The purpose of this study was to question the frequent use of field dependenceindependence (FDI) measures in past cross-cultural learning styles research, to illustrate generalizations this research has made about minority groups, and to examine applications of cross-cultural learning styles studies to composition. In the Part I of the study, 101 undergraduates (37 African-American, 47 Anglo-American, and 17 Hispanic) completed Witkin's Group Embedded Figures Test (GEFT) (designed to measure FDI) and Kolb's Learning Style Inventory (LSI) (an instrument never used in cross-cultural learning styles research). An ANOVA determined a significant difference among mean GEFT scores for ethnicity and gender, results consistent with past research. Results of the LSI, which introduced a new definition of learning styles, however, produced no significant differences among cultural groups. In Part II of this study, six students (one female and one male from each cultural group) volunteered for case studies. The test results and case studies illustrate that past FDI studies have made inaccurate generalizations of minority groups and have ignored individual differences within cultural groups. Further use of Kolb's LSI in future research and in teaching composition is explored. / Department of Anthropology

We propose multigrid methods for solving Hamilton-Jacobi-Bellman (HJB) and Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations. The methods are based on the full approximation scheme. We propose a damped-relaxation method as smoother for multigrid. In contrast with policy iteration, the relaxation scheme is convergent for both HJB and HJBI equations. We show by local Fourier analysis that the damped-relaxation smoother effectively reduces high frequency error. For problems where the control has jumps, restriction and interpolation methods are devised to capture the jump on the coarse grid as well as during coarse grid correction. We will demonstrate the effectiveness of the proposed multigrid methods for solving HJB and HJBI equations arising from option pricing as well as problems where policy iteration does not converge or converges slowly.

[pt] Os transtornos de ansiedade e humor são disfunções emocionais freqüentes que muito repercutem na qualidade de vida humana. Testes psicométricos quantificam esses transtornos e orientam condutas clínicas. A aplicação de ferramentas e conceitos da metrologia pode contribuir para a garantia de confiabilidade dos instrumentos de medição psicométricos. Os objetivos do trabalho são a elaboração de uma proposta de harmonização entre termos utilizados em psicometria e as definições universais para terminologia de medição segundo o Vocabulário Internacional de Termos Fundamentais e Gerais de Metrologia (VIM); e a avaliação da confiabilidade metrológica dos inventários de Beck e Hamilton para quantificação de ansiedade e transtorno de humor. Foram realizados: (i) estudos bibliográficos dos termos da metrologia e psicometria para elaboração da proposta de harmonização; (ii) aplicação dos inventários de Beck (BAI e BDI) e Hamilton (HAM-A e HAM-D) a 100 pacientes (iii) avaliação de validade, de convergência de quantificação de transtornos e cálculo da incerteza de medição entre inventários de Beck e Hamilton; (iv) desenvolvimento e implementação da Análise de Seleção de Parâmetros dos Itens (ASPI), abordagem elaborada para identificação da capacidade de cada item de um inventário, em discernir entre os diferentes níveis de acometimento do constructo. Todos os inventários quantificam de modo equivalente somente os pacientes graves. BAI e BDI apresentaram os maiores percentuais de itens capazes de discernimento entre níveis de acometimento. A boa correspondência obtida entre as abordagens ASPI quantitativa e ASPI qualitativa caracterizou a capacidade preditiva dos indicadores quantitativos propostos na identificação de desempenho de item. / [en] The anxiety disorders and humor are very frequent dysfunctions emotional impact on the quality of human life. Psicométricos quantify these tests and clinical disorders guide ducts. The application of tools and concepts of metrology can contribute to ensuring reliability of psicométricos measurement instruments. Job objectives are the development of a proposal for harmonization between terms used in psicometria and universal definitions for measuring terminology in accordance with the international vocabulary of basic and general terms in Metrology (VIM); and the assessment of the reliability of Beck inventories metrology and Hamilton to quantification of anxiety and mood disorder. Were made: (I) studies bibliographical terms of metrology and psicometria for the establishment of the proposed harmonization; (ii) implementation of inventories of Beck (BAI and BDI) and Hamilton (ham and HAM-d) 100 patients (iii) assessment of validity, quantification of convergence and disorders of the measurement uncertainty calculation between inventories of Beck and Hamilton; (iv) development and implementation of the analysis of selecting parameters of items (ASPI) approach to identification of the capacity of each item of an inventory to discern between different levels of acometimento of construction. All inventories quantify equivalent only serious patients. BAI and BDI submitted the largest percentage of items capable of discernment among acometimento levels. Good match obtained between the approaches quantitative and qualitative ASPI ASPI characterised the predictive capacity of quantitative indicators proposed in item performance identification.

