Dissertations / Theses on the topic 'Hybrid differential equations'

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed.<br>Ph. D.

Cette thèse a pour sujet la stabilisation de systèmes non-linéaires avec des mesures partielles et des entrées contraintes. Les deux premiers chapitres traitent du problème des entrées saturées dans le contexte des systèmes de dimension infinie pour des équations nonlinéaires abstraites et une équation aux dérivées partielles nonlinéaire particulière, l'équation de Korteweg-de Vries. Les outils mathématiques utilisés pour obtenir des résultats Le troisième chapitre propose une méthode de synthèse de retour de sortie pour deux équations de Korteweg-de Vries. Le quatrième chapitre concerne la synthèse d'un retour de sortie pour des systèmes non-linéaires de dimension finie pour lequel il existe un contrôle hybride. Une stratégie basée sur des observateurs grand gain est utilisée<br>This thesis is about the stabilization of nonlinear systems with partial measurements and constrained input. The two first chapters deals with saturated inputs in the contex of infinite-dimensional systems for nonlinear abstract equations and for a particular partial differential equation, the Korteweg-de Vries equation. The third chapter provides an output feedback design for two Korteweg-de Vries equations using the backstepping method. The fourth chapter is about the output feedback design of nonlinear finite-dimensional systems for which there exists a hybrid controller. A high-gain observer strategy is used

Diese Arbeit widmet sich der Simulation von elektrischen/elektronischen Schaltungen welche um elektromagnetische Bauelemente erweitert werden. Im Fokus stehen unterschiedliche Kopplungen der Schaltungsgleichungen, modelliert mit der modifizierten Knotenanalyse, und den elektromagnetischen Bauelementen mit deren verfeinerten Modell basierend auf den vollen Maxwell-Gleichungen in der Lorenz-geeichten A-V Formulierung welche durch Finite-Integrations-Technik räumlich diskretisiert werden. Eine numerische Analyse erweitert die topologischen Kriterien für den Index der resultierenden differential-algebraischen Gleichungen, wie sie bereits in anderen Arbeiten mit ähnlichen Feld/Schaltkreis-Kopplungen hergeleitet wurden. Für die Simulation werden sowohl ein monolithischer Ansatz als auch Waveform-Relaxationsmethoden untersucht. Im Mittelpunkt stehen dabei Zeitintegration, Skalierungsmethoden, strukturelle Eigenschaften und ein hybride Ansatz zur Lösung der zugrundeliegenden linearen Gleichungssysteme welcher den Einsatz spezialisierter Löser für die jeweiligen Teilsysteme erlaubt. Da die vollen Maxwell-Gleichungen zusätzliche Ableitungen in der Kopplungsstruktur verursachen, sind bisher existierende Konvergenzaussagen für die Waveform-Relaxation von gekoppelten differential-algebraischen Gleichungen nicht anwendbar und motivieren eine neue Konvergenzanalyse. Auf dieser Analyse aufbauend werden hinreichende topologische Kriterien entwickelt, welche eine Konvergenz von Gauß-Seidel- und Jacobi-artigen Waveform-Relaxationen für die gekoppelten Systeme garantieren. Schließlich werden numerische Benchmarks zur Verfügung gestellt, um die eingeführten Methoden und Theoreme dieser Abhandlung zu unterstützen.<br>This work is devoted to the simulation of electrical/electronic circuits incorporating electromagnetic devices. The focus is on different couplings of the circuit equations, modeled with the modified nodal analysis, and the electromagnetic devices with their refined model based on full-wave Maxwell's equations in Lorenz gauged A-V formulation which are spatially discretized by the finite integration technique. A numerical analysis extends the topological criteria for the index of the resulting differential-algebraic equations, as already derived in other works with similar field/circuit couplings. For the simulation, both a monolithic approach and waveform relaxation methods are investigated. The focus is on time integration, scaling methods, structural properties and a hybrid approach to solve the underlying linear systems of equations with the use of specialized solvers for the respective subsystems. Since the full-Maxwell approach causes additional derivatives in the coupling structure, previously existing convergence statements for the waveform relaxation of coupled differential-algebraic equations are not applicable and motivate a new convergence analysis. Based on this analysis, sufficient topological criteria are developed which guarantee convergence of Gauss-Seidel and Jacobi type waveform relaxation schemes for introduced coupled systems. Finally, numerical benchmarks are provided to support the introduced methods and theorems of this treatise.

The first contribution of this thesis is a new regularity theorem for time harmonic Maxwell's equations with less than Lipschitz complex anisotropic coefficients. By using the L^p theory for elliptic equations, it is possible to prove H¹ and Hölder regularity results, provided that the coefficients are W^1,p for some p = 3. This improves previous regularity results, where the assumption W^1,∞ for the coefficients was believed to be optimal. The method can be easily extended to the case of bi-anisotropic materials, for which a separate approach turns out to be unnecessary. The second focus of this work is the boundary control of the Helmholtz and Maxwell equations to enforce local constraints inside the domain. More precisely, we look for suitable boundary conditions such that the corresponding solutions and their derivatives satisfy certain local non-zero constraints. Complex geometric optics solutions can be used to construct such illuminations, but are impractical for several reasons. We propose a constructive approach to this problem based on the use of multiple frequencies. The suitable boundary conditions are explicitly constructed and give the desired constraints, provided that a finite number of frequencies, given a priori, are chosen in a fixed range. This method is based on the holomorphicity of the solutions with respect to the frequency and on the regularity theory for the PDE under consideration. This theory finds applications to several hybrid imaging inverse problems, where the unknown coefficients have to be imaged from internal measurements. In order to perform the reconstruction, we often need to find suitable boundary conditions such that the corresponding solutions satisfy certain non-zero constraints, depending on the particular problem under consideration. The multiple frequency approach introduced in this thesis represents a valid alternative to the use of complex geometric optics solutions to construct such boundary conditions. Several examples are discussed.

Modern complex dynamical systems typically possess a multiechelon hierarchical hybrid structure characterized by continuous-time dynamics at the lower-level units and logical decision-making units at the higher-level of hierarchy. Hybrid dynamical systems involve an interacting countable collection of dynamical systems defined on subregions of the partitioned state space. Thus, in addition to traditional control systems, hybrid control systems involve supervising controllers which serve to coordinate the (sometimes competing) actions of the lower-level controllers. A subclass of hybrid dynamical systems are impulsive dynamical systems which consist of three elements, namely, a continuous-time differential equation, a difference equation, and a criterion for determining when the states of the system are to be reset. One of the main topics of this dissertation is the development of stability analysis and control design for impulsive dynamical systems. Specifically, we generalize Poincare's theorem to dynamical systems possessing left-continuous flows to address the stability of limit cycles and periodic orbits of left-continuous, hybrid, and impulsive dynamical systems. For nonlinear impulsive dynamical systems, we present partial stability results, that is, stability with respect to part of the system's state. Furthermore, we develop adaptive control framework for general class of impulsive systems as well as energy-based control framework for hybrid port-controlled Hamiltonian systems. Extensions of stability theory for impulsive dynamical systems with respect to the nonnegative orthant of the state space are also addressed in this dissertation. Furthermore, we design optimal output feedback controllers for set-point regulation of linear nonnegative dynamical systems. Another main topic that has been addressed in this research is the stability analysis of large-scale dynamical systems. Specifically, we extend the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. Furthermore, we present a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii-LaSalle invariant set theorem. Moreover, we develop vector dissipativity theory for large-scale dynamical systems based on vector storage functions and vector supply rates. Finally, using a large-scale dynamical systems perspective, we develop a system-theoretic foundation for thermodynamics. Specifically, using compartmental dynamical system energy flow models, we place the universal energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation laws of thermodynamics on a system-theoretic basis.

Many critical systems are based on the combination of components from different physical domains (e.g. mechanical, electrical, hydraulic), and are mathematically modeled as Switched Multi-Domain Kirchhoff Networks (SMDKN). In this thesis, we tackle a major obstacle to formal verification of SMDKN, namely devising a global model amenable to verification in the form of a Hybrid Automaton. This requires the combination of the local dynamics of the components, expressed as Differential Algebraic Equations, according to Kirchhoff's laws, depending on the (exponentially many) operation modes of the network. We propose an automated SMT-based method to analyze networks from multiple physical domains, detecting which modes induce invalid (i.e. inconsistent) constraints, and to produce a Hybrid Automaton model that accurately describes, in terms of Ordinary Differential Equations, the system evolution in the valid modes, catching also the possible non-deterministic behaviors. The experimental evaluation demonstrates that the proposed approach allows several complex multi-domain systems to be formally analyzed and model checked against various system requirements.

