The numerical treatment of linear-quadratic regulator problems for parabolic partial differential equations (PDEs) on infinite time horizons requires the solution of large scale algebraic Riccati equations (ARE). The Newton-ADI iteration is an efficient numerical method for this task. It includes the solution of a Lyapunov equation by the alternating directions implicit (ADI) algorithm in each iteration step. On finite time intervals the solution of a large scale differential Riccati equation is required. This can be solved by a backward differentiation formula (BDF) method, which needs to solve an ARE in each time step. Here, we study the selection of shift parameters for the ADI method. This leads to a rational min-max-problem which has been considered by many authors. Since knowledge about the complete complex spectrum is crucial for computing the optimal solution, this is infeasible for the large scale systems arising from finite element discretization of PDEs. Therefore several alternatives for computing suboptimal parameters are discussed and compared for numerical examples.

Neste trabalho estudamos as equações de Riccati para a filtragem de sistemas lineares com saltos Markovianos a tempo discreto. Obtemos uma condição geral para estabilidade do filtro ótimo obtido pela equação algébrica de filtragem, e que também é válida para que não haja multiplicidade de soluções. Revisitamos também a questão da existência, chegando a uma condição em termos da sequência de ganhos de um observador de Luenberger. Estes resultados usaram cadeias de Markov em escala reversa de tempo, inspirando a explorar a dualidade entre filtragem e controle em sistemas com reversão na cadeia, chegando a uma relação simples de dualidade. / In this work, we studied Riccati equations for filtering Markovian jump linear systems in discrete time. We found a general condition for the stability of the optimal filter obtained via the coupled algebraic Riccati equation, and it is also valid for uniqueness of solutions. We revisit the topic of existence of solutions of the Riccati and obtain a condition in terms of the sequence of gains of a Luenberger observer. These results used Markov chains in reverse time scale, inspiring us to explore the duality between filtering and control in systems with chain reversion, arriving at a simple relation of duality.

by T. Pappas, A.J. Laub, N.R. Sandell, Jr. / Bibliography: leaf 38. / "May 1979." / Contract ERDA-E(49-18)-2087 Contract No. DAAG29-79-C-0031

This thesis is intended to be a survey on the comparison theorems and oscillation theorems for second order linear differential equations. We shall discuss four comparison theorems in detail: Sturm-Picone, Levin, Reid and Leighton comparison theorems. For oscillation properties, we shall study Hille-Kneser theorems, and Wintner and Leighton oscillation criteria, which involves analysis of a Riccati equation. In 1969, J.S.W. Wong had some deep results about oscillatory and nonoscillatory differential equations. We shall explain these results and some of the examples in detail. This survey is mainly based on the monographs of Swanson [12], and Deng [20], plus a paper of Wong [18]. In some places, we give simplifications and extensions.

Diss.--Math.--Chemnitz-Zwickau--Technische Universität, 1997. / Résumé en allemand. Bibliogr. p. 199-212.

Mes activités de recherche s'inscrivent principalement dans les deux thématiques suivantes : l'étude des équations généralisées de Riccati et l'étude des systèmes commutés à temps discret. Ce mémoire s'attache à présenter ces activités de recherche, de manière non exhaustive, mais surtout le fil conducteur et la cohérence qui les lient. En conséquence le document est organisé de la manière suivante: il est composé, de prolégomènes et de trois chapitres. Le chapitre 1 présente mes contributions de recherche consacrées aux équations algébriques généralisées de Riccati. Le chapitre 2 s'intéresse aux résultats obtenus à propos des systèmes commutés. Le chapitre 3 développe les perspectives de recherches qui me tiennent à cœur et constitueront le squelette de mes activités de recherche pour les quelques années à venir.

Thesis (M.S.)--Missouri University of Science and Technology, 2008. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed April 14, 2008) Includes bibliographical references (p. 119-121).

Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 1983.

Univ., Diss.--Bremen, 1995.

by Norman August Lehtomaki. / Bibliography: p. 161-163. / Originally presented as the author's thesis, (M.S.) in the M.I.T. Dept. of Electrical Engineering and Computer Science, 1978. / Prepared under Dept. of Energy, Division of Electric Energy Systems Grant ERDA-E(49-18)-2087.