A reação de Fenton é o nome dado à oxidação de ferro(II) a ferro(III) pela água oxigenada, uma reação que produz espécies com alto poder oxidante como o radical hidroxila. Neste trabalho, foi desenvolvida uma metodologia espectrofotométrica para o acompanhamento da formação de ferro(III) nos momentos iniciais da reação de Fenton. Esta metodologia foi aplicada a quatro conjuntos de reações: (A) o sistema Fenton simples, contendo apenas ferro(II) e H2O2; (B) o sistema A contendo isopropanol, um substrato orgânico simples que sofre principalmente oxidação a acetona; (C) o sistema A contendo o catalisador 3,5-di-terc-butil-catecol (H2DTBCat); (D) o sistema C mais isopropanol, que corresponde ao sistema catalítico completo. Em cada conjunto, variou-se as concentrações de ferro(II) e H2O2. Um modelo cinético, baseado num conjunto de reações explícitas e as respectivas constantes de velocidade, foi desenvolvido para simular a velocidade de formação de ferro(III) para estes quatro conjuntos de reações. Utilizando reações relatadas na literatura, o modelo forneceu simulações que reproduziram satisfatoriamente os dados experimentais dos conjuntos A e B. No caso dos conjuntos C e D, porém, foi necessário propor uma etapa envolvendo a formação de ferro(IV) ou ferril, estabilizado por complexação com o H2DTBCat. Entretanto, mesmo com a inclusão desta espécie, o modelo não captou a complexidade do sistema em altas concentrações de peróxido e ferro. Esta falha foi atribuída à rápida degradação competitiva do H2DTBCat nestas condições, com a subseqüente participação destes produtos de degradação na reação ou como co-catalisadores ou como inibidores. / The Fenton reaction is the name given to the oxidation of iron(II) to iron(III) by hydrogen peroxide, a reaction that produces highly oxidizing species like the hydroxyl radical. In this work, a spectrophotometric methodology is developed to accompany the formation of iron(III) during the initial moments of the Fenton reaction. This methodology was applied to four sets of reactions: (A) the simple Fenton system, containing only iron(II) and H2O2; (B) system A containing isopropanol, a simple substrate that undergoes principally oxidation to acetone; (C) system A containing the catalyst 3,5-di-tert-butyl-catechol (H2DTBCat); (D) system C plus isopropanol, corresponding to the complete catalytic system. For each set of reactions, the concentrations of iron(II) and H2O2 were varied. A kinetic model, based on explicit chemical reactions and their respective rate constants, was developed to simulate the the rate of formation of iron(III) for these four sets of reactions above. Using reactions described in the literature, the model produced simulations that satisfactorily reproduced the experimental data of sets A and B. In the case of sets C and D, however, it was necessary to propose an additional step involving the formation of iron(IV) or ferril, stabilized by complexation with H2DTBCat. Nonetheless, even with the inclusion of this species, the model failed to capture the complexity of the system at high concentrations of peroxide and iron. This failure was attributed to the rapid competitive degradation of H2DTBCat under these conditions, with the subsequent participation of the degradation products in the reaction as either co-catalysts or inhibitors.

