A distributed delay is a linear input-output operators and appears in many control problems. We investigate distributed delay and its applications. After introducing the definition and the main properties of the distributed delay, the numerical implementation problem of distributed delays is analyzed and a general method for its approximation is given. Then three applications are focused on where distributed delay appears. The first application is the stable inversion and model matching. A new class of stable inversion and model matching problem for finite dimensional linear time-invariant systems is defined. The stable inversion (resp. model matching) is an approximation of the inverse of a given model (resp. model matching), where exact inversion (resp. exact matching) is reached after a time $t=h$, which is a parameter of our procedure. The second application is concerned with stabilization and finite spectrum assignment for a class of infinite dimensional systems. The last application concerns observer synthesis for estimation or output control. For linear finite dimensional systems. A closed-loop memoryless observer by input injection is introduced. Asymptotic convergence as well as finite time convergence of the estimation are analyzed by output injection and input information via distributed delay. At last, we introduce a new class for approximation of distributed parameter systems. We work on the graph topology, and show that under some weak assumptions, such an approximation can be realized using distributed delay.

We employ two efficient parallel approaches to reduce a model arising from a semi-discretization of a controlled heat transfer process for optimal cooling of a steel profile. Both algorithms are based on balanced truncation but differ in the numerical method that is used to solve two dual generalized Lyapunov equations, which is the major computational task. Experimental results on a cluster of Intel Xeon processors compare the efficacy of the parallel model reduction algorithms.

Ce travail a pour but la réalisation d'un outil permettant d'estimer la germination et la distribution en taille de cristaux d'un procédé de cristallisation. Pour ce faire un modèle de cristallisation a été développé et nous a permis de réaliser un observateur de dimension finie afin d'estimer la germination. Par la suite, un observateur de dimension infinie a été mis au point dans le but d'estimer la distribution de taille des cristaux à partir de la germination et de la mesure d'une taille de la distribution. Les deux observateurs ont été validés expérimentalement sur un procédé "batch" de cristallisation de l'oxalate d'ammonium. / The goal of this thesis is the development of an observer to estimate the nucleation and the crystal size distribution of a crystallization process. We first developed a model of crystallization in order to realize an observer for nucleation estimation. Then an infinite dimension observer was developed to estimate the crystal size distribution, using the nucleation estimation and one crystal size distribution measurement. An experimental validation was performed using the crystallization of ammonium oxalate.

This diploma thesis is focused on designing and specifying a linear quadratic regulator for longitudinal and lateral motion of the aircraft. The regulator is designed for a linear system obtained from a linearization of our nonlinear dynamic dynamic model of aircraft around the basic reference state (trimmed state-and-level ight). To nd a complete description of this reference state, some optimality problem has to be solved and the solution of this problem is called a trimmed state.

Dans le cadre du contrôle quantique bilinéaire, cette thèse étudie la possibilité de retrouver l'Hamiltonien et/ou le moment dipolaire à l'aide de mesures d'observables pour un ensemble grand de contrôles. Si l'implémentation du contrôle fait intervenir des bruits alors les mesures prennent la forme de distributions de probabilité. Nous montrons qu'il y a toujours unicité (à des phases près) des Hamiltoniens de du moment dipolaire retrouvés. Plusieurs modèles de bruit sont étudiés: bruit discrète constant additif et multiplicatif ainsi qu'un modèle de bruit dans les phases sous forme de processus Gaussien. Les résultats théoriques sont illustrés par des implémentations numériques. / The problem of recovering the Hamiltonian and dipole moment, termed inversion, is considered in a bilinear quantum control framework. The process uses as inputs some measurable quantities (observables) for each admissible control. If the implementation of the control is noisy the data available is only in the form of probability laws of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to phase factors). Several models of noise are considered including the discrete noise model, the multiplicative amplitude noise model and a Gaussian process phase model. Both theoretical and numerical results are established.