Many critical systems are based on the combination of components from different physical domains (e.g. mechanical, electrical, hydraulic), and are mathematically modeled as Switched Multi-Domain Kirchhoff Networks (SMDKN). In this thesis, we tackle a major obstacle to formal verification of SMDKN, namely devising a global model amenable to verification in the form of a Hybrid Automaton. This requires the combination of the local dynamics of the components, expressed as Differential Algebraic Equations, according to Kirchhoff's laws, depending on the (exponentially many) operation modes of the network. We propose an automated SMT-based method to analyze networks from multiple physical domains, detecting which modes induce invalid (i.e. inconsistent) constraints, and to produce a Hybrid Automaton model that accurately describes, in terms of Ordinary Differential Equations, the system evolution in the valid modes, catching also the possible non-deterministic behaviors. The experimental evaluation demonstrates that the proposed approach allows several complex multi-domain systems to be formally analyzed and model checked against various system requirements.

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

Cette thèse est dédiée au développement d’algorithmes génériques pour l’observation ensembliste de l’état continu et du mode discret des systèmes dynamiques hybrides dans le but de réaliser la détection de défauts. Cette thèse est organisée en deux grandes parties. Dans la première partie, nous avons proposé une méthode rapide et efficace pour le passage ensembliste des gardes. Elle consiste à procéder à la bissection dans la seule direction du temps et ensuite faire collaborer plusieurs contracteurs simultanément pour réduire le domaine des vecteurs d’état localisés sur la garde, durant la tranche de temps étudiée. Ensuite, nous avons proposé une méthode pour la fusion des trajectoires basée sur l'utilisation des zonotopes. Ces méthodes, utilisées conjointement, nous ont permis de caractériser de manière garantie l'ensemble des trajectoires d'état hybride engendrées par un système dynamique hybride incertain sur un horizon de temps fini. La deuxième partie de la thèse aborde les méthodes ensemblistes pour l'estimation de paramètres et pour l'estimation d'état hybride (mode et état continu) dans un contexte à erreurs bornées. Nous avons commencé en premier lieu par décrire les méthodes de détection de défauts dans les systèmes hybrides en utilisant une approche paramétrique et une approche observateur hybride. Ensuite, nous avons décrit deux méthodes permettant d’effectuer les tâches de détection de défauts. Nous avons proposé une méthode basée sur notre méthode d'atteignabilité hybride non linéaire et un algorithme de partitionnement que nous avons nommé SIVIA-H pour calculer de manière garantie l'ensemble des paramètres compatibles avec le modèle hybride, les mesures et avec les bornes d’erreurs. Ensuite, pour l'estimation d'état hybride, nous avons proposé une méthode basée sur un prédicteurcorrecteur construit au dessus de notre méthode d'atteignabilité hybride non linéaire<br>This thesis is dedicated to the development of generic algorithms for the set-membership observation of the continuous state and the discrete mode of hybrid dynamical systems in order to achieve fault detection. This thesis is organized into two parts. In the first part, we have proposed a fast and effective method for the set-membership guard crossing. It consists in carrying out bisection in the time direction only and then makes several contractors working simultaneously to reduce the domain of state vectors located on the guard during the study time slot. Then, we proposed a method for merging trajectories based on zonotopic enclosures. These methods, used together, allowed us to characterize in a guaranteed way the set of all hybrid state trajectories generated by an uncertain hybrid dynamical system on a finite time horizon. The second part focuses on set-membership methods for the parameters or the hybrid state (mode and continuous state) of a hybrid dynamical system in a bounded error framework. We started first by describing fault detection methods for hybrid systems using the parametric approach and the hybrid observer approach. Then, we have described two methods for performing fault detection tasks. We have proposed a method for computing in a guaranteed way all the parameters consistent with the hybrid dynamical model, the actual data and the prior error bound, by using our nonlinear hybrid reachability method and an algorithm for partition which we denote SIVIA-H. Then, for hybrid state estimation, we have proposed a method based on a predictor-corrector, which is also built on top of our non-linear method for hybrid reachability

Les systèmes hybrides sont des systèmes qui combinent des dynamiques discrètes et continues. Cette thèse propose des techniques algorithmiques de vérification et de synthèse pour ces systèmes Le manque de méthodes pour calculer les ensembles atteignables des dynamiques continues est l'obstacle principal vers une méthodologie algorithmique de vérification. Nous développons deux techniques d'atteignabilité approximatives pour les systèmes continus basées sur une méthode efficace pour représenter des ensembles et une combinaison des techniques de la simulation, de la géométrie algorithmique, de l'optimisation, et de la commande optimale. La première technique d'atteignabilité est spécialisée pour les systèmes linéaires et étendue aux systèmes avec entrée incertaine, et la seconde peut être appliquée aux systèmes non-linéaires. En appliquant ces techniques nous développons un algorithme de vérification des propriétés de sûreté pour une large classe des systèmes hybrides avec des dynamiques continues arbitraires et des dynamiques discrètes assez générales. Nous étudions ensuite le problème de la synthèse de contrôleurs de sûreté pour les systèmes hybrides. Nous présentons un algorithme de synthèse des contrôleurs par commutation basé sur le calcul de l'ensemble d'invariance maximal et les techniques d'analyse d'atteignabilité. Nous avons implanté les algorithmes développés dans un outil appelé "d/dt", qui permet la vérification et la synthèse automatique pour les systèmes hybrides avec des inclusions différentielles linéaires. En dehors de nombreux exemples académiques, nous avons appliqué avec succès l'outil pour analyser quelques systèmes pratiques<br>This thesis proposes a practical framework for the verification and synthesis of hybrid systems, that is, systems combining continuous and discrete dynamics. The lack of methods for computing reachable sets of continuous dynamics has been the main obstacle towards an algorithmic verification methodology for hybrid systems. We develop two effective approximate reachability techniques for continuous systems based on an efficient representation of sets and a combination of techniques from simulation, computational geometry, optimization, and optimal control. One is specialized for linear systems and extended to systems with uncertain input, and the other can be applied for non-linear systems. Using these reachability techniques we develop a safety verification algorithm which can work for a broad class of hybrid systems with arbitrary continuous dynamics and rather general switching behavior. We next study the problem of synthesizing switching controllers for hybrid systems with respect to a safety property. We present an effective synthesis algorithm based on the calculation of the maximal invariant set and the approximate reachability techniques. Finally, we describe the experimental tool "d/dt" which provides automatic safety verification and controller synthesis for hybrid systems with linear differential inclusions. Besides numerous academic examples, we have successfully applied the tool to verify some practical systems

La thèse est consacrée à la modélisation discrète et continue des écoulements sanguins et des phénomènes connexes tels que la coagulation du sang et l'athérosclérose. Ce travail comprend l'élaboration des modèles mathématiques et numériques de la coagulation du sang, des simulations numériques et l'analyse mathématique d'un modèle d'inflammation chronique au cours d'athérosclérose. Une partie importante de la thèse est liée à la programmation, la mise en œuvre et l'optimisation des codes numériques. La partie principale de la thèse concerne la modélisation de la coagulation du sang in vivo tenant compte des écoulements sanguins, les réactions biochimiques dans le plasma et l'agrégation de plaquettes. La nouveauté principale de ce travail est l'élaboration d'un modèle hybride (discret-continu) de la coagulation du sang et de la formation de caillot sanguin dans le flux. La partie théorique de la thèse est consacrée à l'analyse mathématique d'un modèle d'inflammation chronique liée à l'athérosclérose. Les simulations numériques réalisées dans le cadre de cette thèse impliquent l'élaboration des algorithmes numériques pour les modèles mathématiques et le d´développement des logiciels. Vu le fait que les simulations numériques ont été coûteuse en temps de calcul, des efforts considérables ont été consacrés à la parallélisation des logiciels et à leur optimisation<br>The thesis is devoted to discrete and continuous modelling of blood flows and related phenomena such as blood coagulation and atherosclerosis. It includes the development of mathematical and numerical models of blood coagulation, numerical simulations and the mathematical analysis of a model problem of chronic inflammation during atherosclerosis. The main part of the thesis concerns modelling of blood coagulation in vivo which takes into account blood flows, biochemical reactions in plasma and platelet aggregation. The main novelty of this work is the development of a hybrid (discrete-continuous) model of blood coagulation and clot formation in flow. The model is used to study several aspects of blood coagulation in flow : platelet aggregation and its interaction with coagulation pathways, influence of the flow speed on the clot development, a possible mechanism by which clot stops growing. The theoretical part of the thesis is devoted to the mathematical analysis of a model of chronic inflammation related to atherosclerosis. In this thesis we study a model problem which describes the propagation of a reaction-diffusion wave in the 2D case with non-linear boundary conditions. For that we use the Leray-Schauder method and a priori estimates of solutions in order to prove the existence of waves in the bistable case. Numerical simulations carried out in the framework of this thesis were based on the numerical implementation of the corresponding models and on the software development. Since the numerical simulations were computationally expensive, a substantial effort was directed to software parallelization and optimization

