An isospectral algorithm is one which manipulates a matrix without changing its spectrum. In this thesis we study three interrelated examples of isospectral algorithms, all pertaining to Toeplitz matrices in some fashion, and one directly involving orthogonal polynomials. The first set of algorithms we study come from discretising a continuous isospectral flow designed to converge to a symmetric Toeplitz matrix with prescribed eigenvalues. We analyse constrained, isospectral gradient flow approaches and an isospectral flow studied by Chu in 1993. The second set of algorithms compute the spectral measure of a Jacobi operator, which is the weight function for the associated orthogonal polynomials and can include a singular part. The connection coefficients matrix, which converts between different bases of orthogonal polynomials, is shown to be a useful new tool in the spectral theory of Jacobi operators. When the Jacobi operator is a finite rank perturbation of Toeplitz, here called pert-Toeplitz, the connection coefficients matrix produces an explicit, computable formula for the spectral measure. Generalisation to trace class perturbations is also considered. The third algorithm is the infinite dimensional QL algorithm. In contrast to the finite dimensional case in which the QL and QR algorithms are equivalent, we find that the QL factorisations do not always exist, but that it is possible, at least in the case of pert-Toeplitz Jacobi operators, to implement shifts to generate rapid convergence of the top left entry to an eigenvalue. A fascinating novelty here is that the infinite dimensional matrices are computed in their entirety and stored in tailor made data structures. Lastly, the connection coefficients matrix and the orthogonal transformations computed in the QL iterations can be combined to transform these pert-Toeplitz Jacobi operators isospectrally to a canonical form. This allows us to implement a functional calculus for pert-Toeplitz Jacobi operators.

Cette thèse porte sur la construction de nouveaux exemples de surfaces minimales dérivées de la famille de surfaces de Costa-Hoffman-Meeks. Il s'agit d'une famille de surfaces minimales complètes plongées avec trois bouts et genre k > 0. Soit M_k la surface de Costa_Hoffman_Meeks de genre k. Dans le chapitre 1, j'ai démontré que M_k est non dégénérée pour k > 37. J'ai donc étendu les résultats de S. Nayatani qui assuraient que la surface M_k est non dégénérée seulement pour k=1,...,37. Ce résultat permet de montrer dans les chapitres 2 et 3 l'existence de nouveaux exemples de surfaces minimales de genre g arbitraire à l'aide d'une procédure de collage d'autres surfaces déjà connues (parmi lesquelles y figure la surface M_k). Sans ceci, ces résultats ne seraient valables que pour k < 38. En particulier dans le chapitre 2, j'ai démontré l'existence, dans H^2 x R, (H^2 étant le plan hyperbolique) d'une famille de surfaces minimales plongées inspirées de M_k, pour tout k > 0. Ce résultat peut être censé un cas particulier d'un théorème générale de désingularisation de l'intersection de deux surfaces minimales annoncé par N. Kapouleas et jamais publié. Le chapitre 3 est consacré à la construction de trois familles de surfaces minimales simplement périodiques plongées dans R^3 dont le quotient a genre arbitraire. Les résultats présentés dans ce chapitre (obtenus en collaborations avec L. Hauswirth et M. Rodríguez) généralisent plusieurs anciennes constructions / This thesis is devoted to the construction of new examples of minimal surfaces derived from the family of surfaces if Costa-Hoffman-Meeks. Surfaces in this family are complete embedded with 3 ends and genus k > 0. Let M_k denote the surface of Costa-Hoffman-Meeks of genus k. In chapter 1 I showed M_k is non degenerate for k > 37. So I extended the results of S. Nayatani which insured M_k is non degenerate only for k=1,...,37. That allows to prove in chapters 2 and 3 the existence of new examples of minimal surfaces by a gluing procedure involving already known surfaces (among which figures M_k). Without it theses results would hold only for k < 38. In particular in chapter 2 I showed the existence in H^2 x R (where H^2 denotes the hyperbolic plane) of a family of surfaces inspired to M_k, for all k > 0, which are complete and embedded. This result can be considered as a particular case of a general theorem of desingularization of the intersection of two minimal surfaces announced by N. Kapouleas and never published. Chapter 3 is devoted to the construction of 3 families of singly periodic minimal surfaces, embedded in R^3, whose quotient has an arbitrary value of the genus. The results showed in this chapter (obtained in collaboration with L. Hauswirth and M. Rodríguez) generalize many previous constructions

