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Metodos para Solução da Equação HJB-Riccati via Famíla de Estimadores Parametricos RLS Simplificados e Dependentes de Modelo. / Methods for Solution of the HJB-Riccati Equation in the Family of Simplified and Model Dependent Parametric RLS Estimators.SANTOS, Watson Robert Macedo 21 August 2014 (has links)
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Previous issue date: 2014-08-21 / Due to the demand for high-performance equipments and the rising cost of energy,
the industrial sector is developing equipments to attend minimization of the theirs
operational costs. The implementation of these requirements generate a demand
for projects and implementations of high-performance control systems. The optimal
control theory is an alternative to solve this problem, because in its design considers
the normative specifications of the system design, as well as those that are related
to the operational costs. Motivated by these perspectives, it is presented the study
of methods and the development of algorithms to the approximated solution of the
Equation Hamilton-Jacobi-Bellman, in the form of discrete Riccati equation, model
free and dependent of the dynamic system. The proposed solutions are developed
in the context of adaptive dynamic programming that are based on the methods for
online design of optimal control systems, Discrete Linear Quadratic Regulator type.
The proposed approach is evaluated in multivariable models of the dynamic systems
to evaluate the perspectives of the optimal control law for online implementations. / Devido a demanda por equipamentos de alto desempenho e o custo crescente da
energia, o setor industrial desenvolve equipamentos que atendem a minimização dos
seus custos operacionais. A implantação destas exigências geram uma demanda por
projetos e implementações de sistemas de controle de alto desempenho. A teoria de
controle ótimo é uma alternativa para solucionar este problema, porque considera
no seu projeto as especificações normativas de projeto do sistema, como também as
relativas aos seus custos operacionais. Motivado por estas perspectivas, apresenta-se
o estudo de métodos e o desenvolvimento de algoritmos para solução aproximada
da Equação Hamilton-Jacobi-Bellman, do tipo Equação Discreta de Riccati, livre
e dependente de modelo do sistema dinâmico. As soluções propostas são desenvolvidas no contexto de programação dinâmica adaptativa (ADP) que baseiam-se nos
métodos para o projeto on-line de Controladores Ótimos, do tipo Regulador Linear
Quadrático Discreto. A abordagem proposta é avaliada em modelos de sistemas
dinâmicos multivariáveis, tendo em vista a implementação on-line de leis de controle
ótimo.
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Μέθοδος Hamilton-Jacobi για τη ρύθμιση μη γραμμικών διεργασιών με ασταθή δυναμική μηδενιστώνΜουσαβερέ, Δήμητρα 13 March 2009 (has links)
Για την αντιμετώπιση του προβλήματος ρύθμισης ενός συστήματος μη ελάχιστης φάσης είναι γνωστοί δύο τρόποι από τη θεωρία των γραμμικών συστημάτων. Ο ένας αφορά στην επιλογή βέλτιστης συνθετικής εξόδου ως προς την οποία το σύστημα είναι ελάχιστης φάσης. Ο δεύτερος τρόπος περιλαμβάνει άμεση κατασκευή βέλτιστου νόμου ανάδρασης καταστάσεων ως προς ένα σύνθετο δείκτη απόδοσης.
