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Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 103-109).

In current economic research, there are many problems that are difficult to solve without powerful computers, unique software, or novel approaches. I wrote this thesis because I believe that a powerful solution technique known as the Riccati Iteration is such a novel approach, and can be applied to complex problems that would otherwise be infeasible to solve. This thesis will demonstrate the power of the Riccati iteration by employing the Riccati iteration with spreadsheet software to solve a difficult dynamic optimization problem - a capital replacement problem posed by Lotfi where multiple objectives have been identified. The Riccati iteration will be shown to be the most practicable method for solving this problem, especially when compared to the Lagrange and Least-Squares solution methods. It is hoped that the demonstration in this thesis is so compelling that others may consider using the Riccati approach in their own research. / Master of Arts

It is often desired to have control over a process or a physical system, to cause it to behave optimally. Optimal control theory deals with analyzing and finding solutions for optimal control for a system that can be represented by a set of differential equations. This thesis examines such a system in the form of a set of matrix differential equations known as a continuous linear time-invariant system. Conditions on the system, such as linearity, allow one to find an explicit closed form finite solution that can be more efficiently computed compared to other known types of solutions. This is done by optimizing a quadratic cost function. The optimization leads to solving a Riccati equation. Conditions are discussed for which solutions are possible. In particular, we will obtain a solution for a stable and controllable system. Numerical examples are given for a simple system with 2x2 matrix coefficients.

We introduce the notion of differential sequences generated by generators of sequences. We discuss the Riccati sequence in terms of symmetry analysis, singularity analysis and identification of the complete symmetry group for each member of the sequence. We provide their invariants and first integrals. We propose a generalisation of the Riccati sequence and investigate its properties in terms of singularity analysis. We find that the coefficients of the leading-order terms and the resonances obey certain structural rules. We also demonstrate the uniqueness of the Riccati sequence up to an equivalence class. We discuss the properties of the differential sequence based upon the equation ww''−2w12 = 0 in terms of symmetry and singularity analyses. The alternate sequence is also discussed. When we analyse the generalised equation ww'' − (1 − c)w12 = 0, we find that the symmetry properties of the generalised sequence are the same as for the original sequence and that the singularity properties are similar. Finally we discuss the Emden-Fowler sequence in terms of its singularity and symmetry properties. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2009.

by Alan J. Laub. / Bibliography: p. 44-46. / Research supported by Contract ERDA-E(49-18)-2087.

Orientador: Celso Pascoli Bottura / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-07-25T16:26:46Z (GMT). No. of bitstreams: 1 DelRealTamariz_Annabell_M.pdf: 3597782 bytes, checksum: e6939ad8e71833c059fabea6a51ca5c6 (MD5) Previous issue date: 1999 / Resumo: Uma proposta de metodologia para a solução da Equação Algébrica de Riccati (EAR) em formas Seqüencial e Paralela e Distribuída é apresentada. O método modifica e propõe uma paralelização do Método de Schur. Transformações de Similaridade Elementares Estabilizadas (TSEE) são utilizadas para transformar a matriz simplética/Hamiltoniana, em uma forma simples. Neste trabalho fazemos uma implementação seqüencial do algoritmo proposto para matrizes densas, bem condicionadas e propomos uma implementação paralela do algoritimo num sistema com memória distribuída e estratégia de paralelização síncrona numa rede de estações de trabalho / Abstract: A proposal of methodology for solving the Algebraic Riccati Equation in Sequential and Parallel and Distriuted forms is presented. The method modifies and proposes a parallelization for the Schur Method. To transform the simpletyc/Hamiltonian matrix in a simple form, Elementary Stabilized Similarity Transformations are utilized. In this work a sequential implementation of the proposed algorithm for dense, well conditioned matrices is made and a parallel implementation on a distributed memory system with a synchronous parallelization strategy over a workstation network is proposed. / Mestrado / Mestre em Engenharia Elétrica

A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

This is the second part of a collection of benchmark examples for the numerical solution of algebraic Riccati equations. After presenting examples for the continuous-time case in Part I, our concern in this paper is discrete-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods.

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.