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 porte sur l'étude d'équation de Hamilton-Jacobi-Bellman associées à des problèmes de contrôle optimal et de jeux à champ moyen avec la particularité qu'on se place sur un réseau (c'est-à-dire, des ensembles constitués d'arêtes connectées par des jonctions) dans les deux problèmes, pour lesquels on autorise différentes dynamiques et différents coûts dans chaque bord d'un réseau. Dans la première partie de cette thèse, on considère un problème de contrôle optimal sur les réseaux dans l'esprit des travaux d'Achdou, Camilli, Cutrì & Tchou (2013) et Imbert, Moneau & Zidani (2013). La principale nouveauté est qu'on rajoute des coûts d'entrée (ou de sortie) aux sommets du réseau conduisant à une éventuelle discontinuité de la fonction valeur. Celle-ci est caractérisée comme l'unique solution de viscosité d'une équation Hamilton-Jacobi pour laquelle une condition de jonction adéquate est établie. L'unicité est une conséquence d'un principe de comparaison pour lequel nous donnons deux preuves différentes, l'une avec des arguments tirés de la théorie du contrôle optimal, inspirée par Achdou, Oudet & Tchou (2015) et l'autre basée sur les équations aux dérivées partielles, d'après Lions & Souganidis (2017). La deuxième partie concerne les jeux à champ moyen stochastiques sur les réseaux. Dans le cas ergodique, ils sont décrits par un système couplant une équation de Hamilton-Jacobi-Bellman et une équation de Fokker- Planck, dont les inconnues sont la densité m de la mesure invariante qui représente la distribution des joueurs, la fonction valeur v qui provient d'un problème de contrôle optimal "moyen" et la constante ergodique ρ. La fonction valeur v est continue et satisfait dans notre problème des conditions de Kirchhoff aux sommets très générales. La fonction m satisfait deux conditions de transmission aux sommets. En particulier, due à la généralité des conditions de Kirchhoff, m est en général discontinue aux sommets. L'existence et l'unicité d'une solution faible sont prouvées pour des Hamiltoniens sous-quadratiques et des hypothèses très générales sur le couplage. Enfin, dans la dernière partie, nous étudions les jeux à champ moyen stochastiques non stationnaires sur les réseaux. Les conditions de transition pour la fonction de valeur v et la densité m sont similaires à celles données dans la deuxième partie. Là aussi, nous prouvons l'existence et l'unicité d'une solution faible pour des Hamiltoniens sous-linéaires et des couplages et dans le cas d'un couplage non-local régularisant et borné inférieurement. La principale difficulté supplémentaire par rapport au cas stationnaire, qui nous impose des hypothèses plus restrictives, est d'établir la régularité des solutions du système posé sur un réseau. Notre approche consiste à étudier la solution de l'équation de Hamilton-Jacobi dérivée pour gagner de la régularité sur la solution de l'équation initiale. / The dissertation focuses on the study of Hamilton-Jacobi-Bellman equations associated with optimal control problems and mean field games problems in the case when the state space is a network. Different dynamics and running costs are allowed in each edge of the network. In the first part of this thesis, we consider an optimal control on networks in the spirit of the works of Achdou, Camilli, Cutrì & Tchou (2013) and Imbert, Monneau & Zidani (2013). The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi equation for which an adequate junction condition is established. The uniqueness is a consequence of a comparison principle for which we give two different proofs. One uses some arguments from the theory of optimal control and is inspired by Achdou, Oudet & Tchou (2015). The other one is based on partial differential equations techniques and is inspired by a recent work of Lions & Souganidis (2017). The second part is about stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton- Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the density m of the invariant measure which represents the distribution of the players, the value function v which comes from an "average" optimal control problem and the ergodic constant ρ. The function v is continuous and satisfies general Kirchhoff conditions at the vertices. The density m satisfies dual transmission conditions. In particular, due to the generality of Kirchhoff’s conditions, m is in general discontinuous at the vertices. Existence and uniqueness are proven for subquadratic Hamiltonian and very general assumptions about the coupling term. Finally, in the last part, we study non-stationary stochastic mean field games on networks. The transition conditions for value function v and the density m are similar to the ones given in second part. Here again, we prove the existence and uniqueness of a weak solution for sublinear Hamiltonian and bounded non-local regularizing coupling term. The main additional difficulty compared to the stationary case, which imposes us more restrictive hypotheses, is to establish the regularity of the solutions of the system placed on a network. Our approach is to study the solution of the derived Hamilton-Jacobi equation to gain regularity over the initial equation.

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 oscillante. / This 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.