Στην παρούσα εργασία αρχικά αναπτύσσεται μέθοδος για τη σύνθεση βέλτιστου νόμου ανάδρασης καταστάσεων για μη γραμμικές διεργασίες, όπου η είσοδος υπεισέρχεται μη γραμμικά στις διαφορικές εξισώσεις, με βάση ένα σύνθετο τετραγωνικό δείκτη απόδοσης. Ο δείκτης αυτός εξαρτάται τόσο από τη ρυθμιστική απόκλιση, όσο και από την απόκλιση της μεταβλητής χειρισμού. Για την επίλυση του προβλήματος δυναμικής βελτιστοποίησης χρησιμοποιούνται οι εξισώσεις Hamilton – Jacobi μέσω των οποίων υπολογίζεται ο βέλτιστος νόμος ανάδρασης καταστάσεων. Η λύση των εξισώσεων Hamilton – Jacobi υπολογίζεται με βάση την επαναληπτική μέθοδο Newton – Kantorovich. Σε κάθε βήμα της επανάληψης επιλύεται προσεγγιστικά μια μερική διαφορική εξίσωση τύπου Zubov με τη βοήθεια αναπτύγματος σε δυναμοσειρά. Στο Νοστό βήμα της επανάληψης η μέθοδος παράγει τη Νοστής τάξης προσέγγιση του αναπτύγματος κατά Taylor του βέλτιστου νόμου ανάδρασης καταστάσεων. Η παραπάνω μέθοδος εφαρμόζεται σε προβλήμα ρύθμισης της συγκέντρωσης προϊόντος σε σύστημα δύο μη ισοθερμοκρασιακών αντιδραστήρων CSTR, όπου λαμβάνει χώρα εξώθερμη αντίδραση, στην περίπτωση που η είσοδος υπεισέρχεται μη γραμμικά στις δυναμικές εξισώσεις της διεργασίας. Επίσης μελετώνται οι ιδιότητες σύγκλισης της επαναληπτικής μεθόδου Newton – Kantorovich, όταν αυτή εφαρμόζεται για την επίλυση της εξίσωσης Hamilton – Jacobi – Bellman που αντιστοιχεί στο πρόβλημα βελτιστοποίησης ενός σύνθετου τετραγωνικού δείκτη απόδοσης υπό τους περιορισμούς μιας μη γραμμικής δυναμικής όπου η είσοδος υπεισέρχεται γραμμικά στις διαφορικές εξισώσεις.
Στη συνέχεια, για τη βέλτιστη ρύθμιση μη γραμμικών συστημάτων με ασταθή δυναμική μηδενιστών (συστήματα μη ελάχιστης φάσης), χρησιμοποιείται ο συνήθης τετραγωνικός δείκτης απόδοσης ISE. Στην περίπτωση αυτή το πρόβλημα δυναμικής βελτιστοποίησης είναι ιδιόμορφο. Για την επίλυση του προβλήματος αυτού το μη γραμμικό σύστημα μετασχηματίζεται στην κανονική μορφή Byrnes-Isidori, εφαρμόζεται η θεωρία Hamilton – Jacobi και υπολογίζεται στατικά ισοδύναμη συνθετική έξοδος με ευσταθή δυναμική μηδενιστών. Η ρύθμιση της συνθετικής εξόδου στο προκαθορισμένο σημείο επιτυγχάνεται με γραμμικοποίηση εισόδου/εξόδου. Για την επίλυση των σχετικών εξισώσεων Hamilton–Jacobi αναπτύσσεται η επαναληπτική μέθοδος Newton – Kantorovich, η οποία περιλαμβάνει την επίλυση μιας μερικής διαφορικής εξίσωσης τύπου Zubov σε κάθε βήμα της επανάληψης. Η μέθοδος εφαρμόζεται σε πρόβλημα ρύθμισης της συγκέντρωσης του επιθυμητού προϊόντος σε μη ισοθερμοκρασιακό αντιδραστήρα CSTR με κινητική Van de Vusse που παρουσιάζει ασταθή δυναμική μηδενιστών.
Τέλος, οι δύο μέθοδοι συγκρίνονται με βάση τους επιμέρους δείκτες απόδοσης ISE και ISC, των οποίων ο γραμμικός συνδυασμός συνιστά το σύνθετο δείκτη απόδοσης της πρώτης μεθόδου, ενώ τα αποτελέσματά τους συγκρίνονται όταν αυτές εφαρμόζονται σε πρόβλημα ρύθμισης της συγκέντρωσης του επιθυμητού προϊόντος σε μη ισοθερμοκρασιακό αντιδραστήρα CSTR με κινητική Van de Vusse. / For the control of nonlinear nonminimum – phase systems, there are two possible lines of attack, originating from linear systems theory: a) direct calculation of the optimal state feedback with respect to a quadratic performance index that represents a combination of an error measure and a control effort measure (composite index), and b) calculation of the ISE-optimal minimum-phase output and subsequent input/output linearization on that output.
This work develops a numerical algorithm for the calculation of an optimal nonlinear state feedback law for nonlinear systems. A quadratic performance index is used, which contains quadratic error terms and quadratic input penalty terms. The optimization problem is solved using the Hamilton-Jacobi equations, which determine the optimal nonlinear state feedback law. A Newton-Kantorovich iteration is developed for the solution of the pertinent Hamilton-Jacobi equations, which involves solving a Zubov partial differential equation at each step of the iteration, using a power series method. At step N of the iteration, the method generates the (N+1)-th order truncation of the Taylor series expansion of the optimal state feedback function. The method is applied to the problem of controlling a system of two non-isothermal continuous stirred tank reactors (CSTR), where an exothermic reaction takes place. Convergence properties of the algorithm are also developed independently of Kantorovich’s theorem, and the results are illustrated in a numerical example.