Resumo: Sistemas dinâmicos híbridos se diferenciam por exibir simultaneamente variados tipos de comportamento dinâmico (contínuo, discreto, eventos discretos) em diferentes partes do sistema. Neste trabalho foram estudados resultados de estabilidade no sentido de Lyapunov para sistemas dinâmicos híbridos gerais, que utilizam uma noção de tempo generalizado, definido em um espaço métrico totalmente ordenado. Mostrou-se que estes sistemas podem ser imersos em sistemas dinâmicos descontínuos definidos em R+, de forma que sejam preservadas suas propriedades qualitativas. Como foco principal, estudou-se resultados de estabilidade para sistemas dinâmicos descontínuos modelados por semigrupos de operadores, em que os estados do sistema pertencem à espaços de Banach. Neste caso, de forma alternativa à teoria clássica de estabilidade, os resultados não utilizam as usuais funções de Lyapunov, sendo portanto mais fáceis de se aplicar, tendo em vista a dificuldade em se encontrar tais funções para muitos sistemas. Além disso, os resultados foram aplicados à uma classe de equações diferenciais com retardo.<br>Abstract: Hybrid dynamical systems are characterized for showing simultaneously a variety of dynamic behaviors (continuous, discrete, discrete events) in different parts of the System. This work discusses stability results in the Lyapunov sense for general hybrid dynamical systems that use a generalized notion of time, defined in a completely ordered metric space. It has been shown that these systems may be immersed in discontinuous dynamical systems defined in R+, so that their quality properties are preserved. As the main focus, it is studied stability results for discontinuous dynamical systems modeled by semigroup operators, in which the states belong to Banach spaces. In this case, an alternative to the classical theory of stability, the results do not make use of the usual Lyapunov functions, and therefore are easier to apply, in view of the difficulty in finding such functions for many systems. Furthermore, the results were applied to a class of time-delay discontinuous differential equations.<br>Orientador: Geraldo Nunes Silva<br>Coorientador: Luís Antônio Fernandes de Oliveira<br>Banca: Carlos Alberto Raposo da Cunha<br>Banca: Waldemar Donizete Bastos<br>Mestre

Conselho Nacional de Desenvolvimento Científico e Tecnológico<br>Optimal Control Problems (OCP), also known as Dynamic Optimization Problems, consist of an Objective Function to be maximized or minimized, associated with a set of differential and algebraic equations which include equality and inequality constraints in the state or control variables and characterize a system of Differential-Algebraic Equations (DAE). The differential-algebraic approach of numerical solution widely used in process simulation due the guarantee of attendance of the implicit algebraic constraints in the original formulation and the elimination of the necessary manipulations to transform the original problem into a purely differential system,was extended to OCP characterizing the called Differential-Algebraic Optimal Control Problem (DAOCP). A category of DAOCP of special interest includes inequality constraints, due the necessity of previous knowledge of the activations and deactivations sequence of these constraints along the trajectory and also of the instants where they occur, named Events. This DAOCPs with inequality constraints is equivalent to a class of hybrid dynamic optimization problems, where continuous and discrete behaviors are associated (FEEHERY, 1998). A particular type of hybrid OCP is that one where continuous state does not present jumps in the Events, called Switched OCP, for which Xu e Antsaklis (2004) considers a solution methodology based on the parameterization of Events with a previous specification of active subsystems sequence, resulting in the solution of a two-point boundary value differential-algebraic problem, formed by the state, co-state and stationarity equations, boundary and continuity conditions and its differentiations, called sensitivity equations. In this work, this indirect approach for Switched OCP was extended for DAOCP with inequality constraints, with the objective to estimate the Events, along the control, state and adjoint variables. The developed approach for Switched OCP described by Xu e Antsaklis (2004) was implemented in a specific code using Maple 9.5, called EVENTS, with the objective to symbolically generate the equations based on the parameterization of Events. This code was incorporated in a interface named OpCol, that collect characterization tools of DAE systems and generation of the optimality conditions extended Pontryagin s Principle for PCOAD of different types. The characterization tools are the INDEX of Murata (1996) that symbolically identifies the index, the resolubility and the consistency of initial conditions and the ACIG of Cunha e Murata (1999) that implements the Gear s algorithm for the index reduction and the index 1 equivalent system generation. The OTIMA (GOMES, 2000; LOBATO, 2004) generates the Euler-Lagrange equations for DAOCP. These tools had been implemented initially in different versions of Maple and all had been update to 9.5 version using the Maplets package that allows the data entry through interactive windows with the user, demanding a little knowledge of the Maple syntax. The OpCol interface was tested for four cases and for each tool a example data bank with typical problems of literature was created to assist the user in its use. Moreover, the direct method implemented in DIRCOL code was extended for multi-phases formulation with estimates of Events and the indirect method with Events Parameterization and differential-algebraic approach implemented in a Matlab code had been used for the numerical solution of three cases: a switched OCP and 2 DAOCP of batch reactors where the control variable is the feed rate of the component B - the first one has parallel reactions and selectivity constraints with 3 phases of index 1, 3 and 1 and the second a safety constraint with 2 phases of index 2 and 1 respectively and had been described by Srinivasan et al. (2003). The methodology used by this authors was applied to attained analytical expressions for the control variable in each phase necessary in indirect method, composing the called Switching Functions, from the optimality conditions based in the Pontryagin s Principle - specifically from the stationarity condition and the active constraint identification that will allow the control variable determination - and of the physical analysis of the problem in order to discard not appropriate activations/deactivations sequences. The results obtained by indirect and direct methods are compared for the 3 cited problems, showing the viability as much of the multiphase formulation using the DIRCOL and also the satisfactory performance of the indirect method with estimates of Events, beyond the utility of the tools of characterization of EADs, of attainment of optimality conditions and parameterization of Events available in Opcol interface.<br>Os Problemas de Controle Ótimo, também chamados Problemas de Otimização Dinâmica, são formados por uma Função Objetivo a ser maximizada ou minimizada, associada a conjuntos de equações algébricas e diferenciais que incluem restrições de igualdade e de desigualdade nas variáveis de estado e de controle que caracterizam um sistema de Equações Algébrico-Diferenciais (EADs). A extensão do ponto de vista algébricodiferencial de solução numérica aos PCOs, já amplamente utilizado na simulação de processos devido à garantia de atendimento às restrições algébricas originais e implícitas na formulação e à eliminação das manipulações necessárias para transformar o problema original num sistema de equações puramente diferenciais, caracteriza o chamado Problema de Controle Ótimo Algébrico-Diferencial (PCOAD). Uma categoria de PCOAD de especial interesse é a dos que incluem restrições de desigualdade, devido à necessidade de conhecimento prévio da seqüência de ativações e desativações destas restrições ao longo da trajetória e também dos instantes em que elas ocorrem, chamados Eventos. As ativações/desativações das restrições causam flutuações no índice diferencial e no número de graus de liberdade dinâmicos do PCOAD, exigindo técnicas especiais de redução deste índice até um e o emprego de métodos numéricos eficientes que garantam a convergência e estabilidade da solução. Estes PCOADs com restrições de desigualdade são equivalentes a uma classe de problemas de otimização dinâmica híbridos, que associam comportamentos contínuos e discretos (FEEHERY, 1998). Um tipo particular de PCO híbrido é aquele cujo estado contínuo não apresenta saltos nos Eventos, chamado PCO Chaveado, para o qual Xu e Antsaklis (2004) propõem uma metodologia de solução baseada na parametrização dos Eventos com a especificação prévia da seqüência de subsistemas ativos, resultando na solução de um problema de valor no contorno algébrico-diferencial em dois pontos, formado pelas equações de estado, co-estado e de estacionariedade, condições de contorno e de continuidade e suas diferenciações, chamadas equações de sensibilidade. Neste trabalho, esta abordagem indireta empregada para PCO Chaveados foi estendida para PCOAD com restrições de desigualdade, com o objetivo de estimar também os Eventos, além das variáveis de controle, de estado e adjuntas. A abordagem desenvolvida por Xu e Antsaklis (2004) para PCO Chaveados foi implementada num código específico utilizando o Maple 9.5, chamado EVENTS, com o objetivo de gerar simbolicamente as equações baseadas na parametrização dos Eventos. Este código foi incorporado a uma interface chamada OpCol, que reúne ferramentas de caracterização de sistemas de EAD e de geração das condições de otimalidade segundo o Princípio de Pontryagin estendidas para PCOAD de diferentes classes. As ferramentas de caracterização são o INDEX de Murata (1996) que identifica simbolicamente o índice, a resolubilidade e a consistência das condições iniciais e o ACIG de Cunha e Murata (1999) que implementa o algoritmo de Gear para a redução do índice e geração do sistema equivalente de índice 1. O OTIMA (GOMES, 2000; LOBATO, 2004) gera as equações de Euler-Lagrange para PCOAD. Estas ferramentas foram inicialmente implementadas em diferentes versões do Maple e todas foram atualizadas para a versão 9.5 utilizando o pacote Maplets que permite a entrada de dados através de janelas interativas com o usuário, exigindo dele pouco conhecimento da sintaxe Maple. A interface OpCol foi testada para quatro casos e para cada ferramenta foi criado um banco de exemplos com problemas típicos da literatura que auxiliam o usuário na sua utilização. Além disto, o método direto implementado no código DIRCOL estendido para formulações multifásicas com estimativa dos Eventos e o método indireto com Parametrização dos Eventos e abordagem algébrico-diferencial implementado num código MATLAB foram utilizados na solução numérica de três estudos de casos: um PCO chaveado e 2 PCOAD de reatores batelada onde a variável de controle é a taxa de alimentação do componente B: o primeiro tem reações paralelas e restrições de seletividade com 3 fases de índices 1, 3 e 1 e o segundo restrições de segurança com 2 fases de índices 2 e 1 e respectivamente e foram descritos por Srinivasan et al. (2003). A mesma metodologia utilizada por estes autores foi empregada na obtenção de expressões analíticas para a variável de controle em cada fase necessárias no método indireto, compondo as chamadas Funções Identificadoras de Fase (FIF), a partir das condições de otimalidade baseadas no Princípio de Pontryagin - especificamente a partir da condição de estacionariedade e da identificação da restrição ativa que permitirá a determinação da variável de controle - e da análise física do problema de modo a descartar seqüências de ativação/desativação não apropriadas. Os resultados obtidos pelo método indireto e pelo método direto são comparados entre si para os 3 problemas citados, mostrando a viabilidade tanto da formulação multifásica empregando o DIRCOL quanto o desempenho satisfatório do método indireto com estimativa de Eventos, além da utilidade das ferramentas de caracterização de EADs, de obtenção das condições de otimalidade e de parametrização dos eventos disponibilizadas na interface Opcol.<br>Mestre em Engenharia Química