Cette thèse est consacrée à l'étude de phénomènes de propagation dans des modèles d’EDP venant de la biologie. On étudie des équations cinétiques inspirées par le déplacement de colonies de bactéries ainsi que des équations de réaction-diffusion importantes en écologie afin de reproduire plusieurs phénomènes de dynamique et d'évolution des populations. La première partie étudie des phénomènes de propagation pour des équations cinétiques. Nous étudions l'existence et la stabilité d'ondes progressives pour des modèles ou la dispersion est donnée par un opérateur hyperbolique et non par une diffusion. Cela fait entrer en jeu un ensemble de vitesses admissibles, et selon cet ensemble, divers résultats sont obtenus. Dans le cas d'un ensemble de vitesses borné, nous construisons des fronts qui se propagent à une vitesse déterminée par une relation de dispersion. Dans le cas d'un ensemble de vitesses non borné, on prouve un phénomène de propagation accélérée dont on précise la loi d'échelle. On adapte ensuite à des équations cinétiques une méthode basée sur les équations de Hamilton-Jacobi pour décrire des phénomènes de propagation. On montre alors comment déterminer un Hamiltonien effectif à partir de l'équation cinétique initiale, et prouvons des théorèmes de convergence.La seconde partie concerne l'étude de modèles de populations structurées en espace et en phénotype. Ces modèles sont importants pour comprendre l'interaction entre invasion et évolution. On y construit d'abord des ondes progressives que l'on étudie qualitativement pour montrer l'impact de la variabilité phénotypique sur la vitesse et la distribution des phénotypes à l'avant du front. On met aussi en place le formalisme Hamilton-Jacobi pour l'étude de la propagation dans ces équations de réaction-diffusion non locales.Deux annexes complètent le travail, l'une étant un travail en cours sur la dispersion cinétique en domaine non-borné, l'autre étant plus numérique et illustre l’introduction. / This thesis is devoted to the study of propagation phenomena in PDE models arising from biology. We study kinetic equations coming from the modeling of the movement of colonies of bacteria, but also reaction-diffusion equations which are of great interest in ecology to reproduce several features of dynamics and evolution of populations. The first part studies propagation phenomena for kinetic equations. We study existence and stability of travelling wave solutions for models where the dispersal part is given by an hyperbolic operator rather than by a diffusion. A set of admissible velocities comes into the game and we obtain various types of results depending on this set. In the case of a bounded set of velocities, we construct travelling fronts that propagate according to a speed given by a dispersion relation. When the velocity set is unbounded, we prove an accelerating propagation phenomena, for which we give the spreading rate. Then, we adapt to kinetic equations the Hamilton-Jacobi approach to front propagation. We show how to derive an effective Hamiltonian from the original kinetic equation, and prove some convergence results.The second part is devoted to studying models for populations structured by space and phenotypical trait. These models are important to understand interactions between invasion and evolution. We first construct travelling waves that we study qualitatively to show the influence of the genetical variability on the speed and the distribution of phenotypes at the edge of the front. We also perform the Hamilton-Jacobi approach for these non-local reaction-diffusion equations.Two appendices complete this work, one deals with the study of kinetic dispersal in unbounded domains, the other one being numerical aspects of competition models.

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2016-12-15T15:01:25Z No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-12-15T17:28:21Z (GMT) No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-12-15T17:28:21Z (GMT). No. of bitstreams: 2 Dissertação - Davi José dos Santos - 2016.pdf: 5567090 bytes, checksum: 5606aa47f640cc5cd4495d2694f38cda (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-08-30 / This research with theoretical approach seeks to investigate inmathematics, octonions,which is a non-associative extension of the quaternions. Its algebra division 8-dimensional formed on the real numbers is more extensive than can be obtained by constructing Cayley-Dickson. In this perspective we have as main goal to answer the following question: "What number systems allow arithmetic operations addition, subtraction, multiplication and division? " In the genesis of octonions is the Irish mathematician William Rowan Hamilton, motivated by a deep belief that quaternions could revolutionize mathematics and physics, was the pioneer of a new theory that transformed the modern world. Today, it is confirmed that the complexs/quaternions/octonions and its applications are manifested in different branches of science such as mechanics, geometry, mathematical physics, with great relevance in 3D animation and robotics. In order to investigate the importance of this issue and make a small contribution, we make an introduction to the theme from the numbers complex and present the rationale and motivations of Hamilton in the discovery of quaternions/octonions. Wemake a presentation of the algebraic structure and its fundamental properties. Then discoremos about constructing Cayley-Dickson algebras that produces a sequence over the field of real numbers, each with twice the previous size. Algebras produced by this process are known as Cayley-Dickson algebras; since they are an extension of complex numbers, that is, hypercomplex numbers. All these concepts have norm, algebra