For the optimal regulation of nonminimum-phase nonlinear systems, the performance index ISE (Integral of the Square of the Error) is used. The problem of minimizing ISE subject to the dynamics of the system and closed-loop stability is singular. The problem of calculation of an ISE-optimal, statically equivalent, minimum-phase output for nonminimum-phase compensation is formulated using Hamilton-Jacobi theory and the Byrnes-Isidori normal form representation of the nonlinear system. An input/output linearizing state feedback law is applied to regulate the synthetic output to a constant set point. A Newton-Kantorovich iteration is developed for the solution of the pertinent Hamilton-Jacobi equations, which involves solving a Zubov equation at each step of the iteration. The method is applied to the problem of controlling a nonisothermal CSTR with Van de Vusse kinetics, which exhibits nonminimum-phase behaviour.
Finally, the two methods are compared with respect to the constituent indexes ISE and ISC (Integral of the Square of the Control), whose linear combination forms the composite performance index. The numerical results from both methods are compared in the control of a nonisothermal CSTR with Van de Vusse kinetics.
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A mean-field game model of economic growth : an essay in regularity theoryLima, Lucas Fabiano 20 December 2016 (has links)
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Previous issue date: 2016-12-20 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / In this thesis, we present a priori estimates for solutions of a mean-field game (MFG) defined
over a bounded domain Ω ⊂ ℝd. We propose an application of these results to a model of capital
and wealth accumulation.
In Chapter 1, an introduction to mean-field games is presented. We also put forward some of
the motivation from Economics and discuss previous developments in the theory of differential
games. These comments aim at indicating the connection between mean-field games theory, its
applications and the realm of Mathematical Analysis.
In Chapter 2, we present an optimal control problem. Here, the agents are supposed to be
undistinguishable, rational and intelligent. Undistinguishable means that every agent is governed
by the same stochastic differential equation. Rational means that all efforts of the agent is to
maximize a payoff functional. Intelligent means that they are able to solve an optimal control
problem. Once we describe this (stochastic) optimal control problem, we produce a heuristic
derivation of the mean-field games system, which is summarized in a Verification Theorem; this
gives rise to the Hamilton-Jacobi equation (HJ). After that, we obtain the Fokker-Plank equation
(FP). Finally, we present a representation formula for the solutions to the (HJ) equation, together
with some regularity results.
In Chapter 3, a specific optimal control problem is described and the associated MFG is
presented. This MFG is prescribed in a bounded domain
Ω ⊂ ℝd, which introduces substantialadditional challenges from the mathematical view point. This is due to estimates for the solutionsat the boundary in Lp. The rest of the chapter puts forward two well known tips of estimates: theso-called Hopf-Lax formula and the First Order Estimate.
In Chapter 4, the wealth and capital accumulation mean-field game model is presented. The
relevance of studying MFG in a bounded domain then becomes clear. In light of the results obtained
in Chapter 3, we close Chapter 4 with the Hopf-Lax formula, and the First Order estimates.
Three appendices close this thesis. They gather elementary material on Stochastic Calculus
and Functional Analysis. / Nesta dissertação são apresentadas algumas estimativas a priori para soluções de sistemas
mean-field games (MFG), definidos em domínios limitados Ω ⊂ ℝd. Tais estimativas são aplicadas
em um modelo mean-field específico, que descreve o acúmulo de riqueza e capital.
No Capítulo 1, é apresentada uma breve introdução histórica sobre os mean-field games.
Nesta introdução, exploramos sua relação com a teoria dos jogos, cujos alicerces foram construídos
por economistas e matemáticos ao longo do século XX. O objetivo do capítulo é transmitir.