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-02-26Bitstream added on 2014-06-13T18:30:53Z : No. of bitstreams: 1 pena_is_me_sjrp.pdf: 488383 bytes, checksum: 40a97f3540caa6b8f6f2691c3a402579 (MD5)<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)<br>Sistemas dinâmicos híbridos se diferenciam por exibir simultaneamente variados tipos de comportamento dinâmico (contínuo, discreto, eventos discretos) em diferentes partes do sistema. Neste trabalho foram estudados resultados de estabilidade no sentido de Lyapunov para sistemas dinâmicos híbridos gerais, que utilizam uma noção de tempo generalizado, definido em um espaço métrico totalmente ordenado. Mostrou-se que estes sistemas podem ser imersos em sistemas dinâmicos descontínuos definidos em R+, de forma que sejam preservadas suas propriedades qualitativas. Como foco principal, estudou-se resultados de estabilidade para sistemas dinâmicos descontínuos modelados por semigrupos de operadores, em que os estados do sistema pertencem à espaços de Banach. Neste caso, de forma alternativa à teoria clássica de estabilidade, os resultados não utilizam as usuais funções de Lyapunov, sendo portanto mais fáceis de se aplicar, tendo em vista a dificuldade em se encontrar tais funções para muitos sistemas. Além disso, os resultados foram aplicados à uma classe de equações diferenciais com retardo.<br>Hybrid dynamical systems are characterized for showing simultaneously a variety of dynamic behaviors (continuous, discrete, discrete events) in different parts of the System. This work discusses stability results in the Lyapunov sense for general hybrid dynamical systems that use a generalized notion of time, defined in a completely ordered metric space. It has been shown that these systems may be immersed in discontinuous dynamical systems defined in R+, so that their quality properties are preserved. As the main focus, it is studied stability results for discontinuous dynamical systems modeled by semigroup operators, in which the states belong to Banach spaces. In this case, an alternative to the classical theory of stability, the results do not make use of the usual Lyapunov functions, and therefore are easier to apply, in view of the difficulty in finding such functions for many systems. Furthermore, the results were applied to a class of time-delay discontinuous differential equations.

Cette thèse porte sur le calcul d'une sur-approximation conservative pour les solutions d'équations différentielles ordinaires en présence d'incertitudes et sur son application à l'estimation et l'analyse de systèmes dynamiques à temps continu. L'avantage principal des méthodes et des algorithmes de calculs présentés dans cette thèse est qu'ils apportent une preuve numérique de résultats. Cette thèse est organisée en deux parties. La première partie est consacrée aux outils mathématiques et aux méthodes d'intégration numérique garantie des équations diff érentielles incertaines. Ces méthodes permettent de caractériser de manière garantie l'ensemble des trajectoires d'état engendrées par un système dynamique incertain dont les incertitudes sont naturellement représentées par des intervalles bornés. Dans cette optique, nous avons développé une méthode d'intégration hybride qui donne de meilleurs résultats que les méthodes d'intégration basées sur les modèles de Taylor intervalles. La seconde partie aborde les problèmes de l'identification et de l'observation dans un contexte à erreurs bornées ainsi que le problème d'atteignabilité continue pour la véri cation de propriétés des systèmes dynamiques hybrides.

