Cette thèse est consacrée au contrôle de quelques équations aux dérivées partielles non linéaires. On s’intéresse notamment à des systèmes paraboliques de réaction-diffusion non linéaires issus de la cinétique chimique. L’objectif principal est de démontrer des résultats de contrôlabilité locale ou globale, en temps petit, ou en temps grand.Dans une première partie, on démontre un résultat de contrôlabilité locale à des états stationnaires positifs en temps petit, pour un système de réaction-diffusion non linéaire.Dans une deuxième partie, on résout une question de contrôlabilité globale à zéro en temps petit pour un système 2 × 2 de réaction-diffusion non linéaire avec un couplage impair.La troisième partie est consacrée au célèbre problème ouvert d’Enrique Fernández-Cara et d’Enrique Zuazua des années 2000 concernant la contrôlabilité globale à zéro de l’équation de la chaleur faiblement non linéaire. On démontre un résultat de contrôlabilité globale à états positifs en temps petit et un résultat de contrôlabilité globale à zéro en temps long.La dernière partie, rédigée en collaboration avec Karine Beauchard et Armand Koenig, est une incursion vers l’hyperbolique. On étudie des systèmes linéaires à coefficients constants, couplant une dynamique transport avec une dynamique parabolique. On identifie leur temps minimal de contrôle et l’influence de leur structure algébrique sur leurs propriétés de contrôle. / This thesis is devoted to the control of nonlinear partial differential equations. We are mostly interested in nonlinear parabolic reaction-diffusion systems in reaction kinetics. Our main goal is to prove local or global controllability results in small time or in large time.In a first part, we prove a local controllability result to nonnegative stationary states in small time, for a nonlinear reaction-diffusion system.In a second part, we solve a question concerning the global null-controllability in small time for a 2 × 2 nonlinear reaction-diffusion system with an odd coupling term.The third part focuses on the famous open problem due to Enrique Fernndez-Cara and Enrique Zuazua in 2000, concerning the global null-controllability of the weak semi-linear heat equation. We show that the equation is globally nonnegative controllable in small time and globally null-controllable in large time.The last part, which is a joint work with Karine Beauchard and Armand Koenig, enters the hyperbolic world. We study linear parabolic-transport systems with constant coeffcients. We identify their minimal time of control and the influence of their algebraic structure on the controllability properties.

The thesis presents a new method of Symmetry Analysis of the Black-Scholes Merton Finance Model through modi ed Local one-parameter transformations. We determine the symmetries of both the one-dimensional and two-dimensional Black-Scholes equations through a method that involves the limit of in nitesimal ! as it approaches zero. The method is dealt with extensively in [23]. We further determine an invariant solution using one of the symmetries in each case. We determine the transformation of the Black-Scholes equation to heat equation through Lie equivalence transformations. Further applications where the method is successfully applied include working out symmetries of both a Gaussian type partial di erential equation and that of a di erential equation model of epidemiology of HIV and AIDS. We use the new method to determine the symmetries and calculate invariant solutions for operators providing them. / Mathematical Sciences / Applied Mathematics / D. Phil. (Applied Mathematics)

A Teoria das equações integrais, desde a segunda metade do século XX, tem assumido um papel cada vez maior no âmbito de problemas aplicados. Com isso, surge a necessidade do desenvolvimento de métodos numéricos cada vez mais eficazes para a resolução deste tipo de equação. Isso tem como consequência a possibilidade de resolução de uma gama cada vez maior de problemas. Nesse sentido, outros tipos de equações integrais estão sendo objeto de estudos, dentre elas as chamadas equações integro-diferenciais. O presente trabalho tem como objetivo o estudo das equações integro-diferenciais singulares lineares e não-lineares. Mais especificamente, no caso linear, apresentamos os principais resultados necessários para a obtenção de um método numérico e a formulação de suas propriedades de convergência. O caso não-linear é apresentado através de um modelo matemático para tubulações em um tipo específico de reator nuclear (LMFBR) no qual origina-se a equação integro-diferencial. A partir da equação integro-diferencial um modelo numérico é proposto com base nas condições físicas do problema / The theory of the integral equations, since the second half of the 20th century, has been assuming an ever more important role in the modelling of applied problems. Consequently, the