No Capítulo 2, apresentamos um problema de controle ótimo em que cada agente é suposto
ser indistinguível, racional e inteligente. Indistinguível no sentido de que cada um é governado
pela mesma equação diferencial estocástica. Racional no sentido de que todos os esforços do
agente são no sentido de maximizar um funcional de recompensa e, inteligente no sentido de que
são capazes de resolver um problema de controle ótimo. Descreve-se este problema de controle
ótimo, e apresenta-se a derivação heurística dos mean-field games; obtém-se através de um
Teorema de Verificação, a equação de Hamilton-Jacobi (HJ) associada, e em seguida, obtémse
a equação de Fokker-Planck. De posse destas equações, apresentamos alguns resultados
preliminares, como uma fórmula de representação para soluções da equação de HJ e alguns
resultados de regularidade.
No Capítulo 3, descreve-se um problema específico de controle ótimo e apresenta-se a respectiva
derivação heurística culminando na descrição de um MFG com condições não periódicas
na fronteira; esta abordagem é original na literatura de MFG. O restante do capítulo é
dedicado à exposição de dois tipos bem conhecidos de estimativas: a fórmula de Hopf-Lax e
estimativa de Primeira Ordem. Uma observação relevante, é a de que o trabalho em obter-se
estimativas a priori é aumentado substancialmente neste caso, devido ao fato de lidarmos com
estimativas para os termos de fronteira com normas em Lp.
ao leitor, as origens da Teoria Econômica contemporânea, que surgem à partir da utilização da
Matemática na formulação e resolução de problemas econômicos. Tal abordagem é motivada
principalmente pelo rigor e clareza da Matemática em tais circunstâncias.
No Capítulo 4, apresenta-se o modelo de jogo do tipo mean-field de acúmulo de capital e
riqueza, o que deixa claro a relevância do estudo dos MFG em um domínio limitado. À luz dos
resultados obtidos no Capítulo 3, encerramos o Capítulo 4 com as estimativas do tipo Hopf-Lax
e de Primeira Ordem.
Três apêndices encerram o texto desta dissertação de mestrado; estes reúnem material elementar
sobre Cálculo Estocástico e Análise Funcional.
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Asymptotic Analysis of Partial Differential Equations Arising in Biological Processes of Anomalous Diffusion / Analyse asymptotique d’équations aux dérivées partielles issues de processus biologiques de diffusion anormaleMateos González, Álvaro 22 September 2017 (has links)
Cette thèse est consacrée à l'analyse asymptotique d'équations aux dérivées partielles issues de modèles de déplacement sous-diffusif en biologie cellulaire. Notre motivation biologique est fondée sur les nombreuses observation récentes de protéinescytoplasmiques dont le déplacement aléatoire dévié de la diffusion Fickienne normale. Dans la première partie, nous étudions la décroissance auto-similaire de la solution d'une équation de renouvellement à queue lourde vers un état stationnaire. Les idéesmises en jeu sont inspirées de méthodes d'entropie relative. Nos principaux apports sont la preuve d'un taux de décroissance en norme L1 vers la loi de l'arc-sinus et l'introduction d'une fonction pivot spécifique dans une méthode d'entropie relative.La seconde partie porte sur la limite hyperbolique d'une équation de renouvellement structurée en âge et à sauts en espace. Nous y prouvons un résultat de « stabilité » : les solutions des problèmes rééchelonnés à ε > 0 convergent lorsque ε --> 0 vers la solution de viscosité de l'équation de Hamilton-Jacobi limite des problèmes à ε > 0. Les outilsmis en jeu proviennent de la théorie des équations de Hamilton-Jacobi.Ce travail présente trois idées intéressantes. La première est celle de prouver le résultat de convergence sur la condition de bord du problème plutôt que d'utiliser des fonctions test perturbées. La deuxième consiste en l'introduction de termes correcteurslogarithmiques en temps dans des estimations a priori ne découlant pas directementdu principe du maximum. Cela est dû à la non-existence d'un équilibre du problèmehomogène en espace. La troisième est une estimation précise de la décroissance de l'influence de la condition initiale sur le terme de renouvellement. Elle correspond à une estimation fine d'une version non-locale de la dérivée temporelle de la solution. Au cours de cette thèse, des simulations numériques de type Monte Carlo, schémas volumes finis, Lax-Friedrichs et Weighted Essentially Non Oscillating ont été réalisées. / This thesis is devoted to the asymptotic analysis of partial differential equations modelling subdiffusive random motion in cell biology. The biological motivation for this work is the numerous recent observations of cytoplasmic proteins whose random motion deviates from normal Fickian diffusion. In the first part, we study the self-similar decay towards a steady state of the solution of a heavy-tailed renewal equation. The ideas therein are inspired from relative entropy methods. Our main contributions are the proof of an L1 decay rate towards the arc-sine distribution and the introduction of a specific pivot function in a relative entropy method.The second part treats the hyperbolic limit of an age-structured space-jump renewal equation. We prove a "stability" result: the solutions of the rescaled problems at ε > 0 converge as ε --> 0 towards the viscosity solution of the limiting Hamilton-Jacobi equation of the ε > 0 problems. The main mathematical tools used come from the theory of Hamilton-Jacobi equations. This work presents three interesting ideas. The first is that of proving the convergence result on the boundary condition of the studied problem rather than using perturbed test functions. The second consists in the introduction of time-logarithmic correction termsin a priori estimates that do not follow directly from the maximum principle. That is due to the non-existence of a suitable equilibrium for the space-homogenous problem. The third is a precise estimate of the decay of the inuence of the initial condition on the renewal term. This is tantamount to a refined estimate of a non-local version of the time derivative of the solution. Throughout this thesis, we have performed numerical simulations of different types: Monte Carlo, finite volume schemes, Lax-Friedrichs schemes and Weighted Essentially Non Oscillating schemes.
