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.