit au second ordre en exprimant ladynamique réduite sous une forme de Lindblad et la paramétrisation définissant la variétélente dans une forme de Kraus (préservant la trace et complètement positif). On obtientainsi des formules explicites, pour calculer le modèle réduit jusqu’au second ordre, dans lecas des systèmes composites faiblement couplés, de façon Hamiltonienne ou en cascade;des premiers résultats au troisième ordre sont présentés. Pour les systèmes possédant unevariété d’états d’équilibre, des formules explicites pour calculer le modèle réduit jusqu’ausecond ordre sont également obtenuesThis thesis addresses the model reduction problem for open quantum systems with differenttime-scales, also called adiabatic elimination. The objective is to derive a generic adiabaticelimination technique preserving the quantum structure for the reduced model.We consider an open quantum system, described by a Lindblad master equation withtwo time-scales, where the fast time-scale drives the system towards an equilibrium state.The cases of a unique steady state and a manifold of steady states (decoherence-free space)are considered. The slow dynamics is treated as a perturbation. Using the time-scaleseparation, we developed a new adiabatic elimination technique to derive at any orderthe reduced model describing the slow variables. The method, based on an asymptoticexpansion and geometric singular perturbation theory, ensures the physical interpretationof the reduced second-order model by giving the reduced dynamics in a Lindblad formand the mapping defining the slow manifold as a completely positive trace-preserving map(Kraus map) form. We give explicit second-order formulas, to compute the reduced model,for composite systems with weak - Hamiltonian or cascade - coupling between the twosubsystems and preliminary results on the third order. For systems with decoherence-freespace, explicit second order formulas are as well derived
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Oudet, Salomé. "Équations de Hamilton-Jacobi sur des réseaux ou des structures hétérogènes." Thesis, Rennes 1, 2015. http://www.theses.fr/2015REN1S051/document.
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Cette thèse porte sur l'étude de problèmes de contrôle optimal sur des réseaux (c'est-à-dire des ensembles constitués de sous-régions reliées entre elles par des jonctions), pour lesquels on autorise différentes dynamiques et différents coûts instantanés dans chaque sous-région du réseau. Comme dans les cas plus classiques, on aimerait pouvoir caractériser la fonction valeur d'un tel problème de contrôle par le biais d'une équation de Hamilton-Jacobi-Bellman. Cependant, les singularités géométriques du domaine, ainsi que les discontinuités des données ne nous permettent pas d'appliquer la théorie classique des solutions de viscosité. Dans la première partie de cette thèse nous prouvons que les fonctions valeurs de problèmes de contrôle optimal définis sur des réseaux 1-dimensionnel sont caractérisées par de telles équations. Dans la seconde partie les résultats précédents sont étendus au cas de problèmes de contrôle définis sur une jonction 2-dimensionnelle. Enfin, dans une dernière partie, nous utilisons les résultats obtenus précédemment pour traiter un problème de perturbation singulière impliquant des problèmes de contrôle optimal dans le plan pour lesquels les dynamiques et les coûts instantanés peuvent être discontinus à travers une frontière oscillanteThis thesis focuses on the study of optimal control problems defined on networks (i.e. sets consisting of sub-regions connected together through junctions), where different dynamics and different running costs are allowed in each sub-region of the network. As in classical cases, we would like to characterize the value function of such an optimal control problem through an Hamilton-Jacobi-Bellman equation. However, the geometrical singularities of the domain and the data discontinuities do not allow us to apply the classical theory of viscosity solutions. In the first part of this thesis, we prove this kind of characterization for the value functions of optimal control problems defined on 1-dimensional networks. In the second part, the previous results are extended to the case of control problems defined on a 2-dimensional junction. Finally, in the last part, we use the results obtained previously to treat a singular perturbation problem involving optimal control problems in the plane for which the dynamics and running costs can be discontinuous through an oscillating border
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Dai, Ping. "Réjection de perturbation sur un système multi-sources - Application à une propulsion hybride." Thesis, Poitiers, 2015. http://www.theses.fr/2015POIT2251/document.