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Comportement en temps long des solutions de quelques équations de Hamilton-Jacobi du premier et second ordre, locales et non-locales, dans des cas non-périodiques / Long time behavior of solutions of some first and second order, local and nonlocal Hamilton-Jacobi equations in non-periodic settingsNguyen, Thi Tuyen 01 December 2016 (has links)
La motivation principale de cette thèse est l'étude du comportement en temps grand des solutions non-bornées d'équations de Hamilton-Jacobi visqueuses dans RN en présence d'un terme d'Ornstein-Uhlenbeck. Nous considérons la même question dans le cas d'une équation de Hamilton-Jacobi du premier ordre. Dans le premier cas, qui constitue le cœur de la thèse, nous généralisons les résultats de Fujita, Ishii et Loreti (2006) dans plusieurs directions. La première est de considérer des opérateurs de diffusion plus généraux en remplaçant le Laplacien par une matrice de diffusion quelconque. Nous considérons ensuite des opérateurs non-locaux intégro-différentiels de type Laplacien fractionnaire. Le second type d'extension concerne le Hamiltonien qui peut dépendre de x et est seulement supposé sous-linéaire par rapport au gradient. / The main aim of this thesis is to study large time behavior of unbounded solutions of viscous Hamilton-Jacobi equations in RN in presence of an Ornstein-Uhlenbeck drift. We also consider the same issue for a first order Hamilton-Jacobi equation. In the first case, which is the core of the thesis, we generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a non-local integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear.
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Financial models and price formation : applications to sport betting / Modèles financiers et formation des prix : applications aux paris sportifsJottreau, Benoît 30 November 2009 (has links)
Cette thèse est composée de quatre chapitres. Le premier chapitre traite de l'évaluation de produits financiers dans un modèle comportant un saut pour l'actif risque. Ce saut représente la faillite de l'entreprise correspondante. On étudie alors l'évaluation des prix d'options par indifférence d'utilité dans un cadre d'utilité exponentielle. Par des techniques de programmation dynamique on montre que le prix d'un Bond est solution d'une équation différentielle et le prix d'options dépendantes de l'actif est solution d'une équation aux dérives partielles d'Hamilton-Jacobi-Bellman. Le saut dans la dynamique de l'actif risque induit des différences avec le modèle de Merton que nous tentons de quantifier. Le second chapitre traite d'un marché comportant des sauts : les paris sur le football. Nous rappelons les différentes familles de modèles pour un match de football et introduisons un modèle complet permettant d'évaluer les prix des différents produits apparus sur ce marché ces dix dernières années. La complexité de ce modèle nous amène à étudier un modèle simplifié dont nous étudions les implications et calculons les prix obtenus que l'on compare à la réalité. On remarque que la calibration implicite obtenue génère de très bons résultats en produisant des prix très proches de la réalité. Le troisième chapitre développe le problème de fixation des prix par un teneur de marche monopolistique dans le marché des paris binaires. Ce travail est un prolongement direct au problème introduit par Levitt [Lev04]. Nous généralisons en effet son travail aux cas des paris européens et proposons une méthode pour estimer la méthode de cotation utilisée par le book-maker. Nous montrons que deux hypothèses inextricables peuvent expliquer cette fixation des prix. D'une part, l'incertitude du public sur la vraie valeur ainsi que le caractère extrêmement risque-averse du bookmaker. Le quatrième chapitre prolonge quant à lui cette approche au cas de produits financiers non binaires. Nous examinons différents modèles d'offre et de demande et en déduisons, par des techniques de programmation dynamique, des équations aux dérivées partielles dictant la formation des prix d'achat et de vente. Nous montrons finalement que l'écart entre prix d'achat et prix de vente ne dépend pas de la position du teneur de marche dans l'actif considère. Cependant le prix moyen dépend lui fortement de la quantité détenue par le teneur de marche. Une approche simplifiée est finalement proposée dans le cas multidimensionnel / This thesis is composed of four chapters. The first one deals with the pricing of financial products