Large scale linear time invariant (LTI) systems arise in many physical and technical fields. An approximation, e.g. with model order reduction techniques, of this large systems is crucial for a cost efficient simulation. In this thesis we focus on a model order reduction method based on modal approximation, where the LTI system is projected onto the left and right eigenspaces corresponding to the dominant poles of the system. These dominant poles are related to the most dominant parts of the residue expansion of the transfer function and usually form a small subset of the eigenvalues of the system matrices. The computation of this dominant poles can be a formidable task, since they can lie anywhere inside the spectrum and the corresponding left eigenvectors have to be approximated as well. We investigate the subspace accelerated dominant pole algorithm and the two-sided and alternating Jacobi-Davidson method for this modal truncation approach. These methods can be seen as subspace accelerated versions of certain Rayleigh quotient iterations. Several strategies that admit an efficient computation of several dominant poles of single-input single-output LTI systems are examined. Since dominant poles can lie in the interior of the spectrum, we discuss also harmonic subspace extraction approaches which might improve the convergence of the methods. Extentions of the modal approximation approach and the applied eigenvalue solvers to multi-input multi-output are also examined. The discussed eigenvalue algorithms and the model order reduction approach will be tested for several practically relevant LTI systems.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24383号 / 理博第4882号 / 新制||理||1699(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 葉廣 和夫, 教授 加藤 毅, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Optimal control problems can in many cases become complicated and difficult to solve. One particular class of difficult control problems to solve are singular control problems. Standard methods for solving optimal control are discussed showing why those methods are difficult to apply to singular control problems. Then standard methods for solving singular control problems are discussed including why the standard methods can be difficult and often impossible to apply without having to resort to numerical techniques. Finally, an alternative method to solving a class of singular optimal control problems is given for a specific class of problems.

The traffic signal affects the life of virtually everyone every day. The effectiveness of signal systems can reduce the incidence of delays, stops, fuel consumption, emission of pollutants, and accidents. The problems related to rapid growth in traffic congestion call for more effective traffic signalization using robust feedback control methodology. Online traffic-responsive signalization is based on real-time traffic conditions and selects cycle, split, phase, and offset for the intersection according to detector data. A robust traffic feedback control begins with assembling traffic demands, traffic facility supply, and feedback control law for the existing traffic operating environment. This information serves the input to the traffic control process which in turn provides an output in terms of the desired performance under varying conditions. Traffic signalization belongs to a class of hybrid systems since the differential equations model the continuous behavior of the traffic flow dynamics and finite-state machines model the discrete state changes of the controller. A complicating aspect, due to the state-space constraint that queue lengths are necessarily nonnegative, is that the continuous-time system dynamics is actually the projection of a smooth system of ordinary differential equations. This also leads to discontinuities in the boundary dynamics of a sort common in queueing problems. The project is concerned with the design of a feedback controller to minimize accumulated queue lengths in the presence of unknown inflow disturbances at an isolated intersection and a traffic network with some signalized intersections. A dynamical system has finite L₂-gain if it is dissipative in some sense. Therefore, the H<SUB>infinity</SUB>-control problem turns to designing a controller such that the resulting closed loop system is dissipative, and correspondingly there exists a storage function. The major contributions of this thesis include 1) to propose state space models for both isolated multi-phase intersections and a class of queueing networks; 2) to formulate H<SUB>infinity</SUB> problems for the control systems with persistent disturbances; 3) to present the projection dynamics aspects of the problem to account for the constraints on the state variables; 4) formally to study this problem as a hybrid system; 5) to derive traffic-actuated feedback control laws for the multi-phase intersections. Though we have mathematically presented a robust feedback solution for the traffic signalization, there still remains some distance before the physical implementation. A robust adaptive control is an interesting research area for the future traffic signalization. / Ph. D.

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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.