development of new numerical methods for integral equations is called for and a larger range of problems has been possible to be solved by these new techniques. In this sense, many types of integral equations have been derived from applications and been the object of studies, among them the so called singular integro-differential equation. The present work has, as its main objective, the study of singular integrodifferential equations, both linear and non-linear. More specifically, in the linear case, we present our main results regarding the derivation of a numerical method and its uniform convergence properties. The non-linear case is introduced through the mathematical model of boiler tubes in a specific type of nuclear reactor (LMFBR) from which the integro-differential equation originates. For this integro-differential equation a numerical method is proposed based on the physical conditions of the problem

Les travaux réalisés dans cette thèse concernent l’étude de problèmes quasi-linéaires paraboliques et elliptiques singuliers. Par singularité, nous signifions que le problème fait intervenir une non linéarité qui explose au bord du domaine où l’équation est posée. La présence du terme singulier entraine un manque de régularité des solutions. Ce défaut de régularité génère en conséquence un manque de compacité qui ne permet pas d’appliquer directement les méthodes classiques d’analyse non linéaires pour démontrer l’existence de solutions et discuter les propriétés de régularité et de comportement asymptotique des solutions. Pour contourner cette difficulté dans le contexte des problèmes que nous avons étudiés, nous sommes amenés à établir des estimations a priori très fines au voisinage du bord en combinant diverses méthodes : méthodes de monotonie (reliées au principe du maximum), méthodes variationnelles, argument de convexité, méthodes d’interpolation dans les espaces de Sobolev, méthodes de point fixe. / In this thesis I have studied the Evolution p-laplacian equation with singular nonlinearity. We start by studying the corresponding elliptic problem and then by defining a proper cone in a suitable Sobolev space find the uniqueness of the solution. Taking that into account and using the semi discretization in time we arrive at the uniqueness and existence result. Next we prove some regularity theorem using tools from Nonlinear Semigroup theory and Interpolation spaces. We also establish some related result for the laplacian case where we improve our result on the existence and regularity, due to the non degeneracy of the laplacian. In another related work we work with a semilinear equation with singular nonlinearity and using the moving plane method prove the symmetry properties of any classical solution. We also give some related apriori estimates which together with the symmetry provide us the existence of solution using the bifurcation result.

The work in this thesis is inspired by the fabrication of Zn₄₅Au₃₀Cu₂₅. This is the first alloy undergoing ultra-reversible martensitic transformations and closely satisfying the cofactor conditions, particular conditions of geometric compatibility between phases, which were conjectured to influence reversibility. With the aim of better understanding reversibility, in this thesis we study the martensitic microstructures arising during thermal cycling in Zn₄₅Au₃₀Cu₂₅, which are complex and different in every phase transformation cycle. Our study is developed in the context of continuum mechanics and nonlinear elasticity, and we use tools from nonlinear analysis. The first aim of this thesis is to advance our understanding of conditions of geometric compatibility between phases. To this end, first, we further investigate cofactor conditions and introduce a physically-based metric to measure how closely these are satisfied in real materials. Secondly, we introduce further conditions of compatibility and show that these are nearly satisfied by some twins in Zn₄₅Au₃₀Cu₂₅. These might influence reversibility as they improve compatibility between high and low temperature phases. Martensitic phase transitions in Zn₄₅Au₃₀Cu₂₅ are a complex phenomenon, especially because the crystalline structure of the material changes from a cubic to a monoclinic symmetry, and hence the energy of the system has twelve wells. There exist infinitely many energy-minimising microstructures, limiting our understanding of the phenomenon as well as our ability to predict it. Therefore, the second aim of this thesis is to find criteria to select physically-relevant energy minimisers. We introduce two criteria or selection mechanisms. The first involves a moving mask approximation, which allows one to describe some experimental observations on the dynamics, while the second is based on using vanishing interface energy. The moving mask approximation reflects the idea of a moving curtain covering and uncovering