La thèse est consacrée au développement de nouvelles méthodes de modélisations mathématiques en biologie et en médecine, du type “off-lattice" modèles hybrides discret-continus, et de leurs applications à l’hématopoïèse et aux maladies sanguines telles la leucémie et l’anémie. Dans cette approche, les cellules biologiques sont considérées comme des objets discrets alors que les réseaux intracellulaire et extracellulaire sont décrits avec des modèles continus régis par des équations aux dérivées partielles et des équations différentielles ordinaires. Les cellules interagissent mécaniquement et biochimiquement entre elles et avec le milieu environnant. Elles peuvent se diviser, mourir par apoptose ou se différencier. Le comportement des cellules est déterminé par le réseau de régulation intracellulaire et influencé par le contrôle local des cellules voisines ou par la régulation globale d’autres organes. Dans la première partie de la thèse, les modèles hybrides du type “off-lattice" dynamiques sont introduits. Des exemples de modèles, spécifiques aux processus biologiques, qui décrivent au sein de chaque cellule la concurrence entre la prolifération et l’apoptose, la prolifération et la différenciation et entre le cycle cellulaire et de l’état de repos sont étudiés. L’émergence des structures biologiques est étudiée avec les modèles hybrides. L’application à la modélisation des filamente de bactéries est illustrée. Dans le chapitre suivant, les modèle hybrides sont appliqués afin de modéliser l’érythropoïèse ou production de globules rouges dans la moelle osseuse. Le modèle inclut des cellules sanguines immatures appelées progéniteurs érythroïdes, qui peuvent s’auto-renouveler, se différencier ou mourir par apoptose, des cellules plus matures appelées les réticulocytes, qui influent les progéniteurs érythroïdes par le facteur de croissance Fas-ligand, et des macrophages, qui sont présents dans les îlots érythroblastiques in vivo. Les régulations intracellulaire et extracellulaire par les protéines et les facteurs de croissance sont précisées et les rétrocontrôles par les hormones érythropoïétine et glucocorticoïdes sont pris en compte. Le rôle des macrophages pour stabiliser les îlots érythroblastiques est montré. La comparaison des résultats de modélisation avec les expériences sur l’anémie chez les souris est effectuée. Le quatrième chapitre est consacré à la modélisation et au traitement de la leucémie. L’érythroleucémie, un sous-type de leucémie myéloblastique aigüe (LAM), se développe à cause de la différenciation insuffisante des progéniteurs érythroïdes et de leur auto-renouvellement excessif. Un modèle de type “Physiologically Based Pharmacokinetics-Pharmacodynamic” du traitement de la leucémie par AraC et un modèle de traitement chronothérapeutique de la leucémie sont examinés. La comparaison avec les données cliniques sur le nombre de blast dans le sang est effectuée. Le dernier chapitre traite du passage d’un modèle hybride à un modèle continu dans le cas 1D. Un théorème de convergence est prouvé. Les simulations numériques confirment un bon accord entre ces deux approches<br>This dissertation is devoted to the development of new methods of mathematical modeling in biology and medicine, off-lattice discrete-continuous hybrid models, and their applications to modelling of hematopoiesis and blood disorders, such as leukemia and anemia. In this approach, biological cells are considered as discrete objects while intracellular and extracellular networks are described with continuous models, ordinary or partial differential equations. Cells interact mechanically and biochemically between each other and with the surrounding medium. They can divide, die by apoptosis or differentiate. Their fate is determined by intracellular regulation and influenced by local control from the surrounding cells or by global regulation from other organs. In the first part of the thesis, hybrid models with off-lattice cell dynamics are introduced. Model examples specific for biological processes and describing competition between cell proliferation and apoptosis, proliferation and differentiation and between cell cycling and quiescent state are investigated. Biological pattern formation with hybrid models is discussed. Application to bacteria filament is illustrated. In the next chapter, hybrid model are applied in order to model erythropoiesis, red blood cell production in the bone marrow. The model includes immature blood cells, erythroid progenitors, which can self-renew, differentiate or die by apoptosis, more mature cells, reticulocytes, which influence erythroid progenitors by means of growth factor Fas-ligand, and macrophages, which are present in erythroblastic islands in vivo. Intracellular and extracellular regulation by proteins and growth factors are specified and the feedback by the hormones erythropoietin and glucocorticoids is taken into account. The role of macrophages to stabilize erythroblastic islands is shown. Comparison of modelling with experiments on anemia in mice is carried out. The following chapter is devoted to leukemia modelling and treatment. Erythroleukemia, a subtype of Acute Myeloblastic Leukemia (AML), develops due to insufficient differentiation of erythroid progenitors and their excessive slef-renewal. A Physiologically Based Pharmacokinetics-Pharmacodynamics (PBPKPD) model of leukemia treatment with AraC drug and chronotherapeutic treatments of leukemia are examined. Comparison with clinical data on blast count in blood is carried out. The last chapter deals with the passage from a hybrid model to a continuous model in the 1D case. A convergence theorem is proved. Numerical simulations confirm a good agreement between these approaches

Stochastic effects in cellular systems are usually modeled and simulated with Gillespie's stochastic simulation algorithm (SSA), which follows the same theoretical derivation as the chemical master equation (CME), but the low efficiency of SSA limits its application to large chemical networks. To improve efficiency of stochastic simulations, Haseltine and Rawlings proposed a hybrid of ODE and SSA algorithm, which combines ordinary differential equations (ODEs) for traditional deterministic models and SSA for stochastic models. In this dissertation, accuracy analysis, efficient implementation strategies, and application of of Haseltine and Rawlings's hybrid method (HR) to a budding yeast cell cycle model are discussed. Accuracy of the hybrid method HR is studied based on a linear chain reaction system, motivated from the modeling practice used for the budding yeast cell cycle control mechanism. Mathematical analysis and numerical results both show that the hybrid method HR is accurate if either numbers of molecules of reactants in fast reactions are above certain thresholds, or rate constants of fast reactions are much larger than rate constants of slow reactions. Our analysis also shows that the hybrid method HR allows for a much greater region in system parameter space than those for the slow scale SSA (ssSSA) and the stochastic quasi steady state assumption (SQSSA) method. Implementation of the hybrid method HR requires a stiff ODE solver for numerical integration and an efficient event-handling strategy for slow reaction firings. In this dissertation, an event-handling strategy is developed based on inverse interpolation. Performances of five wildly used stiff ODE solvers are measured in three numerical experiments. Furthermore, inspired by the strategy of the hybrid method HR, a hybrid of ODE and SSA stochastic models for the budding yeast cell cycle is developed, based on a deterministic model in the literature. Simulation results of this hybrid model match very well with biological experimental data, and this model is the first to do so with these recently available experimental data. This study demonstrates that the hybrid method HR has great potential for stochastic modeling and simulation of large biochemical networks.<br>Ph. D.

Ce mémoire présente une approche basée sur des méthodes garanties pour résoudre des problèmes d’optimisation de systèmes dynamiques multi-physiques. Ces systèmes trouvent des applications directes dans des domaines variés tels que la conception en ingéniérie, la modélisation de réactions chimiques, la simulation de systèmes biologiques ou la prédiction de la performance sportive.La résolution de ces problèmes d’optimisation s’effectue en deux phases. La première consiste à mettre le problème en équations sous forme d’un modèle mathématique constitué d’un ensemble de variables, d’un ensemble de contraintes algébriques et fonctionelles ainsi que de fonctions de coût. Celles-ci sont utilisées lors de la seconde phase qui consiste à d’extraire du modèle les solutions optimales selon plusieurs critères (volume, poids, etc).Les contraintes algébriques permettent de manipuler des grandeurs statiques (quantité, taille, densité, etc). Elles sont non linéaires, non convexes et parfois discontinues.Les contraintes fonctionnelles permettent de manipuler des grandeurs dynamiques. Ces contraintes peuvent être relativement simples comme la monotonie ou la périodicité, mais aussi bien plus complexe par la prise en compte de contraintes différentielles simples ou définies par morceaux. Les équations différentielles sont utilisées pour modéliser des comportements physico-chimiques (magnétiques, thermiques, etc) et d’autres caractéristiques qui varient lors de l’évolution du système.Il existe plusieurs niveaux d’approximation pour chacune de ces deux phases. Ces approximations donnent des résultats pertinents, mais elles ne permettent pas de garantir l’optimalité ni la réalisabilité des solutions.Après avoir présenté un ensemble de méthodes garanties permettant de résoudre de manière garantie des équations différentielles ordinaires, nous formalisons un modèle particulier de systèmes hybrides sous la forme d’équations différentielles ordinaires par morceaux. A l’aide de plusieurs preuves et théorèmes nous étendons la première méthode de résolution pour résoudre de manière garantie ces équations différentielles par morceaux. Dans un second temps, nous intégrons ces deux méthodes au sein d’un module de programmation par contracteurs, que nous avons implémenté. Ce module basé sur des méthodes garantie permet de résoudre des problèmes de satisfaction de contraintes algébriques et fonctionnelles. Ce module est finalement utilisé dans un algorithme d’optimisation globale déterministe modulaire permettant de résoudre les problèmes considérés<br>In this thesis a set of tools based on guaranteed methods are presented in order to solve multi-physics dynamic problems. These systems can be applied in various domains such that engineering design process, model of chemical reactions, simulation of biological systems or even to predict athletic performances.The resolution of these optimization problems is made of two stages. The first one consists in defining a mathematical model by setting up the equations for the problem. The model is made of a set of variables, a set of algebraic and functional constraints and cost functions. The latter are used in the second stage in order to extract the optimal solutions from the model depending on several criteria (volume, weight, etc).Algebraic constraints are used to describe the static properties of the system (quantity, size, density, etc). They are non-linear, non-convex and sometimes discontinuous. Functional constraints are used to manipulate dynamic quantities. These constraints can be quite simple such as monotony or periodicity or they can be more complex such as simple or piecewise differential constraints. Differential equations are used to describe physico-chemical properties (magnetic, thermal, etc) and other features evolving with the component use. Several levels of approximation exist for each of these two stages. These approximations give some relevant results but they do not guarantee the feasibility nor the optimality of the solutions.After presenting a set of guaranteed methods in order to perform the guaranteed integration of ordinary differential equations, a peculiar type of hybrid system that can be modeled with piecewise ordinary differential equation is considered. A new method that computes guaranteed integration of these piecewise ordinary differential equations is developed through an extension of the initial algorithm based on several proofs and theorems. In a second step these algorithms are gathered within a contractor programming module that have been implemented. It is used to solve algebraic and functional constraint satisfaction problems with guaranteed methods. Finally, the considered optimization problems are solved with a modular deterministic global optimization algorithm that uses the previous modules

Signals in chaotic communication systems can be used in secure communications because they are unstable and aperiodic making them difficult to detect or predict. Receivers for conventional chaotic communication systems are complex as chaotic signals are sensitive initial condition and difficult to synchronize. This research developed a method to create a Generalized Multilevel-Hybrid Chaotic Oscillator and derived its generalized fixed basis function leading to the implementation of a simple matched filter receiver that can be synchronized. The proposed system is not sensitive initial condition and can be used effectively for multi-level and multi-access communication systems.

