Functional analysis, calculus of variations and optimal control

Description

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. Front Matter....Pages I-XIV Front Matter....Pages 1-1 Normed Spaces....Pages 3-25 Convex sets and functions....Pages 27-46 Weak topologies....Pages 47-58 Convex analysis....Pages 59-74 Banach spaces....Pages 75-103 Lebesgue spaces....Pages 105-131 Hilbert spaces....Pages 133-155 Additional exercises for Part I....Pages 157-169 Front Matter....Pages 171-171 Optimization and multipliers....Pages 173-191 Generalized gradients....Pages 193-225 Proximal analysis....Pages 227-254 Invariance and monotonicity....Pages 255-272 Additional exercises for Part II....Pages 273-283 Front Matter....Pages 285-285 The classical theory....Pages 287-305 Nonsmooth extremals....Pages 307-318 Absolutely continuous solutions....Pages 319-334 The multiplier rule....Pages 335-346 Nonsmooth Lagrangians....Pages 347-366 Hamilton-Jacobi methods....Pages 367-390 Multiple integrals....Pages 391-414 Front Matter....Pages 285-285 Additional exercises for Part III....Pages 415-432 Front Matter....Pages 433-433 Necessary conditions....Pages 435-471 Existence and regularity....Pages 473-489 Inductive methods....Pages 491-502 Differential inclusions....Pages 503-543 Additional exercises for Part IV....Pages 545-563 Back Matter....Pages 565-591