microstructures during the phase transition, as appears to be the case for Zn₄₅Au₃₀Cu₂₅, and many other materials during thermally induced transformations. We show that the moving mask approximation can be framed in the context of a model for the dynamics of nonlinear elastic bodies. We prove that every macroscopic deformation gradient satisfying the moving mask approximation must be of the form 1 + a(x) ⊗ n(x), for a.e. x. With regards to vanishing interface energy, we consider a one-dimensional energy functional with three wells, which simplifies the physically relevant model for martensitic transformations, but at the same time highlights some key issues. Our energy functional admits infinitely many minimising gradient Young measures, representing energy-minimising microstructures. In order to select the physically relevant ones, we show that minimisers of a regularised energy, where the second derivatives are penalised, generate a unique minimising gradient Young measure as the perturbation vanishes. The results developed in this thesis are motivated by the study of Zn₄₅Au₃₀Cu₂₅, but their relevance is not limited to this material. The results on the cofactor conditions developed here can help for the understanding of new alloys undergoing ultra-reversible transformations, and as a guideline for the fabrication of future materials. Furthermore, the selection mechanisms studied in this work can be useful in selecting physically relevant microstructures not only in Zn₄₅Au₃₀Cu₂₅, but also in other materials undergoing martensitic transformations, and other phenomena where pattern formation is observed.

In this thesis we study Ax-Schanuel type inequalities for abstract differential equations. A motivating example is the exponential differential equation. The Ax-Schanuel theorem states positivity of a predimension defined on its solutions. The notion of a predimension was introduced by Hrushovski in his work from the 1990s where he uses an amalgamation-with-predimension technique to refute Zilber's Trichotomy Conjecture. In the differential setting one can carry out a similar construction with the predimension given by Ax-Schanuel. In this way one constructs a limit structure whose theory turns out to be precisely the first-order theory of the exponential differential equation (this analysis is due to Kirby (for semiabelian varieties) and Crampin, and it is based on Zilber's work on pseudo-exponentiation). One says in this case that the inequality is adequate. Thus, by an Ax-Schanuel type inequality we mean a predimension inequality for a differential equation. Our main question is to understand for which differential equations one can find an adequate predimension inequality. We show that this can be done for linear differential equations with constant coefficients by generalising the Ax-Schanuel theorem. Further, the question turns out to be closely related to the problem of recovering the differential structure in reducts of differentially closed fields where we keep the field structure (which is quite an interesting problem in its own right). So we explore that question and establish some criteria for recovering the derivation of the field. We also show (under some assumptions) that when the derivation is definable in a reduct then the latter cannot satisfy a non-trivial adequate predimension inequality. Another example of a predimension inequality is the analogue of Ax-Schanuel for the differential equation of the modular j-function due to Pila and Tsimerman. We carry out a Hrushovski construction with that predimension and give an axiomatisation of the first-order theory of the strong Fraïssé limit. It will be the theory of the differential equation of j under the assumption of adequacy of the predimension. We also show that if a similar predimension inequality (not necessarily adequate) is known for a differential equation then the fibres of the latter have interesting model theoretic properties such as strong minimality and geometric triviality. This, in particular, gives a new proof for a theorem of Freitag and Scanlon stating that the differential equation of j defines a trivial strongly minimal set.

It is needless to point out the necessity and the importance of the study of Pfaffian differential expressions and equations. While it is interesting to consider from the pure mathematical point of view, their applications in many branches of applied mathematics are well known. To mention a few, one may observe that they arise in connection with line integrals (example, determination of work). They provide a more rational formulation of the foundations of thermodynamics as 'developed by the Greek mathematician Caratheodory. They also arise in the problem of determining the orthogonal trajectories. In many branches of engineering and other physical sciences they appear with problems concerning partial differential equations.

