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Convex Functional Analysis : Systems & Control: Foundations & Applications - Andrew J. Kurdila

Convex Functional Analysis

Systems & Control: Foundations & Applications


Published: 23rd May 2005
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The book is dedicated to functional and convex analysis techniques for the study of recent problems in optimal control and mechanics. It provides a rigorous foundation for the application of variational principles. Through the unifying approach of convex analysis, the text discusses four general types of applications: contact problems in mechanics, control of distributed parameter systems, control of incompressible flow, and identification problems in mechanics. It also considers extensions of convex analysis with application to the homogenization of steady state heat conduction equations, approximation theory in identification problems, and nonconvex variational problems in mechanics. Special emphasis is given to techniques for solving convex optimization problems subject to constraints.

"The book provides not only the bare minimum of the theory required to understand the principles of functional, convex and set-valued analysis, but also a concise summary of definitions and theorems so that the text is self-contained." -Zentralblatt MATH

"This book...is intended as a textbook for classes in variational calculus and applied functional analysis for graduate students in engineering and applied mathematics.... The book is the result of the courses taught by the authors through the years and tries to address the different backgrounds [that] different types of students come in with when taking these courses. The topics covered provide both a treatment of the theoretical aspects of functional analysis, as well as their applications to variational calculus, mechanics and control theory.... I am sure that many instructors seeking a textbook for a course on the applications of functional analysis for engineering or applied mathematics students will find this text very useful." -MAA Reviews

Prefacep. xi
Classical Abstract Spaces in Functional Analysis
Introduction and Notationp. 1
Topological Spacesp. 5
Convergence in Topological Spacesp. 13
Continuity of Functions on Topological Spacesp. 15
Weak Topologyp. 17
Compactness of Sets in Topological Spacesp. 19
Metric Spacesp. 21
Convergence and Continuity in Metric Spacesp. 21
Closed and Dense Sets in Metric Spacesp. 23
Complete Metric Spacesp. 23
The Baire Category Theoremp. 25
Compactness of Sets in Metric Spacesp. 27
Equicontinuous Functions on Metric Spacesp. 30
The Arzela-Ascoli Theoremp. 33
Holder's and Minkowski's Inequalitiesp. 35
Vector Spacesp. 41
Normed Vector Spacesp. 45
Basic Definitionsp. 45
Examples of Normed Vector Spacesp. 46
Space of Lebesgue Measurable Functionsp. 52
Introduction to Measure Theoryp. 52
Lebesgue Integralp. 54
Measurable Functionsp. 57
Hilbert Spacesp. 58
Linear Functionals and Linear Operators
Fundamental Theorems of Analysisp. 65
Hahn-Banach Theoremp. 65
Uniform Boundedness Theoremp. 69
The Open Mapping Theoremp. 71
Dual Spacesp. 75
The Weak Topologyp. 79
The Weak* Topologyp. 80
Signed Measures and Topologyp. 88
Riesz's Representation Theoremp. 91
Space of Lebesgue Measurable Functionsp. 91
Hilbert Spacesp. 94
Closed Operators on Hilbert Spacesp. 95
Adjoint Operatorsp. 97
Gelfand Triplesp. 103
Bilinear Mappingsp. 106
Common Function Spaces in Applications
The L[superscript p] Spacesp. 111
Sobolev Spacesp. 113
Distributional Derivativesp. 114
Sobolev Spaces, Integer Orderp. 117
Sobolev Spaces, Fractional Orderp. 118
Trace Theoremsp. 122
The Poincare Inequalityp. 123
Banach Space Valued Functionsp. 126
Bochner Integralsp. 126
The Space L[superscript p]((0,T), X)p. 131
The Space W[superscript p,q]((0,T), X)p. 133
Differential Calculus in Normed Vector Spaces
Differentiability of Functionalsp. 137
Gateaux Differentiabilityp. 137
Frechet Differentiabilityp. 139
Classical Examples of Differentiable Operatorsp. 143
Minimization of Functionals
The Weierstrass Theoremp. 161
Elementary Calculusp. 163
Minimization of Differentiable Functionalsp. 165
Equality Constrained Smooth Functionalsp. 166
Frechet Differentiable Implicit Functionalsp. 171
Convex Functionals
Characterization of Convexityp. 177
Gateaux Differentiable Convex Functioalsp. 180
Convex Programming in R[superscript n]p. 183
Ordered Vector Spacesp. 188
Positive Cones, Negative Cones, and Orderingsp. 189
Orderings on Sobolev Spacesp. 191
Convex Programming in Ordered Vector Spacesp. 193
Gateaux Differentiable Functionals on Ordered Vector Spacesp. 199
Lower Semicontinuous Functionals
Characterization of Lower Semicontinuityp. 205
Lower Semicontinuous Functionals and Convexityp. 208
Banach Theorem for Lower Semicontinuous Functionalsp. 208
Gateaux Differentiabilityp. 210
Lower Semicontinuity in Weak Topologiesp. 210
The Generalized Weierstrass Theoremp. 212
Compactness in Weak Topologiesp. 213
Bounded Constraint Setsp. 215
Unbounded Constraint Setsp. 215
Constraint Sets on Ordered Vector Spacesp. 217
Referencesp. 221
Indexp. 223
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783764321987
ISBN-10: 3764321989
Series: Systems & Control: Foundations & Applications
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 228
Published: 23rd May 2005
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.5 x 15.5  x 1.27
Weight (kg): 1.16