L’objectif de cette thèse est d’étudier comment la cellule mammifère adapte son métabolisme aux étapes du cycle cellulaire. Le cycle cellulaire est l’ensemble des étapes menant une cellule à se diviser. Le rôle du métabolisme est de fournir à la cellule les éléments et l’énergie dont elle a besoin pour fonctionner. En particulier, à chaque étape du cycle cellulaire, la cellule a besoin de différents éléments pour pouvoir, à terme, se diviser correctement. Il est donc crucial pour la cellule de coordonner le métabolisme et le cycle cellulaire et en particulier de contrôler ce que le métabolisme produit au cours du cycle cellulaire. Pour mieux comprendre ce lien entre ces deux processus, nous avons étudié comment un modèle mathématique du métabolisme répondait à différentes variations imposées par le cycle cellulaire et nous avons comparé ces réponses à la littérature. Satisfaits des résultats obtenus, nous avons alors construit un modèle hybride représentant l’évolution du métabolisme au cours du cycle cellulaire. Nous retrouvons dans ce modèle hybride les grandes variations connues des voies métaboliques au cours des phases du cycle cellulaire ainsi que des variations expérimentales des métabolites énergétiques et redox. Encouragés par ces résultats, nous avons finalement perturbé notre modèle hybride pour retrouver des tendances du métabolisme dues au cancer, un ensemble de maladies touchant à la fois le cycle cellulaire et le métabolisme<br>The goal of this thesis is to study how the mammal cell adjusts its metabolism to the steps of the cell cycle. The cell cycle is the series of events leading a cell to divide itself. The purpose of the metabolism is to supply the cell with all the elements and the energy it needs to work. In particular, at every step of the cell cycle, the cell needs different elements to properly divide itself. So, it is crucial for the cell to coordinate the metabolism and the cell cycle and in particular to control what the metabolism produces through the cell cycle. To have a better understanding of the links between these two processes, we studied how a mathematical model representing the metabolism answered to different variations imposed by the cell cycle and we compared those answers to the literature. Satisfied by the results, we therefore built a hybrid model representing the evolution of the metabolism through the cell cycle. We recover in this hybrid model the main known variations of the metabolism through the cycle’s phases as well as experimental variations of the energetic and redox metabolites. Encouraged by these results, we finally disturbed our hybrid model to recover metabolic tendencies due to cancer, a set of diseases affecting both the metabolism and the cell cycle

Plants are essential for the existence of most living things on this planet. Plants are used for providing food, shelter, and medicine. The ability to identify plants is very important for several applications, including conservation of endangered plant species, rehabilitation of lands after mining activities and differentiating crop plants from weeds. In recent times, many researchers have made attempts to develop automated plant species recognition systems. However, the current computer-based plants recognition systems have limitations as some plants are naturally complex, thus it is difficult to extract and represent their features. Further, natural differences of features within the same plant and similarities between plants of different species cause problems in classification. This thesis developed a novel hybrid intelligent system based on a neuro-genetic model for automatic recognition of plants using leaf image analysis based on novel approach of combining several image descriptors with Cellular Neural Networks (CNN), Genetic Algorithm (GA), and Probabilistic Neural Networks (PNN) to address classification challenges in plant computer-based plant species identification using the images of plant leaves. A GA-based feature selection module was developed to select the best of these leaf features. Particle Swam Optimization (PSO) and Principal Component Analysis (PCA) were also used sideways for comparison and to provide rigorous feature selection and analysis. Statistical analysis using ANOVA and correlation techniques confirmed the effectiveness of the GA-based and PSO-based techniques as there were no redundant features, since the subset of features selected by both techniques correlated well. The number of principal components (PC) from the past were selected by conventional method associated with PCA. However, in this study, GA was used to select a minimum number of PC from the original PC space. This reduced computational cost with respect to time and increased the accuracy of the classifier used. The algebraic nature of the GA’s fitness function ensures good performance of the GA. Furthermore, GA was also used to optimize the parameters of a CNN (CNN for image segmentation) and then uniquely combined with PNN to improve and stabilize the performance of the classification system. The CNN (being an ordinary differential equation (ODE)) was solved using Runge-Kutta 4th order algorithm in order to minimize descritisation errors associated with edge detection. This study involved the extraction of 112 features from the images of plant species found in the Flavia dataset (publically available) using MATLAB programming environment. These features include Zernike Moments (20 ZMs), Fourier Descriptors (21 FDs), Legendre Moments (20 LMs), Hu 7 Moments (7 Hu7Ms), Texture Properties (22 TP) , Geometrical Properties (10 GP), and Colour features (12 CF). With the use of GA, only 14 features were finally selected for optimal accuracy. The PNN was genetically optimized to ensure optimal accuracy since it is not the best practise to fix the tunning parameters for the PNN arbitrarily. Two separate GA algorithms were implemented to optimize the PNN, that is, the GA provided by MATLAB Optimization Toolbox (GA1) and a separately implemented GA (GA2). The best chromosome (PNN spread) for GA1 was 0.035 with associated classification accuracy of 91.3740% while a spread value of 0.06 was obtained from GA2 giving rise to improved classification accuracy of 92.62%. The PNN-based classifier used in this study was benchmarked against other classifiers such as Multi-layer perceptron (MLP), K Nearest Neigbhour (kNN), Naive Bayes Classifier (NBC), Radial Basis Function (RBF), Ensemble classifiers (Adaboost). The best candidate among these classifiers was the genetically optimized PNN. Some computational theoretic properties on PNN are also presented.

L'objectif principal de cette thèse consiste à étudier plusieurs thématiques : l'étude de l'observation et la commande d'un système de structure flexible et l'étude de la stabilité asymptotique d'un système d'échangeurs thermiques. Ce travail s'inscrit dans le domaine du contrôle des systèmes décrits par des équations aux dérivées partielles (EDP). On s'intéresse au système du corps-poutre en rotation dont la dynamique est physiquement non mesurable. On présente un observateur du type Luenberger de dimension infinie exponentiellement convergent afin d'estimer les variables d'état. L'observateur est valable pour une vitesse angulaire en temps variant autour d'une constante. La vitesse de convergence de l'observateur peut être accélérée en tenant compte d'une seconde étape de conception. La contribution principale de ce travail consiste à construire un simulateur fiable basé sur la méthode des éléments finis. Une étude numérique est effectuée pour le système avec la vitesse angulaire constante ou variante en fonction du temps. L'influence du choix de gain est examinée sur la vitesse de convergence de l'observateur. La robustesse de l'observateur est testée face à la mesure corrompue par du bruit. En mettant en cascade notre observateur et une loi de commande stabilisante par retour d'état, on souhaite obtenir une stabilisation globale du système. Des résultats numériques pertinents permettent de conjecturer la stabilité asymptotique du système en boucle fermée. Dans la seconde partie, l'étude est effectuée sur la stabilité exponentielle des systèmes d'échangeurs thermiques avec diffusion et sans diffusion. On établit la stabilité exponentielle du modèle avec diffusion dans un espace de Banach. Le taux de décroissance optimal du système est calculé pour le modèle avec diffusion. On prouve la stabilité exponentielle dans l'espace Lp pour le modèle sans diffusion. Le taux de décroissance n'est pas encore explicité dans ce dernier cas.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein.<br>Mathematical Sciences<br>M.Sc. (Applied Mathematics)

Thesis (Ph. D.) -- University of Tennessee, Knoxville, 2006.<br>Title from title page screen (viewed on September 14, 2006). Thesis advisor: Suzanne Lenhart. Vita. Includes bibliographical references.