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Ce mémoire porte sur l'étude d'un système de gestion d'énergie électrique dans un système multi-sources soumis à des perturbations exogènes. L'application visée est l'alimentation d'une propulsion hybride diesel/électrique équipée d'un système d'absorption des pulsations de couple. Les perturbations exogènes considérées peuvent être transitoires ou persistantes. Une perturbation transitoire correspond à une variation rapide du couple de charge, due par exemple à une accélération ou une décélération du véhicule. Une perturbation persistante provient du système de compensation des pulsations de couple générées par le moteur thermique. Le premier objectif du contrôle est de maintenir constante la tension du bus continu. Le deuxième objectif est d'absorber dans un système de stockage rapide constitué de super condensateur ces perturbations qui peuvent à terme provoquer une usure prématurée de la batterie. Le troisième objectif est de compenser l'auto-décharge dans le super condensateur en maintenant constante sa tension nominale. Les deux sources (batterie et super condensateur) sont reliées au bus continu par l'intermédiaire de deux convertisseurs boost DC/DC. La commande consiste à piloter les rapports cycliques de chaque convertisseur. C'est un système non linéaire où la commande est multiplicative de l'état. L'approche classique consistant à résoudre les équations Francis-Byrnes-Isidori ne s'applique pas directement dans ce cas où la sortie et la matrice d'interconnection dépendent de la commande. De plus, si cette approche est bien adaptée au rejet de perturbations persistantes, elle montre ces limites pour le rejet de perturbations non persistantes combiné à des objectifs de régulation. Notre approche a consisté à écrire le système sous un formalisme Port-Controlled Hamiltonian et à s'affranchir de la contrainte de la dépendance de la matrice d'interconnection avec la commande en utilisant la théorie des perturbations singulières. La commande du système dégénéré peut ensuite être calculée par une approche passive. Les performances de cette commande ont été testées en simulation et à l'aide d'un banc d'essai expérimental. Les résultats montrent l'efficacité du système d'absorption des différents types de perturbation tout en respectant les deux objectifs de régulationThis thesis presents the research of energy management in a battery/ultracapacitor hybrid energy storage system with exogenous disturbance in hybrid electric vehicular application. Transient and harmonic persistent disturbances are the two kinds of disturbances considered in this thesis. The former is due to the transient load power demand during acceleration and deceleration, and the latter is introduced from the process of the internal combustion engine torque ripples compensation. Our control objective is to absorb the disturbances causing battery wear via the ultracapacitor, and meanwhile, to maintain a constant DC voltage and to compensate the self-discharge in the ultracapacitor to maintain it operating at the nominal state of charge. The object system is nonlinear due to the multiplicative relation between the input and the state. The traditional approach to solve Francis-Byrnes-Isidori equations cannot be directly applied in this case since the interconnect matrix depends on the control input. Besides, even if this approach is well suited to the rejection of persistent disturbances, it shows the limits for the case of non-persistent disturbances which is also our object. Our contributed control method is realized through a cascade control structure based on the singular perturbation theory. The ultracapacitor current with the fastest motion rate is controlled in the inner fast loop through which we impose the desired dynamic to the system. The reduced system controlled in the outer slow loop is a Hamiltonian system and the controller is designed via interconnection and damping assignment. Simulations and experiments have been carried out to evaluate the control performance. A contrast of the system responses with and without the control algorithm shows that, with the control algorithm, the ultracapacitor effectively absorbs the disturbances; and verifies the effectiveness of the control algorithm
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Franz, Sebastian. "Uniform Error Estimation for Convection-Diffusion Problems." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2014. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-133017.
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Let us consider the singularly perturbed model problem
Lu := -epsilon laplace u-bu_x+cu = f
with homogeneous Dirichlet boundary conditions on the unit-square (0,1)^2. Assuming that b > 0 is of order one, the small perturbation parameter 0 < epsilon << 1 causes boundary layers in the solution.