Nesta dissertação de Mestrado será apresentada uma poderosa técnica numérica, conhecida como método Level Set, capaz de simular e analisar movimentos de curvas em diferentes cenários físicos. Tal método - formulado por Osher e Sethian [1] - está sedimentado na seguinte idéia: representar uma determinada curva (ou superfície) Γ como a curva de nível zero (zero level set) de uma função Φ de maior dimensão (denominada função Level Set). A equação diferencial do tipo Hamilton-Jacobi que descreve a evolução da função Level Set é discretizada através da utilização de acurados esquemas hiperbólicos e, como resultado de tal acurácia, obtém-se uma formulação numérica capaz de tratar eficazmente mudanças topológicas e/ou descontinuidades que, eventualmente, podem surgir no decorrer da propagação da curva (ou superfície) de nível zero. Em virtude da eficácia e versatilidade do método Level Set, esta técnica numérica está sendo amplamente aplicada à diversas áreas científicas, incluindo mecânica dos fluidos, processamento de imagens e visão computacional, crescimento de cristais, geometria computacional e ciência dos materiais. Particularmente, o propósito deste trabalho equivale ao estudo dos fundamentos do método Level Set e, por fim, visa-se aplicar tal modelo numérico à problemas existentes na área de crescimento de cristais. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988. / In this dissertation, we present a powerful numerical technique known as Level Set Method for computing and analyzing moving fronts in different physical settings. The method -formulated by Osher and Sethian [1] - is based on the following idea: a curve (or surface) is embedded as the zero level set of a higher-dimensional function Φ (called level set function). Then, we can link the evolution of this function Φ to the propagation of the curve itself through a time-dependent initial value problem. At any time, the curve is given by the zero level set of the time-dependent level set function Φ. The evolution of the level set function Φ is described by a Hamilton-Jacobi type partial differential equation, which can be discretised by the use of accurate methods for hyperbolic equations. As a result, the Level Set Method is able to track complex curves that can develop large spikes, sharp corners or change its topology as they evolve. Because of its versatility and efficacy, this numerical technique has found applications in a large number of areas, including fluid mechanics, image processing and computer vision, crystal growth, computational geometry and materials science. Particularly, the aim of this dissertation has been to understand the fundamentals of Level Set Method and its final goal is compute the motion of bondaries in crystal growth using this numerical model. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.

Nesta dissertação de Mestrado será apresentada uma poderosa técnica numérica, conhecida como método Level Set, capaz de simular e analisar movimentos de curvas em diferentes cenários físicos. Tal método - formulado por Osher e Sethian [1] - está sedimentado na seguinte idéia: representar uma determinada curva (ou superfície) Γ como a curva de nível zero (zero level set) de uma função Φ de maior dimensão (denominada função Level Set). A equação diferencial do tipo Hamilton-Jacobi que descreve a evolução da função Level Set é discretizada através da utilização de acurados esquemas hiperbólicos e, como resultado de tal acurácia, obtém-se uma formulação numérica capaz de tratar eficazmente mudanças topológicas e/ou descontinuidades que, eventualmente, podem surgir no decorrer da propagação da curva (ou superfície) de nível zero. Em virtude da eficácia e versatilidade do método Level Set, esta técnica numérica está sendo amplamente aplicada à diversas áreas científicas, incluindo mecânica dos fluidos, processamento de imagens e visão computacional, crescimento de cristais, geometria computacional e ciência dos materiais. Particularmente, o propósito deste trabalho equivale ao estudo dos fundamentos do método Level Set e, por fim, visa-se aplicar tal modelo numérico à problemas existentes na área de crescimento de cristais. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988. / In this dissertation, we present a powerful numerical technique known as Level Set Method for computing and analyzing moving fronts in different physical settings. The method -formulated by Osher and Sethian [1] - is based on the following idea: a curve (or surface) is embedded as the zero level set of a higher-dimensional function Φ (called level set function). Then, we can link the evolution of this function Φ to the propagation of the curve itself through a time-dependent initial value problem. At any time, the curve is given by the zero level set of the time-dependent level set function Φ. The evolution of the level set function Φ is described by a Hamilton-Jacobi type partial differential equation, which can be discretised by the use of accurate methods for hyperbolic equations. As a result, the Level Set Method is able to track complex curves that can develop large spikes, sharp corners or change its topology as they evolve. Because of its versatility and efficacy, this numerical technique has found applications in a large number of areas, including fluid mechanics, image processing and computer vision, crystal growth, computational geometry and materials science. Particularly, the aim of this dissertation has been to understand the fundamentals of Level Set Method and its final goal is compute the motion of bondaries in crystal growth using this numerical model. [1] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79:12, 1988.

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.