This work investigates energy-efficient approximate computation for solving differential equations. It extends the analog computing techniques to a new paradigm: continuous-time hybrid computation, where both analog and digital circuits operate in continuous time. In this approach, the time intervals in the digital signals contain important information. Unlike conventional synchronous digital circuits, continuous-time digital signals offer the benefits of adaptive power dissipation and no quantization noise. Two prototype chips have been fabricated in 65 nm CMOS technology and tested successfully. The first chip is capable of solving nonlinear differential equations up to 4th order, and the second chip scales up to 16th order based on the first chip. Nonlinear functions are generated by a programmable, clockless, continuous-time 8-bit hybrid architecture (ADC+SRAM+DAC). Digitally-assisted calibration is used in all analog/mixed-signal blocks. Compared to the prior art, our chips makes possible arbitrary nonlinearities and achieves 16 times lower power dissipation, thanks to technology scaling and extensive use of class-AB analog blocks. Typically, the unit achieves a computational accuracy of about 0.5% to 5% RMS, solution times from a fraction of 1 micro second to several hundred micro seconds, and total computational energy from a fraction of 1 nJ to hundreds of nJ, depending on equation details. Very significant advantages are observed in computational speed and energy (over two orders of magnitude and over one order of magnitude, respectively) compared to those obtained with a modern MSP430 microcontroller for the same RMS error.

The Monte Carlo forward Euler method with uniform time stepping is the standard technique to compute an approximation of the expected payoff of a solution of an Itô SDE. For a given accuracy requirement TOL, the complexity of this technique for well behaved problems, that is the amount of computational work to solve the problem, is O(TOL-3). A new hybrid adaptive Monte Carlo forward Euler algorithm for SDEs with non-smooth coefficients and low regular observables is developed in this thesis. This adaptive method is based on the derivation of a new error expansion with computable leading-order terms. The basic idea of the new expansion is the use of a mixture of prior information to determine the weight functions and posterior information to compute the local error. In a number of numerical examples the superior efficiency of the hybrid adaptive algorithm over the standard uniform time stepping technique is verified. When a non-smooth binary payoff with either GBM or drift singularity type of SDEs is considered, the new adaptive method achieves the same complexity as the uniform discretization with smooth problems. Moreover, the new developed algorithm is extended to the MLMC forward Euler setting which reduces the complexity from O(TOL-3) to O(TOL-2(log(TOL))2). For the binary option case with the same type of Itô SDEs, the hybrid adaptive MLMC forward Euler recovers the standard multilevel computational cost O(TOL-2(log(TOL))2). When considering a higher order Milstein scheme, a similar complexity result was obtained by Giles using the uniform time stepping for one dimensional SDEs. The difficulty to extend Giles' Milstein MLMC method to the multidimensional case is an argument for the flexibility of our new constructed adaptive MLMC forward Euler method which can be easily adapted to this setting. Similarly, the expected complexity O(TOL-2(log(TOL))2) is reached for the multidimensional case and verified numerically.

<p>We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.</p><p>Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.</p><p>Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.</p><br>Dissertation

Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts. In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches. Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed. Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.

The hyperbolic conservation laws model the phenomena of nonlinear waves including discontinuities. The coupled nonlinear equations representing such conservation laws may lead to discontinuous solutions even for smooth initial data. To solve such equations, developing numerical methods which are accurate, robust, and resolve all the wave structures appearing in the solutions is a challenging task. Among several discretization techniques developed for solving hyperbolic conservation laws numerically, Finite Volume Method (FVM) is the most popular. Numerical algorithms, in the framework of FVM, are broadly classified as upwind and central discretization methods. Upwind methods mimic the features of hyperbolic conservation laws very well. However, most of the popular upwind schemes are known to suffer from the shock instabilities. Many upwind methods are heavily dependent on eigen-structure, therefore methods developed for one system of conservation laws are not straightforwardly extended to other systems. On the contrary, central discretization methods are simple, independent of eigen-structure, and therefore, are easily extended to other systems. In the first part of the thesis, a hybrid central discretization method is introduced for Euler equations of gas dynamics. This hybrid scheme is then extended to other hyperbolic conservation laws namely, shallow water equations of oceanography and ideal magnetohydrodynamics equations. The baseline solver for the new hybrid scheme, Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), is an accurate scheme capable of capturing grid aligned steady discontinuities exactly. This central scheme is free from complicated Riemann solvers and therefore is easy to implement. This low diffusive algorithm produces sonic glitches at the expansion regions involving sonic points and is prone to shock instabilities. Therefore it requires an entropy fix to avoid these problems. With the use of entropy fix the exact discontinuity capturing property of the scheme is lost, although sonic glitches and shock instabilities are avoided. The motivation for this work is to develop a numerical method which exactly preserves the steady contacts, is accurate, free of multi-dimensional shock instabilities and yet avoids the entropy fix. This is achieved by constructing a coefficient of numerical diffusion based on pressure gradient sensor. The pressure gradients are known to detect shocks and they vanish across contact discontinuities. This property of pressure sensor is utilized in constructing the coefficient of numerical diffusion. In addition to the numerical diffusion of the baseline solver, a numerical diffusion based on the pressure sensor, scaled by the maximum of eigen-spectrum, is used to avoid shock instabilities. At contact discontinuities, pressure gradients vanish and coefficient of numerical diffusion of MOVERS is automatically retained to capture steady contact discontinuities exactly. This simple hybrid central solver is accurate, captures steady contact discontinuities exactly and is free of multi-dimensional shock instabilities. This novel method is extended to shallow water and ideal magnetohydrodynamics equations in a similar way. In the second part of the thesis, an entropy stable central discretization method for hyperbolic conservation laws is introduced. In a quest for optimal numerical viscosity, development of entropy stable schemes gained importance in recent times. In this work, the entropy conservation equation is used as a guideline to fix the coefficient of numerical diffusion for smooth regions of the flow. At the large gradients, coefficient of numerical diffusion of baseline solver is used. Switch over between smooth and large gradients of the flow is done using limiter functions which are known to distinguish between smooth and high gradient regions of the flow. This simple and stable central scheme termed MOVERS-LE captures grid aligned steady discontinuities exactly and is free of shock instabilities in multi-dimensions. Both the above algorithms are tested on various well established benchmark test problems.

The hyperbolic conservation laws model the phenomena of nonlinear waves including discontinuities. The coupled nonlinear equations representing such conservation laws may lead to discontinuous solutions even for smooth initial data. To solve such equations, developing numerical methods which are accurate, robust, and resolve all the wave structures appearing in the solutions is a challenging task. Among several discretization techniques developed for solving hyperbolic conservation laws numerically, Finite Volume Method (FVM) is the most popular. Numerical algorithms, in the framework of FVM, are broadly classified as upwind and central discretization methods. Upwind methods mimic the features of hyperbolic conservation laws very well. However, most of the popular upwind schemes are known to suffer from the shock instabilities. Many upwind methods are heavily dependent on eigen-structure, therefore methods developed for one system of conservation laws are not straightforwardly extended to other systems. On the contrary, central discretization methods are simple, independent of eigen-structure, and therefore, are easily extended to other systems. In the first part of the thesis, a hybrid central discretization method is introduced for Euler equations of gas dynamics. This hybrid scheme is then extended to other hyperbolic conservation laws namely, shallow water equations of oceanography and ideal magnetohydrodynamics equations. The baseline solver for the new hybrid scheme, Method of Optimal Viscosity for Enhanced Resolution of Shocks (MOVERS), is an accurate scheme capable of capturing grid aligned steady discontinuities exactly. This central scheme is free from complicated Riemann solvers and therefore is easy to implement. This low diffusive algorithm produces sonic glitches at the expansion regions involving sonic points and is prone to shock instabilities. Therefore it requires an entropy fix to avoid these problems. With the use of entropy fix the exact discontinuity capturing property of the scheme is lost, although sonic glitches and shock instabilities are avoided. The motivation for this work is to develop a numerical method which exactly preserves the steady contacts, is accurate, free of multi-dimensional shock instabilities and yet avoids the entropy fix. This is achieved by constructing a coefficient of numerical diffusion based on pressure gradient sensor. The pressure gradients are known to detect shocks and they vanish across contact discontinuities. This property of pressure sensor is utilized in constructing the coefficient of numerical diffusion. In addition to the numerical diffusion of the baseline solver, a numerical diffusion based on the pressure sensor, scaled by the maximum of eigen-spectrum, is used to avoid shock instabilities. At contact discontinuities, pressure gradients vanish and coefficient of numerical diffusion of MOVERS is automatically retained to capture steady contact discontinuities exactly. This simple hybrid central solver is accurate, captures steady contact discontinuities exactly and is free of multi-dimensional shock instabilities. This novel method is extended to shallow water and ideal magnetohydrodynamics equations in a similar way. In the second part of the thesis, an entropy stable central discretization method for hyperbolic conservation laws is introduced. In a quest for optimal numerical viscosity, development of entropy stable schemes gained importance in recent times. In this work, the entropy conservation equation is used as a guideline to fix the coefficient of numerical diffusion for smooth regions of the flow. At the large gradients, coefficient of numerical diffusion of baseline solver is used. Switch over between smooth and large gradients of the flow is done using limiter functions which are known to distinguish between smooth and high gradient regions of the flow. This simple and stable central scheme termed MOVERS-LE captures grid aligned steady discontinuities exactly and is free of shock instabilities in multi-dimensions. Both the above algorithms are tested on various well established benchmark test problems.

Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física, 2011.<br>En este trabajo se da una breve introducción y algunas primeras herramientas para la teoría de control y los sistemas de ecuaciones diferenciales con delay. Se utilizan estas herramientas para analizar dos modelos, con miradas diferentes del crecimiento tumoral existentes en la literatura y se propone un nuevo modelo híbrido que contemple no solo la dinámica continua sino también elementos discretos, de forma de buscar un protocolo óptimo de tratamientos de quimioterapia. Para esto, se propone y prueba un teorema donde se caracteriza la derivada del valor final de unavariable con respecto a las duraciones de un sistema con sitches. En la última sección, se encuentra una experiencia numérica mostrando la factibilidad de implementar el teorema anterior al problema propuesto.<br>Matías Gonzalo Delgadino.

Abstract in English<br>Mathematical modeling of the spread of infectious diseases in a population has always been recognized as a powerful tool that can help decision-makers understand how a disease evolves over time. With the evolution of science and humanity, it has become evident that Mathematical models are too simplistic and have some limitations in modeling environmental phenomena, such as the spread of epidemics in a population, when they are applied without combining them with other sciences. In understanding the dynamics of epidemics in a population, the weakness of these models is their difficulty in grasping the complexity inherent in the spread of diseases in real life because, life is supported by human interactions and behaviors that are understood through networks of social and spatial interactions. Modeling the spread of epidemics which takes this reality into account requires the implementation of new tools to refine the results already obtained by mathematical models. The aim of this thesis is to explore and attempt to extend new developments in mathematical modeling of the spread of infectious diseases by proposing new tools based on mathematical models from differential equations and agent-based models from intelligent agents derived from artificial intelligence. To achieve this objective, the study starts from a comparative study of two ways of modeling and simulation of the spread of infectious diseases in the population, namely mathematical modeling and agent-based modeling with a concrete case study of the spread of tuberculosis based on data from the Democratic Republic of the Congo (DRC). Then comes a coupling study of these two approaches in a single model and its implementation in a multi-scale environment. The results show that the coupled model is more realistic compared to mathematical models generally implemented in the literature. Four case studies are presented in this thesis. Mathematical modeling based on differential equations is used in the first and second cases. The third case is based on intelligent agents model while the last one is based on the coupling of mathematical models and agent-based models. Application of implemented models to the spread of tuberculosis reveals that detection of people with latent tuberculosis and their treatment are among the actions to be taken into account in addition to those currently carried out by the Congolese health system. The models assert that the current TB situation in DRC remains endemic and that the necessary measures need to be taken to reduce the burden of TB, especially to control it, through the tuberculosis elimination strategy and its elimination in the future in accordance with the Sustainable Development Goals. Our hybrid model benefiting from the advantages of EBM and ABM confirms that taking the individual into account as a fully-fledged entity and managing their behavior gives the microscopic aspect of the model set up and brings it closer as much as possible to reality. Mathematical management of the spread of the disease in cities gives a macroscopic aspect to the model. Numerical simulations of this last model on a multi-scale virtual environment affirm that the mobility of individuals from city to city has a significant impact on the spread of tuberculosis in the population. Controlling the rate of population mobility from one city to another is one of the most important measures for large-scale disease control. This model therefore draws its richness from this dynamic at two different scales (two time scales modeling approaches: at the microscopic/individual level (ABM) and macroscopic/city level (ODE)), which gives the emergence of the model at the global level. As a result, it seems that the coupling of mathematical models to agent-based models should be applied when the dynamics of the complex system under consideration is at different scales. Based on our research results, it seems that the choice of an approach must depend on how the modeler would like to achieve the expected results. Mathematical models remain essential due to their analytical and synthetic aspect, but their coupling with intelligent agent-based models makes it possible to refine known results and thus reflect the reality of real life, because the resulting model integrate interactions of individuals and their heterogeneous behaviors that are necessary for understanding the spread of infectious diseases in the population that only mathematical models based on differential equations can not capture.<br>Mathematical Sciences<br>Ph D. (Applied Mathematics)

Дмитренко В. В. Динаміка обертового осцилятора із нелінійною функцією демпфування : кваліфікаційна робота магістра спеціальності 113 "Прикладна математика" / наук. керівник В. З. Грищак. Запоріжжя : ЗНУ, 2020. 51 с.<br>UA : Робота викладена на 51 сторінці друкованого тексту, містить 7 рисунків, 29 джерел. Об’єкт дослідження: задача динаміки обертового математичного маятника із нелінійною функцією демпфування. Мета роботи: отримання нових наближених аналітико-чисельних розв’язків актуальних нелінійних задач динаміки систем із нелінійною функцією демпфування. Метод дослідження: аналітичний. У кваліфікаційній роботі на базі гібридного асимптотичного підходу запропонована нова математична модель з аналітико-чисельним алгоритмом дослідження і візуалізацією досліджуваних процесів, а також отриманими наближеними аналітичними розв’язками задачі, яка описується сингулярним диференціальним рівнянням зі змінними параметрами. Отримані наближені аналітичні розв’язки дозволяють рекомендувати здобуті у кваліфікаційній роботі залежності для подальшого розвитку теорії математичного моделювання, а також практичного використання при проектуванні конструкцій нової техніки.<br>EN : The work is presented on 51 pages of printed text, 7 figures, 29 references. The object of the study is nonlinear problem of dynamics for rotating mathematical oscillation with nonlinear damping function under the action of a periodic external loading. The aim of the study is obtaining a new approximate analytic-numerical solution of actual nonlinear problems of dynamics of systems with nonlinear damping function. The method of research is analytical. In this qualification paper on the basis of hybrid asymptotic approach a new mathematical model with analytical and numerical algorithm for the study and visualization of the are proposed. An approximate analytical solution of problems, which are described by singular differential equations with variable coefficients and their systems are obtained. The approximate analytical solutions obtained allows to recommend the dependence obtained in the qualification work for the further development of the theory of mathematical modeling, as well as practical use in the design of structures of new technology.

Руденко Д. О. Асимптотико-чисельний підхід до розв’язання задач математичної фізики зі змінними коефіцієнтами : кваліфікаційна робота магістра спеціальності 113 "Прикладна математика" / наук. керівник В. З. Грищак. Запоріжжя : ЗНУ, 2020. 69 с.<br>UA : Робота викладена на 69 сторінках друкованого тексту, містить 13 рисунків, 46 джерел. Об’єкт дослідження – неоднорідне нелінійне диференціальне рівняння із змінними коефіцієнтами та 𝛿-функцією у правій частині. Мета роботи: створення алгоритму наближеного аналітичного розв’язку. Метод дослідження – аналітичний на базі асимптотичного підходу, прямий чисельний метод інтегрування із застосуванням комп’ютерної алгебри і системи «Mathematica». У кваліфікаційній роботі запропоновано наближений аналітичний розв’язок деяких задач математичної фізики, які зводяться до інтегрування сингулярних нелінійних диференціальних рівнянь із змінними розривними коефіцієнтами, нелінійною першою похідною і 𝛿-функцією у правій частині.<br>EN : The work is presented on 69 pages of printed text, 13 figures, 46 references. The object of the study is an inhomogeneous nonlinear differential equation with variable coefficients and 𝛿-function in the right part. The aim of the study ‒ to create an algorithm for approximate analytical solution. The methods of research are analytical based on the asymptotic approach, a irect numerical method of integration using computer algebra and the system "Mathematica". The thesis proposes an approximate analytical solution of some problems of mathematical physics, which are reduced to the need to integrate singular nonlinear differential equations with variable discontinuities, nonlinear first derivative and 𝛿-function in the right part.