and conjugate. The general idea is that the multiplication of an element and its conjugate should be the square of its norm. The surprise is that, in addition to larger, the following algebra loses some specific algebraic property. Finally, we describe and analyze certain symmetry groups with multiple representations through matrixes and applications to show that This content has a value in the evolution of technology. / Esta pesquisa com abordagem teórica busca investigar na matemática, os octônios, que é uma extensão não-associativa dos quatérnios. Sua álgebra com divisão formada de 8 dimensões sobre os números reais é a mais extensa que pode ser obtida através da construção de Cayley-Dickson. Nessa perspectiva temos comometa principal responder a seguinte questão: "Que sistemas numéricos permitemas operações aritméticas de adição, subtração, multiplicação e divisão?" Na gênese dos octônios está o matemático irlandêsWilliam Rowan Hamilton que, motivado por uma profunda convicção de que os quatérnios poderiam revolucionar a Matemática e a Física, foi o pioneiro de uma nova teoria que transformou o mundo moderno. Hoje, confirma-se que os complexos/quatérnios/octônios e suas aplicações se manifestam em diferentes ramos da ciências tais como a mecânica, a geometria, a física matemática, com grande relevância na animação 3D e na robótica. Com o propósito de investigar a importância deste tema e dar uma pequena contribuição, fazemos uma introdução ao tema desde os números complexos e apresentamos o raciocínio e motivações de Hamilton na descoberta dos quatérnios/octônios. Fazemos uma apresentação da estrutura algébrica, bem como as suas propriedades fundamentais. Emseguida discoremos sobre a construção de Cayley-Dickson que produz uma sequência de álgebras sobre o campo de números reais, cada uma com o dobro do tamanho anterior. Álgebras produzidas por este processo são conhecidas como álgebras Cayley-Dickson; uma vez que elas são uma extensão dos números complexos, isto é, os números hipercomplexos. Todos esses conceitos têm norma, álgebra e conjugado. A idéia geral é que o produto de um elemento e seu conjugado deve ser o quadrado de sua norma. A surpresa é que, além de maior dimensão, a álgebra seguinte perde alguma propriedade álgebrica específica. Por fim, descrevemos e analisamos alguns grupos de simetria, com várias representações através de matrizes e aplicações que demonstram que este conteúdo tem uma utilidade na evolução da tecnologia.

Esta dissertação aborda a teoria dos jogos diferenciais em sua estreita relação com a teoria das equações de Hamilton-Jacobi (HJ). Inicialmente, uma revisão da noção de solução em teoria dos jogos é empreendida. Discutem-se nesta ocasião as idéias de equilíbrio de Nash e alguns de seus refinamentos. Em seguida, tem lugar uma introdução à teoria dos jogos diferenciais, onde noções de solução como a função de valor de Isaacs e de Friedman são discutidas. É nesta altura do trabalho que fica evidente a conexão entre este conceito de solução e a teoria das equações de Hamilton-Jacobi. Por ocasião desta conexão, é explorada a noção de solução clássica e é exposta uma demonstração do fato de que se um jogo diferencial possuir uma função de valor pelo menos continuamente diferenciável, esta será uma solução da equação de Hamilton-Jacobi associada ao jogo. Este resultado faz uso do princípio da programação dinâmica, devido a Bellman, e cuja demonstração está presente no texto. No entanto, quando a função de valor do jogo é apenas contínua, então embora esta não seja uma solução clássica da equação HJ associada a jogo, vemos que ela será uma solução viscosa, ou solução no sentido da viscosidade - e a esta altura são discutidos os elementos e propriedades desta classe de soluções, um teorema de existência e unicidade e alguns exemplos. Por fim, retomamos o estudo dos jogos diferenciais à luz das soluções viscosas da equação de Hamilton-Jacobi e, assim, expomos uma demonstração de existência da função de valor e do princípio da programação dinâmica a partir das noções da viscosidade / This dissertation aims to address the topic of Differential Game Theory in its connection with the Hamilton-Jacobi (HJ) equations framework. Firstly we introduce the idea of solution for a game, through the discussion of Nash equilibria and its refinements. Secondly, the solution concept is then translated to the context of Differential Games and the idea of value function is introduced in its Isaacs\'s as well as Friedman\'s version. As the value function is discussed, its relationship with the Hamilton-Jacobi equations theory becomes self-evident. Due to such relation, we investigate the HJ equation from two distinct points of view. First of all, we discuss a statement according to which if a differential game has a continuously differentiable value function, then such function is a classical solution of the HJ equation associated to the game. This result strongly relies on Bellman\'s Dynamic Programming Principle - and this is the reason why we devote an entire chapter to this theme. Furthermore, HJ is still at our sight from the PDE point of view. Our motivation is simple: under some lack of regularity - a value function which is continuous, but not continuously differentiable - a game may still have a value function represented as a solution of the associated HJ equation. In this case such a solution will be called a solution in the viscosity sense. We then discuss the properties of viscosity solutions as well as provide an existence and uniqueness theorem. Finally we turn our attention back to the theory of games and - through the notion of viscosity - establish the existence and uniqueness of value functions for a differential game within viscosity solution theory.