in a single jump model for the risky asset. This jump represents the bankrupcy of the quoted firm. We study the pricing of derivatives in the context of indifference of utility with an exponential utility. By means of dynamic programming we show that the bond price is solution of an ordinary differential equation and that stock price dependent options are solutions of an equation with partial derivatives of Hamilton-Jacobi-Bellman type generalizing the Black-Scholes one. We then try to quantify differences in the price obtained here and the one from Merton model without jump. The second chapter deals with a specific jump market : the soccer betting market. We recall the different model families for a soccer match and introduce some full model which allows to price the products recently born in this market in last ten years. Nevertheless the model complexity leads us to study a simplified model introduced by Dixon and Robinson from which we are able to derive closed formulas and simulate prices that we compare to market prices. We remark that implicit calibration gives pretty goof fit of market data. Third chapter developps the approach of Levitt [Lev04] on price formation in binary betting market held by a monopolistic market-maker operating in a one time step trading. We generalize Levitt results with european format of betting. We show that prices are distorded on the pressure of demand and offer, that phenomena introducing a market probability that allows to price products under this new measure. We identify some best model for demand and offer and market maker strategy and show that probability change is obvious in case of imperfect information about the value of the product. Fourth chapter generalizes this approach to the case of general payoffs and continuous time. The task is more complex and we just derive partial derivative equations from dynamic programming that enable us to give the bid-ask prices of the product traded by the market-maker. One result is that, in most models, bid-ask spread does not depend on the inventory held by the dealer whereas mid-quote price strongly reflects the unbalance of the dealer
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Solutions variationnelles et solutions de viscosité de l'équation de Hamilton-Jacobi / Variational and viscosity solutions of the Hamilton-Jacobi equationRoos, Valentine 30 June 2017 (has links)
On étudie l'équation de Hamilton-Jacobi évolutive du premier ordre, couplée avec une donnée initiale lipschitzienne. Le but est de comparer les solutions de viscosité et les solutions variationnelles pour cette équation, deux notions de solutions faibles qui coïncident en dynamique hamiltonienne convexe. Pour travailler dans un cadre pertinent pour les deux types de solutions, on doit d’abord construire une solution variationnelle sans hypothèse de compacité sur la variété ou le hamiltonien étudiés. On retrace dans ce cas la construction historique des solutions variationnelles, en détaillant les propriétés de la famille génératrice obtenue par la méthode des géodésiques brisées. Il en découle des estimées permettant d’obtenir la solution de viscosité à partir de la solution variationnelle par un procédé d’itération. Après avoir vérifié que la solution variationnelle construite coïncide effectivement avec la solution de viscosité pour un Hamiltonien convexe, on caractérise les Hamiltoniens intégrables pour lesquels cette propriété persiste, en étudiant attentivement des exemples élémentaires en dimension 1 et 2. / We study the first order Hamilton-Jacobi equation associated with a Lipschitz initial condition. The purpose of this thesis is to compare two notions of weak solutions for this equation, namely the viscosity solution and the variational solution, that are known to coincide in convex Hamiltonian dynamics. In order to work in a relevant framework for both notions, we first need to build a variational solution without compactness assumption on the manifold or the Hamiltonian. To do so, we follow the historical construction, detailing properties of the generating family obtained via the broken geodesics method. Local estimates allow to prove that the viscosity solution can be obtained from the variational solution via an iterative process. We then check that this construction gives effectively the viscosity solution for a convex Hamiltonian, and characterize the integrable Hamiltonians for which this property persists by carefully studying elementary examples in dimension 1 and 2.