[pt] Os objetivos deste trabalho são três: O primeiro é fazer uma revisão sucinta da teoria de sistemas lineares e de estimação de parâmetros que é usada nos problemas de estruturas mecânicas; o segundo consiste no estudo do algoritmo ERA (Eigensystem Realization Algorithm) que fornece uma realização mínima, usando os sinais de entrada (forças, momentos, impulsos) e saída (acelerações); e o terceiro objetivo é desenvolver algumas aplicações tanto numéricas quanto experimentais. Assim, é realizado um estudo dos sistemas dinâmicos determinísticos e estocásticos para fornecer alguns conhecimentos básicos. Da mesma forma, são apresentados características e propriedades dos estimadores Bayesianos,de máxima verossimilhança e de quadrados mínimos. Com esta base, o algoritmo ERA e o algoritmo alternativo chamado ERA/OKID (Observer Kalman Filter Identication) são descritos. Como o uso de um observador faz com que o problema tenha uma estrutura estatística, o que permite ao estimador trabalhar com ruído, foi estudado o comportamento de ERA e ERA/OKID aplicados numa treliça bidimensional afetada por diferentes níveis de ruído. A última parte deste trabalho corresponde as aplicações, que estão divididas em parte experimental e parte numérica: Na parte experimental, foram gerados dois conjuntos de dados a partir de dois experimentos. Em cada um deles, uma treliça tridimensional foi excitada por uma força randômica e foram medidas as acelerações em dois pontos da estrutura. Com um dos conjuntos de dados, identificou-se um modelo de espaço de estado de ordem mínima, usando ERA e ERA/OKID. Compararam-se os sinais de saída gerados pelo modelo identificado e pela estrutura real para cada um dos sinais de entrada. A idéia é obter um modelo de espaço de estado de ordem mínima sem ter que fazer uma modelagem matemática previa. Na parte numérica é feita uma aplicação do algoritmo RSS (Remote Sensor System) usando um sistema de massas, molas e amortecedores. Este algoritmo, proposto em [42] é famoso porque foi usado para identificar se ocorria algum dano na estação espacial MIR depois de cada processo de acoplamento. No processo de identicação do algoritmo RSS, foram usados ERA e ERA/OKID. / [en] The aims of this work are three: First, to make a brief review of linear system and estimation theory used in mechanical structure problems. Second, to study an algorithm called ERA (Eigensystem Realization Algorithm) that provides a minimal realization using the input (forces, moments, impulses) and output signals (accelerations). Third, to develop some numerical and experimental applications. To achieve these goals, a study of deterministic and stochastic linear systems is performed in order to provide some basic insights. In the same way, characteristics and properties of the Bayesian, maximum likelihood and least square estimators are presented. With this background, the original ERA and an alternative algorithm called ERA/OKID (Observer Kalman Filter Identification) are described. Since the use of an observer gives a system a statistical framework that allows the estimators to deal with noise, the behavior of ERA and ERA/OKID when applied to a bidimensional truss a®ected by di®erent levels of noise is studied . The last part of this work is concerned with the applications. That consist of an experimental and a numerical part: In the first part, two sets of data are generated by performing two experiments. In each one, a tridimensional truss was excited by a randomic force and the accelerations of two points of the structure were measured. With one set of data, it is found a state space model of minimal order using ERA and ERA/OKID. Comparisons are made between the output signals generated by the identified model and the real truss for each input signal. The goal is to achieve a state space model of minimal order without performing any mathematical model process. In the second part, it is performed an application of the RSS (Remote Sensor System) algorithm using a damped mass-spring system. This algorithm, proposed in [42] is famous because it was used to identify whether the MIR space station was damaged after a docking process. In the identification stage of the RSS algorithm, ERA and ERA/OKID is used. / [es] Los objetivos de este trabajo son tres: El primero es hacer una revisión sucinta de la teoría de sistemas lineales y de estimación de parámetros que se utiliza en los problemas de extructuras mecánicas; el segundo consiste en el estudio del algoritmo ERA (Eigensysten Realization Algorithm) que obtiene una realización mínima, usando señales de entrada (fuerzas, momentos, impulsos) y salida (aceleraciones); y el tercer objetivo es dearrollar algunas aplicaciones tanto numéricas como experimentales. Se realiza un estudio de los sistemas dinámicos determinísticos y estocásticos para discutir algunos conocimentos básicos. De la misma forma, son presentados características y propriedades de los estimadores Bayesianos,de máxima verosimilitud y de mínimos cuadrados. Con esta base, se describen los algoritmos ERA y el algoritmo alternativo llamado ERA/OKID (Observer Kalman Filter Identication). Como el uso de un observador hace con que el problema tenga una extructura estadística, que permite al estimador trabajar con ruido, se estudio el comportamento de ERA y ERA/OKID aplicados en una celosía bidimensional afectada por diferentes nívelesde ruido. La última parte de este trabajo corresponde a las aplicaciones, que están divididas en parte experimental y parte numérica: En la parte experimental, fueron generados dos conjuntos de datos a partir de dos experimentos. En cada un deles, se aplicó una fuerza randómica a una celosía tridimensional y posteriormente se midieron las aceleraciones en dos puntos de la extructura. Con uno de los conjuntos de datos, se identificó un modelo de espacio de estado de orden mínima, usando ERA y ERA/OKID. Se compararon los señales de salida generados por el modelo identificado y por la extructura real para cada uno de los señales de entrada. La idea es obtener un modelo de espacio de estado de orden mínima sin tener que hacer una modelaje matemática previa. En la parte numérica se aplica el algoritmo RS (Remote Sensor System) usando un sistema de masas, molas y amortecedores. Este algoritmo, proposto en [42] es famoso porque fue utilizado para identificar si ocurría algun daño en la estación espacial MIR después de cada proceso de acoplamiento. En el proceso de identicación del algoritmo RS, fueron usados ERA y ERA/OKID.