Positioned at the metaphysical divide between civilization and wilderness, this thesis investigates the potential for symbiotic relationships among cultural, ecological and industrial systems in an effort to suggest alternative modes for human sustainability. The City of Hamilton, where steel and iron industries continue to scar the landscape, serves as the location for a speculative design intervention. Amongst existing urban structures, a hybrid form of industrial production is proposed to acknowledge society’s reliance on artificial devices. In turn, this hybrid form is integrated with natural ecological processes to demonstrate humanity’s dependence on the natural world. The first chapter positions the thesis within a discourse regarding the boundary between civilization and wilderness and their conventional dichotomy. The thesis is aligned to themes of ecological-artificial hybridization, which include the scientific application of biological metaphors, economic and manufacturing theories of industrial ecology, and architectural and design methodology. Chapter two employs Complex Systems methodology to structure an analysis of Hamilton’s ‘intrinsic’ and ‘extrinsic’ systems. The city is considered within ecological, historical, cultural, industrial and economic contexts, at local and regional scales. Chapter three proposes an urban plan for Hamilton that seeks to regenerate and integrate ecological, cultural and industrial systems. Within the framework of this plan, industrial ecosystems can coexist with public function and ecological infrastructure in close proximity. Though designed for long term application, the plan is intended to provide context for a more detailed and immediate intervention within the scope of the thesis. Chapter four proposes the implementation of a speculative urban design, as a central component of the urban plan. Sited on the Stelco pier, one of the largest and oldest steel producers in Canada, the design would reclaim a pivotal historical and physical location along the Hamilton waterfront. Regeneration of the heavily contaminated industrial site will be initiated with a phased program of remediation and managed ecological succession. The new science of industrial ecology will inform this new development. This approach is based on a shift from ‘open loop’ systems, in which material and energy flows dissipate through processes of waste creation, towards ‘closed loop’ systems in which energy and material are recycled. A new Instrumental Matrix is proposed where decentralized cultural, ecological and industrial systems are interwoven to create diverse and sustainable habitats for wildlife, people and industry.

Positioned at the metaphysical divide between civilization and wilderness, this thesis investigates the potential for symbiotic relationships among cultural, ecological and industrial systems in an effort to suggest alternative modes for human sustainability. The City of Hamilton, where steel and iron industries continue to scar the landscape, serves as the location for a speculative design intervention. Amongst existing urban structures, a hybrid form of industrial production is proposed to acknowledge society’s reliance on artificial devices. In turn, this hybrid form is integrated with natural ecological processes to demonstrate humanity’s dependence on the natural world. The first chapter positions the thesis within a discourse regarding the boundary between civilization and wilderness and their conventional dichotomy. The thesis is aligned to themes of ecological-artificial hybridization, which include the scientific application of biological metaphors, economic and manufacturing theories of industrial ecology, and architectural and design methodology. Chapter two employs Complex Systems methodology to structure an analysis of Hamilton’s ‘intrinsic’ and ‘extrinsic’ systems. The city is considered within ecological, historical, cultural, industrial and economic contexts, at local and regional scales. Chapter three proposes an urban plan for Hamilton that seeks to regenerate and integrate ecological, cultural and industrial systems. Within the framework of this plan, industrial ecosystems can coexist with public function and ecological infrastructure in close proximity. Though designed for long term application, the plan is intended to provide context for a more detailed and immediate intervention within the scope of the thesis. Chapter four proposes the implementation of a speculative urban design, as a central component of the urban plan. Sited on the Stelco pier, one of the largest and oldest steel producers in Canada, the design would reclaim a pivotal historical and physical location along the Hamilton waterfront. Regeneration of the heavily contaminated industrial site will be initiated with a phased program of remediation and managed ecological succession. The new science of industrial ecology will inform this new development. This approach is based on a shift from ‘open loop’ systems, in which material and energy flows dissipate through processes of waste creation, towards ‘closed loop’ systems in which energy and material are recycled. A new Instrumental Matrix is proposed where decentralized cultural, ecological and industrial systems are interwoven to create diverse and sustainable habitats for wildlife, people and industry.

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.