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Homogénéisation stochastique de quelques problèmes de propagations d'interfaces / Stochastic homogenization of some front propagation problemsHajej, Ahmed 01 July 2016 (has links)
Dans ce travail, on étudie l'homogénéisation de quelques problèmes de propagations de fronts dans des milieux stationnaires et ergodiques. Dans la première partie, on étudie l'homogénéisation stochastique de quelques problèmes de propagations de fronts non-locaux. En particulier, on donne une version non-locale de la méthode de la fonction test perturbée d'Evans. La deuxième partie est consacrée à l'approximation numérique du Hamiltonien effectif qui découle de l'homogénéisation stochastique des équations de Hamilton-Jacobi. On établit des estimations d'erreurs entre les solutions numériques et l'Hamiltonien effectif. Dans la troisième partie, on s'intéresse à l'homogénéisation stochastique de problèmes de propagations de fronts qui évoluent dans la direction normale avec une vitesse qui peut être non bornée. On montre des résultats d'homogénéisation dans le cas des milieux i.i.d. / In this work, we study the homogenization of some front propagation problems in stationary ergodic media. In the first part, we study the stochastic homogenization of non-local front propagation problems. In particular, we give a non-local variation of the perturbed test function method of Evans. The second part is devoted to numerical approximations of the effective Hamiltonian arising in stochastic homogenization of Hamilton-Jacobi equations. We establish error estimates between numerical solutions and the effective Hamiltonian. In the third part, we are interested in the stochastic homogenization of front propagation problems moving in the normal direction with possible unbounded velocity. Assuming that the media satisfies a finite range of dependence condition, we prove homogenization results.
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Analytical study of complex quantum trajectoriesChou, Chia-Chun 03 September 2009 (has links)
Quantum trajectories are investigated within the complex quantum Hamilton-Jacobi formalism. A unified description is presented for complex quantum trajectories for one-dimensional time-dependent and time-independent problems. Complex quantum trajectories are examined for the free Gaussian
wave packet, the coherent state in the harmonic potential, and the the barrier scattering problems. We analyze the variations of the complex-valued kinetic energy, the classical potential, and the quantum potential along the complex
quantum trajectories. For one-dimensional time-independent scattering problems, we demonstrate general properties and similar structures of the complex quantum trajectories and the quantum potentials. In addition, it is shown that a quantum vortex forms around a node in the wave function in complex space, and the quantized circulation integral originates from the discontinuity
in the real part of the complex action. Although the quantum momentum field displays hyperbolic flow around a node, the corresponding Polya vector field
displays circular flow. Moreover, local topologies of the quantum momentum function and the Polya vector field are thoroughly analyzed near a stagnation point or a pole (including circular, hyperbolic, and attractive or repulsive structures). The local structure of the quantum momentum function and the Polya vector field around a stagnation point are related to the first derivative of the quantum momentum function. However, the
magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Finally, quantum interference is investigated and it leads to the formation of the topological structure of quantum caves in space-time Argand plots. These caves consist of the vortical
and stagnation tubes originating from the isosurfaces of the amplitude of the wave function and its first derivative. Complex quantum trajectories display
helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. Moreover, the wrapping time for a specific trajectory is
determined by the divergence and vorticity of the quantum momentum field. The lifetime for interference features is determined by the rotational dynamics of the nodal line in the complex plane. Therefore, these results demonstrate that the complex quantum trajectory method provides a novel perspective for
analysis and interpretation of quantum phenomena. / text
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The optimal control of a Lévy processDiTanna, Anthony Santino 23 October 2009 (has links)
In this thesis we study the optimal stochastic control problem of the drift of a Lévy process. We show that, for a broad class of Lévy processes, the partial integro-differential Hamilton-Jacobi-Bellman equation for the value function admits classical solutions and that control policies exist in feedback form. We then explore the class of Lévy processes that satisfy the requirements of the theorem, and find connections between the uniform integrability requirement and the notions of the score function and Fisher information from information theory. Finally we present three different numerical implementations of the control problem: a traditional dynamic programming approach, and two iterative approaches, one based on a finite difference scheme and the other on the Fourier transform. / text
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