This paper studies the endogenous evolution of investment behaviour under the various macroeconomic circumstances, which might be relatively constructed by free-float, fixed and target zone regimes as the economic stability policy. It applies the issues of stock price target zone policy to a simultaneous stochastic differential equation system. We construct the stochastic macro model which utilized the basic conception of Dornbusch [1976] with the different price adjustment mechanism. In addition, we intend to apply the topological method which used by Miller and Weller [1991] to analyze the general economic property from the non-recursive model. The main purpose of this paper is to discuss how the public’s expectation affects the dynamic loci of commodity and stock price when the public agents have the perfect or imperfect credibility. We utilize this model to investigate whether stock price target zone regime will have honeymoon effect or not, when the government announce to execute the stock price target zone policy in the various situations. Moreover, we discuss whether stock price target zone can simultaneously stabilize other variables in the different situations.

<p>Stochastic information, to be understood as &quot;information gained by the application of stochastic methods&quot;, is proposed as a tool in the assessment of changes in climate.</p> <p>This thesis aims at demonstrating that stochastic information can improve the consideration and reduction of uncertainty in the assessment of changes in climate. The thesis consists of three parts. In part one, an indicator is developed that allows the determination of the proximity to a critical threshold. In part two, the tolerable windows approach (TWA) is extended to a probabilistic TWA. In part three, an integrated assessment of changes in flooding probability due to climate change is conducted within the TWA.</p> <p>The thermohaline circulation (THC) is a circulation system in the North Atlantic, where the circulation may break down in a saddle-node bifurcation under the influence of climate change. Due to uncertainty in ocean models, it is currently very difficult to determine the distance of the THC to the bifurcation point. We propose a new indicator to determine the system's proximity to the bifurcation point by considering the THC as a stochastic system and using the information contained in the fluctuations of the circulation around the mean state. As the system is moved closer to the bifurcation point, the power spectrum of the overturning becomes &quot;redder&quot;, i.e. more energy is contained in the low frequencies. Since the spectral changes are a generic property of the saddle-node bifurcation, the method is not limited to the THC, but it could also be applicable to other systems, e.g. transitions in ecosystems. </p> <p>In part two, a probabilistic extension to the tolerable windows approach (TWA) is developed. In the TWA, the aim is to determine the complete set of emission strategies that are compatible with so-called guardrails. Guardrails are limits to impacts of climate change or to climate change itself. Therefore, the TWA determines the &quot;maneuvering space&quot; humanity has, if certain impacts of climate change are to be avoided. Due to uncertainty it is not possible to definitely exclude the impacts of climate change considered, but there will always be a certain probability of violating a guardrail. Therefore the TWA is extended to a probabilistic TWA that is able to consider &quot;probabilistic uncertainty&quot;, i.e. uncertainty that can be expressed as a probability distribution or uncertainty that arises through natural variability.</p> <p>As a first application, temperature guardrails are imposed, and the dependence of emission reduction strategies on probability distributions for climate sensitivities is investigated. The analysis suggests that it will be difficult to observe a temperature guardrail of 2&deg;C with high probabilities of actually meeting the target.