Dans les installations nucléaires, les explosions, qu’elles soient d’origine interne ou externe, peuvent entrainer la rupture du confinement et le rejet de matières radioactives dans l’environnement. Il est donc fondamental, dans un cadre de sûreté de modéliser ce phénomène. L’objectif de cette thèse est de contribuer à l’élaboration de schémas numériques performants pour résoudre ces modèles complexes. Les travaux présentés s’articule autour de deux axes majeurs : le développement de schémas volumes finis consistants pour les équations d’Euler compressible qui modélise les ondes de choc et celui de schémas performants pour la propagation d’interfaces comme le front de flamme lors d'une déflagration. La discrétisation spatiale est de type mailles décalées pour tous les schémas développés. Les schémas pour les équations d'Euler se basent sur une formulation en énergie interne qui permet de préserver sa positivité ainsi que celle de la masse volumique. Un bilan d'énergie cinétique discret peut être obtenu et permet de retrouver un bilan d'énergie totale par l'ajout d'un terme de correction dans le bilan d'énergie interne. Le schéma ainsi construit est consistant au sens de Lax avec les solutions faibles entropiques des équations continues. On utilise les propriétés des équations de type Hamilton-Jacobi pour construire une classe de schémas volumes finis performants sur une large variété de maillages modélisant la propagation du front de flamme. Ces schémas garantissent un principe du maximum et possèdent des propriétés importantes de monotonie et consistance qui permettent d'obtenir un résultat de convergence. / In nuclear facilities, internal or external explosions can cause confinement breaches and radioactive materials release in the environment. Hence, modeling such phenomena is crucial for safety matters. The purpose of this thesis is to contribute to the creation of efficient numerical schemes to solve these complex models. The work presented here focuses on two major aspects: first, the development of consistent schemes for the Euler equations which model the blast waves, then the buildup of reliable schemes for the front propagation, like the flame front during the deflagration phenomenon. Staggered discretization is used in space for all the schemes. It is based on the internal energy formulation of the Euler system, which insures its positivity and the positivity of the density. A discrete kinetic energy balance is derived from the scheme and a source term is added in the discrete internal energy balance equation to preserve the exact total energy balance. High order, MUSCL-like interpolators are used in the discrete momentum operators. The resulting scheme is consistent (in the sense of Lax) with the weak entropic solutions of the continuous problem. We use the properties of Hamilton-Jacobi equations to build a class of finite volume schemes compatible with a large number of meshes to model the flame front propagation. These schemes satisfy a maximum principle and have important consistency and monotonicity properties. These latters allows to derive a convergence result for the schemes based on Cartesian grids.

