Functional analysis, calculus of variations and optimal control
 Responsibility
 Francis Clarke.
 Language
 English.
 Imprint
 London ; New York : Springer, c2013.
 Physical description
 xiv, 591 p. : ill. (some col.) ; 24 cm.
 Series
 Graduate texts in mathematics ; 264.
Access
Available online
 dx.doi.org SpringerLink
Math & Statistics Library

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QA320 .C53 2013

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QA320 .C53 2013
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Creators/Contributors
 Author/Creator
 Clarke, Francis, 1948
Contents/Summary
 Bibliography
 Includes bibliographical references (p. 583584) and index.
 Contents

 Functional Analysis
 Normed spaces
 Basic definitions
 Linear mappings
 The dual space
 Derivatives, tangents, and normals
 Convex sets and functions
 Properties of convex sets
 Extendedvalued functions, semicontinuity
 Convex functions
 Separation of convex sets
 Weak topologies
 Induced topologies
 The weak topology of a normed space
 The weak* topology
 Separable spaces
 Convex analysis
 Subdifferential calculus
 Conjugate functions
 Polarity
 The minimax theorem
 Banach spaces
 Completeness of normed spaces
 Perturbed minimization
 Open mappings and surjectivity
 Metric regularity
 Reflexive spaces and weak compactness
 Lebesgue spaces
 Uniform convexity and duality
 Measurable multifunctions
 Integral functionals and semicontinuity
 Weak sequential closures
 Hilbert spaces
 Basic properties
 A smooth minimization principle
 The proximal subdifferential
 Consequences of proximal density
 Additional exercises
 Optimization and Nonsmooth Analysis
 Optimization and multipliers
 The multiplier rule
 The convex case
 Convex duality
 Generalized gradients
 Definition and basic properties
 Calculus of generalized gradients
 Tangents and normals
 A nonsmooth multiplier rule
 Proximal analysis
 Proximal calculus
 Proximal geometry
 A proximal multiplier rule
 Dini and viscosity subdifferentials
 Invariance and monotonicity
 Weak invariance
 Weakly decreasing Systems
 Strong invariance
 Additional exercises
 Calculus of Variations
 The classical theory
 Necessary conditions
 Conjugate points
 Two variants of the basic problem
 Nonsmooth extremals
 The integral Euler equation
 Regularity of Lipschitz Solutions
 Sufficiency by convexity
 The Weierstrass necessary condition
 Absolutely continuous solutions
 Tonelli's theorem and the direct method
 Regularity via growth conditions
 Autonomous Lagrangians
 The multiplier rule
 A classic multiplier rule
 A modern multiplier rule
 The isoperimetric problem
 Nonsmooth Lagrangians
 The Lipschitz problem of Bolza
 Proof of Theorem "the Lipschitz problem of Bolza"
 Sufficient conditions by convexity
 Generalized TonelliMorrey conditions
 HamiltonJacobi methods
 Verification functions
 The logarithmic Sobolev inequality
 The HamiltonJacobi equation
 Proof of Theorem "Verification functions"
 Multiple integrals
 The classical context
 Lipschitz Solutions
 HilbertHaar theory
 Additional exercises
 Optimal control
 Necessary conditions
 The maximum principle
 A problem affine in the control
 Problems with variable time
 Unbounded control sets
 A hybrid maximum principle
 The extended maximum principle
 Existence and regularity
 Relaxed trajectories
 Three existence theorems
 Regularity of optimal controls
 Inductive methods
 Sufficiency by the maximum principle
 Verification functions in control
 Use of the HamiltonJacobi equation
 Diffferential inclusions
 A theorem for Lipschitz multifunctions
 Proof of the extended maximum principle
 Stratified necessary conditions
 The multiplier rule and mixed constraints
 Additional exercises
 Notes, Solutions, and hints
 References
 Index.
 Publisher's Summary
 Functional analysis owes much of its early impetus to problems that arise in the calculus of variations. In turn, the methods developed there have been applied to optimal control, an area that also requires new tools, such as nonsmooth analysis. This selfcontained textbook gives a complete course on all these topics. It is written by a leading specialist who is also a noted expositor. This book provides a thorough introduction to functional analysis and includes many novel elements as well as the standard topics. A short course on nonsmooth analysis and geometry completes the first half of the book whilst the second half concerns the calculus of variations and optimal control. The author provides a comprehensive course on these subjects, from their inception through to the present. A notable feature is the inclusion of recent, unifying developments on regularity, multiplier rules, and the Pontryagin maximum principle, which appear here for the first time in a textbook. Other major themes include existence and HamiltonJacobi methods. The many substantial examples, and the more than three hundred exercises, treat such topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality, periodic trajectories, and systems theory. They also touch lightly upon several fields of application: mechanics, economics, resources, finance, control engineering. Functional Analysis, Calculus of Variations and Optimal Control is intended to support several different courses at the firstyear or secondyear graduate level, on functional analysis, on the calculus of variations and optimal control, or on some combination. For this reason, it has been organized with customization in mind. The text also has considerable value as a reference. Besides its advanced results in the calculus of variations and optimal control, its polished presentation of certain other topics (for example convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will be appreciated by researchers in these and related fields.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2013
 Series
 Graduate texts in mathematics ; 264
 ISBN
 1447148193
 9781447148197
 9781447148203 (ebk.)