</p> <p>In part three, an integrated assessment of changes in flooding probability due to climate change is conducted. A simple hydrological model is presented, as well as a downscaling scheme that allows the reconstruction of the spatio-temporal natural variability of temperature and precipitation. These are used to determine a probabilistic climate impact response function (CIRF), a function that allows the assessment of changes in probability of certain flood events under conditions of a changed climate. </p> <p>The assessment of changes in flooding probability is conducted in 83 major river basins. Not all floods can be considered: Events that either happen very fast, or affect only a very small area can not be considered, but large-scale flooding due to strong longer-lasting precipitation events can be considered. Finally, the probabilistic CIRFs obtained are used to determine emission corridors, where the guardrail is a limit to the fraction of world population that is affected by a predefined shift in probability of the 50-year flood event. This latter analysis has two main results. The uncertainty about regional changes in climate is still very high, and even small amounts of further climate change may lead to large changes in flooding probability in some river systems.</p> / <p>Stochastische Information, zu verstehen als &quot;Information, die durch die Anwendung stochastischer Methoden gewonnen wird&quot;, wird als Hilfsmittel in der Bewertung von Klima&auml;nderungen vorgeschlagen.</p> <p>Das Ziel dieser Doktorarbeit ist es, zu zeigen, dass stochastische Information die Ber&uuml;cksichtigung und Reduktion von Unsicherheit in der Bewertung des Klimawandels verbessern kann. Die Arbeit besteht aus drei Teilen. Im ersten Teil wird ein Indikator entwickelt, der die Bestimmung des Abstandes zu einem kritischen Grenzwert ermöglicht. Im zweiten Teil wird der &quot;tolerable windows approach&quot; (TWA) zu einem probabilistischen TWA erweitert. Im dritten Teil wird eine integrierte Absch&auml;tzung der Ver&auml;nderung von &Uuml;berflutungswahrscheinlichkeiten im Rahmen des TWA durchgeführt.</p> <p>Die thermohaline Zirkulation (THC) ist ein Zirkulationssystem im Nordatlantik, in dem die Zirkulation unter Einfluss des Klimawandels in einer Sattel-Knoten Bifurkation abreißen kann. Durch Unsicherheit in Ozeanmodellen ist es gegenw&auml;rtig kaum m&ouml;glich, den Abstand des Systems zum Bifurkationspunkt zu bestimmen. Wir schlagen einen neuen Indikator vor, der es erm&ouml;glicht, die N&auml;he des Systems zum Bifurkationspunkt zu bestimmen. Dabei wird die THC als stochastisches System angenommen, und die Informationen, die in den Fluktuationen der Zirkulation um den mittleren Zustand enthalten sind, ausgenutzt. Wenn das System auf den Bifurkationspunkt zubewegt wird, wird das Leistungsspektrum &quot;roter&quot;, d.h. die tiefen Frequenzen enthalten mehr Energie. Da diese spektralen Ver&auml;nderungen eine allgemeine Eigenschaft der Sattel-Knoten Bifurkation sind, ist die Methode nicht auf die THC beschränkt, sondern weitere Anwendungen k&ouml;nnten m&ouml;glich sein, beispielsweise zur Erkennung von &Uuml;berg&auml;ngen in &Ouml;kosystemen.</p> <p>Im zweiten Teil wird eine probabilistische Erweiterung des &quot;tolerable windows approach&quot; (TWA) entwickelt. Das Ziel des TWA ist die Bestimmung der Menge der Emissionsreduktionsstrategien, die mit sogenannten Leitplanken kompatibel sind. Diese Leitplanken sind Begrenzungen der Auswirkungen des Klimawandels, oder des Klimawandels selber. Der TWA bestimmt daher den Spielraum, den die Menschheit hat, wenn bestimmte Auswirkungen des Klimawandels vermieden werden sollen. Durch den Einfluss von Unsicherheit ist es aber nicht m&ouml;glich, die betrachteten Auswirkungen des Klimawandels mit Sicherheit auszuschlie&szlig;en, sondern es existiert eine gewisse Wahrscheinlichkeit, dass die Leitplanke verletzt wird. Der TWA wird daher zu einem probabilistischen TWA weiterentwickelt, der es erm&ouml;glicht, &quot;probabilistische Unsicherheit&quot;, also Unsicherheit, die durch eine Wahrscheinlichkeitsverteilung ausgedr&uuml;ckt werden kann, oder die durch den Einfluß von nat&uuml;rlicher Variabilit&auml;t entsteht, zu ber&uuml;cksichtigen.