L’implémentation d’un agent structurant impliquant un macrocycle à 31membres et intégrant un récepteur de type Hamilton / bis(2,6-diamidopyridine) a permis deconcevoir des [2]rotaxanes via une réaction click catalysée par du cuivre(I), soit à partir d’unpseudorotaxane en présence d’un barbiturique fonctionnalisé, soit par une méthode de« gabarit actif ». Ces structures supramoléculaires ont été rendues photochimiquement activesen exploitant des analogues de récepteurs Hamilton incorporant des groupementsphotodimérisables de type 9-anthracène. La photodimérisation et la retrodimérisation de cesrécepteurs en présence d’un fil barbiturique comportant des groupements terminauxencombrants (bouchons) permettent l’assemblage et le désassemblage de rotaxanes via unprocessus de photocapture. Ces unités 9-anthracène jouent également le rôle de bouchonsphotoactifs dans la formation d’un [2]rotaxane composé d’un plus petit anneau de typedibenzo-24-couronne-8, dont la photoirradiation résulte en une interconversion topologiqueinhabituelle entre un rotaxane et un caténane. Deux approches photochimiques ont permis deréguler des machines moléculaires distantes par une communication chimiqueintermoléculaire, c’est à dire soit par un transfert de molécule photoguidée, soit par untransfert d’électron photoinduit exalté par la présence d’un transfert d’énergie électroniqueréversible. / A templating motif involving a 31-member macrocycle integrating a bis(2,6-diamidopyridine) / Hamilton-type receptor aided [2]rotaxane sythesis, via a copper(I)catalyzed Huisgen reaction, in the presence of a designer barbiturate or by functionalizationusing an active template synthesis. Homologous supramolecular structures were madephotochemically-active, harnessing Hamilton receptors incorporating photodimerizable 9-anthracene groups. Photodimerization and retrodimerisation of these receptors in the presence of a barbiturate thread bearing terminal bulky stopper groups permitted rotaxane assembly /disassembly. The 9-anthracene units serve as stoppers in the formation of a [2]rotaxanecompound comprising a smaller dibenzo-24-crown-8 ring. Photoirradiation of these photoactive stoppers results in an unusual all-optical topological rotaxane - catenaneinterconversion. Two approaches to photoregulate remote molecular machines byintermolecular chemical communication involving a photoguided molecule or by photoinduced electron transfer aided by reversible electronic energy transfer are considered.

Cette thèse se développe sur deux fronts: nous nous concentrons sur les méthodes numériques des problèmes de contrôle optimal, en particulier sur le Principe de la Programmation Dynamique et sur le Model Predictive Control (MPC) et nous présentons des applications de techniques de contrôle en biologie. Dans la première partie, nous considérons l'approximation d'un problème de contrôle optimal avec horizon infini, qui combine une première étape, basée sur MPC permettant d'obtenir rapidement une bonne approximation de la trajectoire optimal, et une seconde étape, dans la quelle l¿équation de Bellman est résolue dans un voisinage de la trajectoire de référence. De cette façon, on peux réduire une grande partie de la taille du domaine dans lequel on résout l¿équation de Bellman et diminuer la complexité du calcul. Le deuxième sujet est le contrôle des méthodes Level Set: on considère un problème de contrôle optimal, dans lequel la dynamique est donnée par la propagation d'un graphe à une dimension, contrôlé par la vitesse normale. Un état finale est fixé, l'objectif étant de le rejoindre en minimisant une fonction coût appropriée. On utilise la programmation dynamique grâce à une réduction d'ordre de l'équation utilisant la Proper Orthogonal Decomposition. La deuxième partie est dédiée à l'application des méthodes de contrôle en biologie. On présente un modèle décrit par une équation aux dérivées partielles qui modélise l'évolution d'une population de cellules tumorales. On analyse les caractéristiques du modèle et on formule et résout numériquement un problème de contrôle optimal concernant ce modèle, où le contrôle représente la quantité du médicament administrée. / This thesis is divided in two parts: in the first part we focus on numerical methods for optimal control problems, in particular on the Dynamic Programming Principle and on Model Predictive Control (MPC), in the second part we present some applications of the control techniques in biology. In the first part of the thesis, we consider the approximation of an optimal control problem with an infinite horizon, which combines a first step based on MPC, to obtain a fast but rough approximation of the optimal trajectory and a second step where we solve the Bellman equation in a neighborhood of the reference trajectory. In this way, we can reduce the size of the domain in which the Bellman equation can be solved and so the computational complexity is reduced as well. The second topic of this thesis is the control of the Level Set methods: we consider an optimal control, in which the dynamics is given by the propagation of a one dimensional graph, which is controlled by the normal velocity. A final state is fixed and the aim is to reach the trajectory chosen as a target minimizing an appropriate cost functional. To apply the Dynamic Programming approach we firstly reduce the size of the system using the Proper Orthogonal Decomposition. The second part of the thesis is devoted to the application of control methods in biology. We present a model described by a partial differential equation that models the evolution of a population of tumor cells. We analyze the mathematical and biological features of the model. Then we formulate an optimal control problem for this model and we solve it numerically.