Asymptotic stability is a key notion of system stability for controlled dynamical systems as it guarantees that the system trajectories are bounded in a neighborhood of a given isolated equilibrium point and converge to this equilibrium over the infinite horizon. In some applications, however, asymptotic stability is not the appropriate notion of stability. For example, for systems with a continuum of equilibria, every neighborhood of an equilibrium contains another equilibrium and a nonisolated equilibrium cannot be asymptotically stable. Alternatively, in stabilization of spacecraft dynamics via gimballed gyroscopes, it is desirable to find state- and output-feedback control laws that guarantee partial-state stability of the closed-loop system, that is, stability with respect to part of the system state. Furthermore, we may additionally require finite-time stability of the closed-loop system, that is, convergence of the system's trajectories to a Lyapunov stable equilibrium in finite time. The Hamilton-Jacobi-Bellman optimal control framework provides necessary and sufficient conditions for the existence of state-feedback controllers that minimize a given performance measure and guarantee asymptotic stability of the closed-loop system. In this research, we provide extensions of the Hamilton-Jacobi-Bellman optimal control theory to develop state-feedback control laws that minimize nonlinear-nonquadratic performance criteria and guarantee semistability, partial-state stability, finite-time stability, and finite-time partial state stability of the closed-loop system.

L'objectif de cette thèse est de développer la théorie du contrôle stochastique et ses applications en dynamique des populations. D'un point de vue théorique, nous présentons l'étude de problèmes de contrôle stochastique à horizon fini sur des processus de diffusion, de branchement non linéaire et de branchement-diffusion. Dans chacun des cas, nous raisonnons par la méthode de la programmation dynamique en veillant à démontrer soigneusement un argument de conditionnement analogue à la propriété de Markov forte pour les processus contrôlés. Le principe de la programmation dynamique nous permet alors de prouver que la fonction valeur est solution (régulière ou de viscosité) de l'équation de Hamilton-Jacobi-Bellman correspondante. Dans le cas régulier, nous identifions également un contrôle optimal markovien par un théorème de vérification. Du point de vue des applications, nous nous intéressons à la modélisation mathématique du cancer et de ses stratégies thérapeutiques. Plus précisément, nous construisons un modèle hybride de croissance de tumeur qui rend compte du rôle fondamental de l'acidité dans l'évolution de la maladie. Les cibles de la thérapie apparaissent explicitement comme paramètres du modèle afin de pouvoir l'utiliser comme support d'évaluation de stratégies thérapeutiques.

The existence of pathwise stationary solutions of this stochastic partial differential equation (SPDE, for abbreviation) is demonstrated. In Part II, a connection between certain kind of state constrained controlled Forward-Backward Stochastic Differential Equations (FBSDEs) and Hamilton-Jacobi-Bellman equations (HJB equations) are demonstrated. The special case provides a probabilistic representation of some geometric flows, including the mean curvature flows. Part II includes also a probabilistic proof of the finite time existence of the mean curvature flows.

Thèse diffusée initialement dans le cadre d'un projet pilote des Presses de l'Université de Montréal/Centre d'édition numérique UdeM (1997-2008) avec l'autorisation de l'auteur.

O ensaio de simples reconhecimento com o SPT é um dos mais utilizados para investigação do subsolo nos EUA e no Brasil. Os dados obtidos com este ensaio dinâmico, tais como o índice de resistência à penetração NSPT, classificação do solo e nível freático são usados por diferentes métodos para estimativa de capacidade de carga de fundações rasas e profundas. Para um mesmo solo, o índice NSPT pode ser diferente se diferentes equipamentos forem usados devido ao fato de que diferentes equipamentos de SPT mostram diferentes níveis de eficiência. Em vista disso, uma alternativa para estimar a eficiência do ensaio SPT é utilizando-se o ensaio de prova de carga estática sobre o amostrador. Este ensaio foi implementado para poder avaliar a eficiência do ensaio SPT. Para este fim, foram executados 13 ensaios dinâmicos e 13 ensaios estáticos no campus experimental do Laboratório de Engenharia Civil da UNESP (Bauru). Para estimar a eficiência do ensaio SPT através do ensaio dinâmico, um sistema de aquisição analógico-digital foi utilizado para obter registros de aceleração e força na seção localizada junto ao amostrador SPT. O ensaio dinâmico foi modificado para utilizar várias energias potenciais nominais do martelo e sistemas de elevação. A determinação da eficiência do ensaio através da execução da prova de carga estática foi obtida com base no princípio de Hamilton. Adicionalmente, realizando-se a comparação entre o comportamento de carregamento dinâmico e estático, pode-se obter a resistência estática do sistema amostrador-solo. Esta resistência é calculada utilizando a base de dados obtidas no campo e a aplicação de vários métodos de estimativa como: Aoki et al.(2007), De Souza (2009) e Odebrecht (2003). Os resultados da estimativa da eficiência através do ensaio de prova de carga estática sobre o amostrador mostraram-se consistentes em relação aos valores obtidos com o ensaio dinâmico. Concluiu-se que a execução do ensaio de prova de carga estática sobre, o amostrador é uma alternativa simples e econômica para obter a eficiência do SPT. / The Standard Penetration Test, SPT, is one of the most used one for subsoil information in the USA and Brazil. The data obtained in this dynamic test, such as the penetration resistance index, NSPT, the soil classification and phreatic level are used by different methods to estimate the shallow and deep foundations load capacity. For the same soil, the index NSPT can be different if different equipments are used, because different SPT equipments show different efficiency levels. Having this in mind, an alternative to estimate the essay efficiency of the SPT test is using the static load test (SLT) to the sampler. This test was implemented in order to evaluate the efficiency of SPT. To do so, 13 dynamic tests and 13 static tests were performed in the experimental campus of the State University of São Paulo at Bauru (Unesp- Bauru-Brazil). To estimate the efficiency of the SPT test through the dynamic test, a data acquisition system was used to register accelerations and forces at the located section close to the sampler SPT. The dynamic test was modified so that it would be possible to use many nominal potential energy of the hammer and elevation systems. The estimation of the efficiency made through the SLT tests was based on Hamilton\'s principle. Furthermore, by comparing the dynamic and static behavior, we can get the soil-sampler interface static resistance. This resistance is calculated by using the field data through various estimation methods such as Aoki\'s (2007), De Souza\'s (2009) and Odebrecht\'s (2003). The estimates of the efficiency obtained through SLT test were found to be consistent with those obtained through the SPT modified tests. We conclude that SLT is a simple and non-expensive alternative to obtain the efficiency of SPT equipment.

