Control Theory and Optimization I : Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

By: S.A. Vakhrameev (Translator), M.I. Zelikin

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Hardcover | 1 July 2009

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This book is devoted to the development of geometrie methods for studying and revealing geometrie aspects of the theory of differential equations with quadratie right-hand sides (Riccati-type equations), which are closely related to the calculus of variations and optimal control theory. The book contains the following three parts, to each of which aseparate book could be devoted: 1. the classieal calculus of variations and the geometrie theory of the Riccati equation (Chaps. 1-5), 2. complex Riccati equations as flows on Cartan-Siegel homogeneity da- mains (Chap. 6), and 3. the minimization problem for multiple integrals and Riccati partial dif- ferential equations (Chaps. 7 and 8). Chapters 1-4 are mainly auxiliary. To make the presentation complete and self-contained, I here review the standard facts (needed in what folIows) from the calculus of variations, Lie groups and algebras, and the geometry of Grass- mann and Lagrange-Grassmann manifolds. When choosing these facts, I pre- fer to present not the most general but the simplest assertions. Moreover, I try to organize the presentation so that it is not obscured by formal and technical details and, at the same time, is sufficiently precise. Other chapters contain my results concerning the matrix double ratio, com- plex Riccati equations, and also the Riccati partial differential equation, whieh the minimization problem for a multiple integral. arises in The book is based on a course of lectures given in the Department of Me- and Mathematics of Moscow State University during several years.

Industry Reviews

".... This book was written by a master expositor and is required reading for anyone who is intersted in pursuing a serious study of the Riccati equation. The first four chapters should be required reading for every graduate student who is thinking about studying geometric or mathematical control theory. I do not know of a better overview of the matheamtics required to do modern geometric control theory. Every control theorist should have a well-worn copy of this book on his bookshelf. ... Zelikin has written a book that will be well read for many years...."

Siam Review, Vol. 43/1, March 2001

"... The text requires good background, but will be a useful reference."

Mathematika 2002, Issue 93-94

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Non-Fiction Mathematics Calculus & Mathematical Analysis Calculus Differential Calculus & Equations Geometry Differential & Riemannian Geometry Medicine Medicine in General Optimisation Linear Programming

Introduction	p. 1
Classical Calculus of Variations	p. 5
Euler Equation	p. 5
Hamiltonian Formalism	p. 12
Theory of the Second Variation	p. 25
Riccati Equation	p. 30
Morse Index	p. 35
Jacobi Envelope Theorem	p. 38
Strong Minimum	p. 42
Poincare-Cartan Integral Invariant	p. 45
Fields of Extremals	p. 53
Riccati Equation in the Classical Calculus of Variations	p. 60
Riccati Equation as a Sufficient Condition for Positivity of the Second Variation	p. 60
Riccati Equation for a Problem with Differential Constraints	p. 62
Riccati Equation and the Grassmann Manifold	p. 68
Grassmann Manifolds of Lower Dimension	p. 72
Lie Groups and Lie Algebras	p. 80
Lie Groups: Definition and Examples	p. 80
Lie Algebras	p. 84
Lie Groups of Lower Dimension	p. 88
Adjoint Representation and Killing Form	p. 91
Semisimple Lie Groups	p. 96
Homogeneous and Symmetrical Spaces	p. 100
Totally Geodesic Submanifolds	p. 107
Grassmann Manifolds	p. 112
Three Approaches to the Description of the Grassmann Manifolds	p. 112
Lagrange-Grassmann Manifolds	p. 117
Riccati Equation as a Flow on the Manifold G[subscript n](R[superscript 2n])	p. 123
Systems Associated with a Linear System of Differential Equations	p. 126
Matrix Double Ratio	p. 130
Matrix Double Ratio on the Grassmann Manifold	p. 130
Clifford Algebras	p. 135
Totally Geodesic Submanifolds of Grassmann Manifolds	p. 137
Curves with a Scalar Double Ratio	p. 143
Fourth Harmonic as a Geodesic Symmetry	p. 147
Clifford Parallels	p. 150
Connection Between Clifford Parallels and Isoclinic Planes	p. 154
Matrix Double Ratio on the Lagrange-Grassmann Manifold	p. 155
Morse-Maslov-Arnol'd Index in the Leray-Kashivara Form	p. 158
Fourth Harmonic as an Isometry of the Lagrange-Grassmann Manifold	p. 162
Application of the Matrix Double Ratio to the Study of the Riccati Equation	p. 162
Complex Riccati Equations	p. 166
Cartan-Siegel Domains	p. 166
Klein - Poincare Upper Half-Plane and Generalized Siegel Upper Half-Plane	p. 174
Complexified Riccati Equation as a Flow on the Generalized Siegel Upper Half-Plane	p. 187
Flow on Cartan-Siegel Homogeneity Domains	p. 189
Matrix Analog of the Schwarz Differential Operator	p. 198
Higher-Dimensional Calculus of Variations	p. 208
Minimal Surfaces	p. 208
Necessary Optimality Conditions for a Multiple Integral	p. 212
Vector Bundles	p. 217
Distributions and the Frobenius Theorem	p. 219
Connection in a Linear Bundle	p. 227
Levi-Civita Connection	p. 230
Nonnegativity Conditions of the Second Variation	p. 236
Field Theory in the Weyl Form	p. 240
Caratheodory Transformation	p. 244
Field Theory in the Caratheodory Form	p. 248
On the Quadratic System of Partial Differential Equations Related to the Minimization Problem for a Multiple Integral	p. 254
Riccati Equation in the Case of the Degenerate Legendre Condition	p. 254
Reducing the Dirichlet Integral to the Integral of Its Principal Part	p. 257
Relation of the Riccati Partial Differential Equation to the Euler Equation	p. 261
Connection Defined by a Solution to the Riccati Partial Differential Equation	p. 264
Epilogue	p. 272
Appendix to the English Edition	p. 273
References	p. 276
Index	p. 282
Table of Contents provided by Blackwell. All Rights Reserved.

Control Theory and Optimization I : Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

Homogeneous Spaces and the Riccati Equation in the Calculus of Variations

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