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Vienas iš svarbiausių šiuolaikinės kriptografijos uždavinių yra saugių vienkrypčių funkcijų paieška. Dabartiniai mokslininkai skiria šiam klausimui ypatingą demėsį. Šiame darbe yra nagrinėjama viena iš naujausių vienkrypčių funkcijų – matricinio laipsnio funkcija. Ši funkcija yra panaudota netiesinės algebrinės lygčių sistemos sudarymui. Pagrindinis demėsys darbe yra skirtas šios lygčių sistemos analizei bei jos praktiniam taikymui. Nustatysime ar matricinio laipsnio funkcija gali būti panaudota kriptografijoje. Taip pat nustatysime lygčių sistemos sprendinių skaičiaus priklausomybę nuo jos parametrų: matricų eilės m bei grupės Z_p parametro p. / Since the introduction of Diffie-Hellman key agreement protocol in 1976 computer technology has made a giant step forward. Nowadays there is not much time left before quantum computers will be in every home. However it was theoretically proven that discrete logarithm problem which is the basis for Diffie-Hellman protocol could be solved in polynomial time using such computers. Such possibility would make D-H protocol insecure. Thus cryptologists are searching for different ways to improve the security of the protocol by using hard problems. One of the ways to do so is to introduce secure one-way functions (OWF). In this paper a new kind of OWF called the matrix power function will be analyzed. Professor Eligijus Sakalauskas introduced this function in 2007 and later used this function to construct a Diffie-Hellman type key agreement protocol using square matrices. This protocol is not only based on matrix power function but also on commutative matrices which are defined in finite fields or rings. Thus an algebraic non-linear system of equations is formed. The security of this system will be analyzed. It will be shown that we can use matrix power function in cryptography. We will also be analyzing how does the solution of the system depend on system parameters: the order of matrices and a parameter p which defines a finite group Z_p. We will also briefly discuss the usage of this system in real life and the algebraic properties of the suggested OWF.