In this dissertation, certain problems of stochastic optimal control and relevant analysis of random vibrations are considered. Dynamic Programming approach is used to find an optimal control law for a linear single-degree-of-freedom system subjected to Gaussian white-noise excitation. To minimize a system's mean response energy, a bounded in magnitude control force is applied. This approach reduces the problem of finding the optimal control law to a problem of finding a solution to the Hamilton-Jacobi-Bellman (HJB) partial differential equation. A solution to this partial differential equation (PDE) is obtained by developed 'hybrid' solution method. The application of bounded in magnitude control law will always introduce a certain type of nonlinearity into the system's stochastic equation of motion. These systems may be analyzed by the Energy Balance method, which introduced and developed in this dissertation. Comparison of analytical results obtained by the Energy Balance method and by stochastic averaging method with numerical results is provided. The comparison of results indicates that the Energy Balance method is more accurate than the well-known stochastic averaging method.

O objetivo desta tese é investigar aspectos práticos que facilitem a aplicação da teoria de controle H1 não linear em projetos de sistemas de controle. A primeira contribuição deste trabalho é a proposta do uso de funções ponderação com dinâmica no projeto de controladores H1 não lineares. Essas funções são usadas no projeto de controladores H1 lineares para rejeição de perturbações, ruídos, atenuação de erro de rastreamento, dentre outras especificações. O maior obstáculo para aplicação prática da teoria de controle H1 não linear é a dificuldade para resolver simultaneamente as duas equações de Hamilton-Jacobi-Isaacs relacionadas ao problema de realimentação de estados e injeção da saída. Não há métodos sistematicos para resolver essas duas equações diferenciais parciais não lineares, equivalentes µas equações de Riccati da teoria de controle H1 linear. A segunda contribuição desta tese é um método para obter a injeção da saída transformando a equação de Hamilton-Jacobi-Isaacs em uma sequencia de equações diferenciais parciais lineares, que são resolvidas usando o método de Galerkin. Controladores H1 não lineares para um sistema de levitação magnética são obtidos usando o método clássico de expansão em série de Taylor e o método de proposto para comparação. / The purpose of this thesis is to investigate practical aspects to facilitate the ap- plication of nonlinear H1 theory in control systems design. Firstly, it is shown that dynamic weighting functions can be used to improve the performance and robustness of the nonlinear H1 controller such as in the design of H1 controllers for linear plants. The biggest bottleneck to the practical applications of nonlinear H1 control theory has been the di±culty in solving the Hamilton-Jacobi-Isaacs equations associated with the design of a state feedback and an output injection gain. There is no systematic numerical approach for solving this ¯rst order, nonlinear partial di®erential equations, which reduces to Riccati equations in the linear context. In this work, successive ap- proximation and Galerkin approximation methods are combined to derive an algorithm that produces an output injection gain. Design of nonlinear H1 controllers obtained by the well established Taylor approximation and by the proposed Galerkin approxi- mation method applied to a magnetic levitation system are presented for comparison purposes.

This thesis focuses on optimization on a generic objective function based on reflected Brownian motion (RBM). We investigate in several approaches including the partial differential equation approach where we write our objective function in terms of a Hamilton-Jacobi-Bellman equation using the dynamic programming principle and the gradient descent approach where we use two different gradient estimators. We provide extensive numerical results with the gradient descent approach and we discuss the difficulties and future study opportunities for this problem.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

Although the ‘death of God’ theology attracted considerable attention during the 1960s, in recent decades it has fallen into neglect. Nonetheless, the issues raised by the ‘death of God’ theology were important ones and it remains an interesting question whether the ‘death of God’ theologians were able to make substantial contributions to them. This thesis re-examines the work of the ‘death of God’ theologians. It argues that the popular view – that the ‘death of God’ theology represented a common tendency, or movement, towards atheism among certain prominent American Protestant theologians – is mistaken. Through a series of detailed studies of Thomas J.J. Altizer (chapters 3 and 4), William Hamilton (Chapter 5), Paul van Buren (Chapter 6), and Harvey Cox (Chapter 7), the thesis shows not only that the significance of the ‘death of God’ theologians has been widely misinterpreted, but that their work contains a number of features which have been under-emphasised or even overlooked. The aim of the thesis is to provide a more balanced contemporary reading of their work. The work of Altizer receives special attention and a case is made for the view that he should be read as a Protestant mystic of a peculiar sort.