In order to solve above problem numerically, it is beneficial to resolve these layers. On properly layer-adapted meshes we can apply finite element methods and observe convergence.
We will consider standard Galerkin and stabilised FEM applied to above problem. Therein the polynomial order p will be usually greater then two, i.e. we will consider higher-order methods.
Most of the analysis presented here is done in the standard energy norm. Nevertheless, the question arises: Is this the right norm for this kind of problem, especially if characteristic layers occur? We will address this question by looking into a balanced norm.
Finally, a-posteriori error analysis is an important tool to construct adapted meshes iteratively by solving discrete problems, estimating the error and adjusting the mesh accordingly. We will present estimates on the Green’s function associated with L, that can be used to derive pointwise error estimators.
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Barles, Guy. "Contribution à la théorie des solutions de viscosité des équations de Hamilton-Jacobi du premier ordre et applications à des problèmes de contrôle optimal et de perturbations singulières." Paris 9, 1988. https://portail.bu.dauphine.fr/fileviewer/index.php?doc=1988PA090004.
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Nous présentons dans ce travail divers résultats concernant les équations de Hamilton-Jacobi du premier ordre ainsi que leurs applications à certains problèmes de contrôle optimal déterministe et de perturbations singulières. La première partie est consacrée à l'étude des solutions continues: nous donnons divers résultats d'existence, d'unicité et de régularité à la fois locale et globale). La deuxième partie décrit une étude systématique des solutions discontinues: elle fournit une approche générale très simple des problèmes de temps de sortie, de contrôle non-borne et de perturbations singulières, avec, en particulier, des applications dans le cadre des grandes déviations
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Tidefelt, Henrik. "Differential-algebraic equations and matrix-valued singular perturbation." Doctoral thesis, Linköpings universitet, Reglerteknik, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51653.
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With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero. The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions. Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations. While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly. One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution. One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.
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Heck, Bonnie S. "On singular perturbation theory for piecewise-linear systems." Diss., Georgia Institute of Technology, 1988. http://hdl.handle.net/1853/15054.
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Yang, James Ting Feng. "Singular Perturbation of Stochastic Control and Differential Games." Thesis, University of Sydney, 2020. https://hdl.handle.net/2123/22979.
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The defining trait of singular perturbation problems in dynamical systems is the degeneracy of the highest order differential term when a small parameter is formally set to be zero. The implication is that the limiting solution does not entirely coincide with the solution to the degenerated system. It then becomes apparent that the derivation of the limiting solution is non-trivial. However, under certain circumstances, this has been resolved by the Tikhonov's theorem. On the other hand, these problems arise naturally in slow-fast or multiscale models where the small parameter represents the ratio between the evolutionary speeds. Moreover, the limiting solution is often of lower dimensionality and offers a viable method for dimension reduction. For these reasons, singular perturbation techniques have been widely applied in optimisation theory to disciplines such as ecology, robotics, finance and physics, to name a few. In this thesis, we propose three optimisation problems described by a quadratic cost function and a coupled pair of slow-fast linear state equations driven by Brownian motion. The first one is an optimal control problem when the state and control appear in the diffusion coefficient of the noise. The second one is a two-player zero-sum differential game with a constant diffusion coefficient. And third, is an optimal control problem with processes taking values in infinite dimensional spaces. Our aim is to investigate the limiting behaviour of these optimisation problems and its value functions. The general approach is to convert the stochastic and controlled singular perturbation problem into a classical and deterministic singular perturbation problem via the Riccati equation. Consequently, a version of Tikhonov's theorem has to be formulated.
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Oueslati, Soumaya. "Une nouvelle formulation variationnelle pour le problème de diffusion en électromagnétisme utilisant une méthode intégrale avec une condition aux limites d'impédance d'ordre élevé - Petites perturbations d'une interface pour le système de Stokes." Thesis, Cergy-Pontoise, 2019. http://www.theses.fr/2019CERG1046.