</p> <p>Als erste Anwendung werden Temperaturleitplanken betrachtet, und die Abh&auml;ngigkeit der Emissionsreduktionsstrategien von Wahrscheinlichkeitsverteilungen über die Klimasensitivit&auml;t wird bestimmt. Die Analyse ergibt, dass die Einhaltung einer Temperaturleitplanke von 2&deg;C sehr schwierig wird, wenn man hohe Wahrscheinlichkeiten des Einhaltens der Leitplanke fordert.</p> <p>Im dritten Teil wird eine integrierte Absch&auml;tzung der &Auml;nderungen von &Uuml;berflutungswahrscheinlichkeiten unter Einfluss des Klimawandels durchgef&uuml;hrt. Ein einfaches hydrologisches Modell wird vorgestellt, sowie ein Skalierungsansatz, der es erm&ouml;glicht, die raum-zeitliche nat&uuml;rliche Variabilit&auml;t von Temperatur und Niederschlag zu rekonstruieren. Diese werden zur Bestimmung einer probabilistischen Klimawirkungsfunktion genutzt, einer Funktion, die es erlaubt, die Ver&auml;nderungen der Wahrscheinlichkeit bestimmter &Uuml;berflutungsereignisse unter Einfluss von Klima&auml;nderungen abzusch&auml;tzen.</p> <p>Diese Untersuchung der Ver&auml;nderung von &Uuml;berflutungswahrscheinlichkeiten wird in 83 großen Flusseinzugsgebieten durchgef&uuml;hrt. Nicht alle Klassen von &Uuml;berflutungen k&ouml;nnen dabei ber&uuml;cksichtigt werden: Ereignisse, die entweder sehr schnell vonstatten gehen, oder die nur ein kleines Gebiet betreffen, k&ouml;nnen nicht ber&uuml;cksichtigt werden, aber großfl&auml;chige &Uuml;berflutungen, die durch starke, langanhaltende Regenf&auml;lle hervorgerufen werden, k&ouml;nnen ber&uuml;cksichtigt werden. Zuguterletzt werden die bestimmten Klimawirkungsfunktion dazu genutzt, Emissionskorridore zu bestimmen, bei denen die Leitplanken Begrenzungen des Bev&ouml;lkerungsanteils, der von einer bestimmten Ver&auml;nderung der Wahrscheinlichkeit eines 50-Jahres-Flutereignisses betroffen ist, sind. Letztere Untersuchung hat zwei Hauptergebnisse. Die Unsicherheit von regionalen Klima&auml;nderungen ist immer noch sehr hoch, und au&szlig;erdem k&ouml;nnen in einigen Flusssystemen schon kleine Klima&auml;nderungen zu gro&szlig;en &Auml;nderungen der &Uuml;berflutungswahrscheinlichkeit f&uuml;hren.</p>

In almost all works in the literature there are several results showing asymptotic relationships between the solutions of x&prime / &prime / = f (t, x) (0.1) and the solutions 1 and t of x&prime / &prime / = 0. More specifically, the existence of a solution of (0.1) asymptotic to x(t) = at + b, a, b &isin / R has been obtained. In this thesis we investigate in a systematic way the asymptotic behavior as t &rarr / &infin / of solutions of a class of differential equations of the form (p(t)x&prime / )&prime / + q(t)x = f (t, x), t &ge / t_0 (0.2) and (p(t)x&prime / )&prime / + q(t)x = g(t, x, x&prime / ), t &ge / t_0 (0.3) by the help of principal u(t) and nonprincipal v(t) solutions of the corresponding homogeneous equation (p(t)x&prime / )&prime / + q(t)x = 0, t &ge / t_0. (0.4) Here, t_0 &ge / 0 is a real number, p &isin / C([t_0,&infin / ), (0,&infin / )), q &isin / C([t_0,&infin / ),R), f &isin / C([t_0,&infin / ) &times / R,R) and g &isin / C([t0,&infin / ) &times / R &times / R,R). Our argument is based on the idea of writing the solution of x&prime / &prime / = 0 in terms of principal and nonprincipal solutions as x(t) = av(t) + bu(t), where v(t) = t and u(t) = 1. In the proofs, Banach and Schauder&rsquo / s fixed point theorems are used. The compactness of the operator is obtained by employing the compactness criteria of Riesz and Avramescu. The thesis consists of three chapters. Chapter 1 is introductory and provides statement of the problem, literature review, and basic definitions and theorems. In Chapter 2 first we deal with some asymptotic relationships between the solutions of (0.2) and the principal u(t) and nonprincipal v(t) solutions of (0.4). Then we present existence of a monotone positive solution of (0.3) with prescribed asimptotic behavior. In Chapter 3 we introduce the existence of solution of a singular boundary value problem to the Equation (0.2).