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Functional analysis, calculus of variations and optimal control / Francis Clarke.

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Author/Creator:
Clarke, Francis, 1948-
Language:
English.
Publication date:
2013
Imprint:
London ; New York : Springer, c2013.
Format:
  • Book
  • xiv, 591 p. : ill. (some col.) ; 24 cm.
Bibliography:
Includes bibliographical references (p. 583-584) 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
  • Extended-valued 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 Tonelli-Morrey conditions
  • Hamilton-Jacobi methods
  • Verification functions
  • The logarithmic Sobolev inequality
  • The Hamilton-Jacobi equation
  • Proof of Theorem "Verification functions"
  • Multiple integrals
  • The classical context
  • Lipschitz Solutions
  • Hilbert-Haar 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 Hamilton-Jacobi 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.
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 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"--P. [4] of cover.
Series:
Graduate texts in mathematics ; 264.
Subjects:
ISBN:
1447148193
9781447148197

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