Η μερική διαφορική εξίσωση Hamilton-Jacobi-Bellman παράγει τη λύση στο πρόβλημα του υπολογισμού της βέλτιστης ανάδρασης καταστάσεων σε μη γραμμικά δυναμικά συστήματα. Η προσπάθεια ανάπτυξης εύχρηστων και αξιόπιστων μεθόδων αριθμητικής ή προσεγγιστικής επίλυσης της εξίσωσης Hamilton-Jacobi-Bellman έχει τεράστια σημασία στη ρύθμιση διεργασιών γιατί μπορεί να οδηγήσει άμεσα σε εργαλεία σχεδιασμού μη γραμμικών ρυθμιστών. Ειδικότερα, στη ρύθμιση διεργασιών, η απόδοση ενός ρυθμιστικού συστήματος αξιολογείται βάσει ενός τετραγωνικού δείκτη απόδοσης σε άπειρο χρονικό ορίζοντα, και η βέλτιστη ανάδραση καταστάσεων μπορεί να υπολογισθεί μέσω της λύσης της εξίσωσης Hamilton-Jacobi-Bellman, μη εξαρτώμενης από το χρόνο. Στο πρόβλημα της επίλυσης της παραπάνω εξίσωσης παρουσιάζονται σοβαρές δυσκολίες, κυρίως λόγω υπολογιστικής πολυπλοκότητας. Για το λόγο αυτό, οι μέχρι στιγμής πρακτικές εφαρμογές υπήρξαν περιορισμένες. Στην παρούσα εργασία αναπτύσσεται υπολογιστική μέθοδος, βασισμένη στον αλγόριθμο επαναλήψεων Newton-Kantorovich, η οποία επιτυγχάνει πολυωνυμική προσέγγιση της λύσης της μερικής διαφορικής εξίσωσης Hamilton-Jacobi-Bellman υπό μορφή αναπτύγματος σε δυναμοσειρά Taylor. Με τον τρόπο αυτό επιταχύνονται σημαντικά οι υπολογισμοί για τον προσδιορισμό της βέλτιστης ανάδρασης καταστάσεων. Η μέθοδος εφαρμόζεται αρχικά σε ένα παράδειγμα ισοθερμοκρασιακού αντιδραστήρα συνεχούς λειτουργίας με ανάδευση, ο οποίος παρουσιάζει δυναμική συμπεριφορά μη-ελάχιστης φάσης, με μία είσοδο, μία έξοδο και δύο καταστάσεις. Στη συνέχεια, εφαρμόζεται σε παραδείγματα μη ισοθερμοκρασιακού αντιδραστήρα αντίστοιχης δυναμικής συμπεριφοράς, τριών καταστάσεων, πρωτίστως με μία είσοδο και μία έξοδο και κατόπιν με δύο εισόδους και δύο εξόδους. Με ανάπτυξη και εφαρμογή κώδικα MAPLE για κάθε μία περίπτωση χωριστά, υπολογίζονται προσεγγιστικά οι βέλτιστοι νόμοι ανάδρασης και σχεδιάζονται οι βέλτιστες αποκρίσεις των εισόδων και των εξόδων κάθε ενός από τα παραπάνω συστήματα, ενώ ταυτόχρονα γίνεται και καταγραφή των αντίστοιχων χρόνων εκτέλεσης κάθε κώδικα. Τέλος, στην περίπτωση του ισοθερμοκρασιακού αντιδραστήρα, γίνεται σύγκριση της προτεινόμενης μεθόδου με προϋπάρχουσες, κατά κύριο λόγο σε ζητήματα χρόνων εκτέλεσης, αλλά και σε ζητήματα απόδοσης στη ρύθμιση. / The partial differential equation Hamilton-Jacobi-Bellman produces the solution in the problem of calculation of optimal state feedback in non-linear dynamic systems. The effort of designing functional and reliable, numerical or approximate, methods for solving Hamilton-Jacobi-Bellman equation has enormous importance in process control because it can lead directly to tools of planning non-linear regulators. More specifically, in process control, the attribution of a regulating system is evaluated using a quadratic performance index in infinite time horizon, and the optimal state feedback can be calculated by the solution of the non time depended Hamilton-Jacobi-Bellman equation. The problem of solving the equation above encounters serious difficulties, mainly because of the calculation complexity. For this reason, the practical applications existed until now were very few. In the present work a calculating method is developed, based in the iterative algorithm Newton-Kantorovich, which achieves polynomial approach of the solution of partial differential equation Hamilton-Jacobi-Bellman under the form of Taylor series expansion. Thus the calculations for the determination of optimal state feedback are considerably accelerated. The method is initially applied in an example of continuous stirred tank reactor, with non-minimum phase dynamic behavior, with one input, one output and two state variables. Afterwards, it is applied in examples of not isothermal reactor of the same dynamic behavior, three state variables, firstly with one input and one output variables and then with two input and two output variables. Using the symbolic program MAPLE, a code was developed for each case separately, which calculates approximately the optimal feedback laws and designs the optimal responses of the inputs and outputs of each of the systems above, while the corresponding times of implementation of each code are simultaneously recording. Finally, in the case of isothermal reactor, a comparison is made between the proposed and preexisting methods, mainly in the base of the time of implementations and the regulation performance.