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Cette thèse contient deux parties principales. Dans la première partie, nous proposons une nouvelle formulation variationnelle pour le problème de diffusion en électromagnétisme qui s’obtient avec une méthode intégrale utilisant une condition aux limites d’impédance d’ordre élevé (HOIBC). Cette méthode améliore la précision du calcul de la surface équivalente radar (SER) par rapport à la condition limite d’impédance de Leontovich dans de nombreux cas. Ensuite, nous donnons la discrétisation de cette formulation avec des fonctions de base de Rao-Wilton-Glisson. Par suite, une approximation de la condition aux limites d’impédance d’ordre élevé est donnée. A cette fin, des formules de saut et la théorie des distributions pour surmonter la difficulté de la discrétisation de div(n×w) et de rot(w) pour tout w dans H(div) est utilisée. De plus, nous mettons en oeuvre trois méthodes pour évaluer certaines intégrales singulières qui apparaissent dans les éléments de matrice de notre formulation. Cette méthode numérique est validée par des cas tests sur des sphères où l'on compare les résultats numériques et analytiques pour le calcul de la SER. Dans la deuxième partie, on a étudié un problème de transmission pour le système de Stokes. Tout d’abord, on a trouvé une représentation de la solution en appliquant la théorie du potentiel. Par la suite, on a obtenu un développement asymptotique de cette solution en fonction du paramètre de la déformation du bord de l’inclusion. Puis, on a donné un développement asymptotique pour le tenseur de viscosité du système de StokesThis thesis contains two main parts. In the first one, we propose a new variational formulation for the electromagnetic scattering problem which derives from an integral method with the use of high order impedance boundary condition (HOIBC) to improve the accuracy of Leontovich impedance boundary condition. Then, we give the discretization for this formulation with Rao-Wilton-Glisson basis functions. Therefore, we propose an approximation of the high order impedance boundary condition which is Hodge operator. We use the jump formulas and the theory of distributions to overcome the difficulty of the discretization of div(n × w) and rot(w) for all w in H(div). Moreover, we implement three methods to evaluate some singular integrals that appear in the boundary integral equation. The performances of the HOIBC are evaluated by calculating the radar cross section (RCS) with different meshes for the unit sphere. We also compare the numerical and analytical results. In the second part, we have considered the Stokes system for a viscous medium consisting of an inclusion immerged in a background medium. We derive the asymptotic expansion of the perturbed velocity field due to the presence of small perturbations in the interface of an inclusion using the layer potential theory. Further, we use these techniques to determine a relationship between Stokes solutions measurements and the shape of the object. Besides, we prove an asymptotic expansion for the perturbation in the viscosity moment tensors caused by the presence of an inhomogeneity
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Sensi, Mattia. "A Geometric Singular Perturbation approach to epidemic compartmental models." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/286191.
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We study fast-slow versions of the SIR, SIRS and SIRWS epidemiological models, and of the SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. The multiple time scale behavior is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast-slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov's method, we are able to not only analyze the stability of the endemic equilibria of the SIR and SIRS models, but also to show that in the remaining models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model and the SIRS on homogeneous graphs, for which we provide a detailed description of their dynamics by combining analytic and numerical techniques. In particular, for the latter we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.
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Sensi, Mattia. "A Geometric Singular Perturbation approach to epidemic compartmental models." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/286191.
Full textAbstract:
We study fast-slow versions of the SIR, SIRS and SIRWS epidemiological models, and of the SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. The multiple time scale behavior is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast-slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov's method, we are able to not only analyze the stability of the endemic equilibria of the SIR and SIRS models, but also to show that in the remaining models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model and the SIRS on homogeneous graphs, for which we provide a detailed description of their dynamics by combining analytic and numerical techniques. In particular, for the latter we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.
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Hsu, Ting-Hao. "A Geometric Singular Perturbation Theory Approach to Viscous Singular Shocks Profiles for Systems of Conservation Laws." The Ohio State University, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=osu1437144893.
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