Functional analysis, calculus of variations and optimal control

Medientyp: Buch

Titel: Functional analysis, calculus of variations and optimal control

Enthält: Functional Analysis, Calculus of Variations and Optimal Control; Preface; Contents; Part I Functional Analysis; 1 Normed Spaces; 1.1 Basic definitions; 1.2 Linear mappings; 1.3 The dual space; 1.4 Derivatives, tangents, and normals; 2 Convex sets and functions; 2.1 Properties of convex sets; 2.2 Extended-valued functions, semicontinuity; 2.3 Convex functions; 2.4 Separation of convex sets; 3 Weak topologies; 3.1 Induced topologies; 3.2 The weak topology of a normed space; 3.3 The weak* topology; 3.4 Separable spaces; 4 Convex analysis; 4.1 Subdifferential calculus; 4.2 Conjugate functions
4.3 Polarity4.4 The minimax theorem; 5 Banach spaces; 5.1 Completeness of normed spaces; 5.2 Perturbed minimization; 5.3 Open mappings and surjectivity; 5.4 Metric regularity; 5.5 Reflexive spaces and weak compactness; 6 Lebesgue spaces; 6.1 Uniform convexity and duality; 6.2 Measurable multifunctions; 6.3 Integral functionals and semicontinuity; 6.4 Weak sequential closures; 7 Hilbert spaces; 7.1 Basic properties; 7.2 A smooth minimization principle; 7.3 The proximal subdifferential; 7.4 Consequences of proximal density; 8 Additional exercises for Part I
Part II Optimization and Nonsmooth Analysis9 Optimization and multipliers; 9.1 The multiplier rule; 9.2 The convex case; 9.3 Convex duality; 10 Generalized gradients; 10.1 Definition and basic properties; 10.2 Calculus of generalized gradients; 10.3 Tangents and normals; 10.4 A nonsmooth multiplier rule; 11 Proximal analysis; 11.1 Proximal calculus; 11.2 Proximal geometry; 11.3 A proximal multiplier rule; 11.4 Dini and viscosity subdifferentials; 12 Invariance and monotonicity; 12.1 Weak invariance; 12.2 Weakly decreasing systems; 12.3 Strong invariance; 13 Additional exercises for Part II
Part III Calculus of Variations14 The classical theory; 14.1 Necessary conditions; 14.2 Conjugate points; 14.3 Two variants of the basic problem; 15 Nonsmooth extremals; 15.1 The integral Euler equation; 15.2 Regularity of Lipschitz solutions; 15.3 Sufficiency by convexity; 15.4 The Weierstrass necessary condition; 16 Absolutely continuous solutions; 16.1 Tonelli's theorem and the direct method; 16.2 Regularity via growth conditions; 16.3 Autonomous Lagrangians; 17 The multiplier rule; 17.1 A classic multiplier rule; 17.2 A modern multiplier rule; 17.3 The isoperimetric problem
18 Nonsmooth Lagrangians18.1 The Lipschitz problem of Bolza; 18.2 Proof of Theorem 18.1; 18.3 Sufficient conditions by convexity; 18.4 Generalized Tonelli-Morrey conditions; 19 Hamilton-Jacobi methods; 19.1 Verification functions; 19.2 The logarithmic Sobolev inequality; 19.3 The Hamilton-Jacobi equation; 19.4 Proof of Theorem 19.11; 20 Multiple integrals; 20.1 The classical context; 20.2 Lipschitz solutions; 20.3 Hilbert-Haar theory; 20.4 Solutions in Sobolev space; 21 Additional exercises for Part III; Part IV Optimal Control; 22 Necessary conditions; 22.1 The maximum principle
22.2 A problem affine in the control
Normed Spaces -- Convex sets and functions -- Weak topologies -- Convex analysis -- Banach spaces -- Lebesgue spaces -- Hilbert spaces -- Additional exercises for Part I -- Optimization and multipliers -- Generalized gradients -- Proximal analysis -- Invariance and monotonicity -- Additional exercises for Part II -- The classical theory -- Nonsmooth extremals -- Absolutely continuous solutions -- The multiplier rule -- Nonsmooth Lagrangians -- Hamilton-Jacobi methods -- Additional exercises for Part III -- Multiple integrals -- Necessary conditions -- Existence and regularity -- Inductive methods -- Differential inclusions -- Additional exercises for Part IV.

Beteiligte: Clarke, Francis H. [VerfasserIn]

Erschienen: New York; Heidelberg; New York; Dordrecht: Springer, [2013]

Umfang: xiv, 591 Seiten; Diagramme

Sprache: Englisch

DOI: 10.1007/978-1-4471-4820-3

ISBN: 9781447148197

Identifikator:

RVK-Notation: SI 990 : Sonstige (CSN + Bandzählung)
SK 600 : Funktionalanalysis

Schlagwörter: Funktionalanalysis > Variationsrechnung > Optimale Kontrolle

Entstehung:

Anmerkungen: Literaturverzeichnis: Seiten: 583-584

Beschreibung: 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 self-contained 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 Hamilton-Jacobi 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 first-year or second-year 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

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