A clear and succinct presentation of the essentials of this subject, together with some of its applications and a generous helping of interesting exercises. Following an introductory chapter with a taste of what is to come, the next three chapters constitute a course in nonsmooth analysis and identify a coherent and comprehensive approach to the subject, leading to an efficient, natural, and powerful body of theory. The whole is rounded off with a self-contained introduction to the theory of control of ordinary differential equations. The authors have incorporated a number of new results which clarify the relationships between the different schools of thought in the subject, with the aim of making nonsmooth analysis accessible to a wider audience. End-of-chapter problems offer scope for deeper understanding.
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4 of 5 people found the following review helpful:
5.0 out of 5 stars
Clarke et al.'s Nonsmooth Analysis,
This review is from: Nonsmooth Analysis and Control Theory (Graduate Texts in Mathematics) (Hardcover)
Of all the scientific or mathematical books that I have reviewed or even read, I would place this book at the position of number one (1) in excellence, creativity, genius, inspiration, intuition, and usefulness. It has inspired some of my own best research and I often cite it in presenting papers at conferences and publishing papers. In my opinion, Nonsmooth Analysis is one of the 20 main research areas in mathematics of the last 5 years (others include rare events/large deviations, solutions of Navier Stokes/Einstein field equations/Schrodinger equation, fractals/chaos/entropy, fuzzy sets/fuzzy logic/multivalued logic/other logics, semigroups/Clifford algebras/spacetime algebras,etc.). Perhaps the most astonishing finding of Clarke et al., book here and in their journal papers (and those of their colleagues), is that equations become inequalities and subset relationships when one goes from smooth physics to disconnected and sharp-bend physics. The latter types of physics may seem difficult to visualize at first, but think of what happens when ice suddenly changes phase to water, or water changes phase suddenly to vapor/steam. Or think of what happens when a runner or a racecar or a plane suddenly makes a 180 degree about-face (runners might be able to do this, but planes can only do it approximately at usual speeds). Ordinary physics and mathematics cannot handle these situations. Other examples are catastrophes, sudden strokes of good fortune, etc. You can see that these are often related to rare events, which I have reviewed elsehwhere. It turns out that the usual mathematics which involves equations becomes inequalities (less than, greater than, etc.) and subset relationships (A is inside B or is a subset of B) in the new situations. Clarke et. al. prove theorems quite rigorously in this area. If you have any hesitation in reading this book because of its mathematical content, hire a reputable consultant or tutor to translate the results into an approximation to ordinary English. If you don't, you'll miss out on opportunities to apply the results to your own area and maybe even your own daily life.
2 of 4 people found the following review helpful:
5.0 out of 5 stars
an excellent book,
By A Customer
This review is from: Nonsmooth Analysis and Control Theory (Graduate Texts in Mathematics) (Hardcover)
In general, we assume the differentiability of functions. But this is the very strict assumption and nondifferentiable phenomena are common. So if you want to drop the assumption of differentiability, I'm convinced that you have to read this book. This is one of the best books.
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Inside This Book
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Key Phrases - Statistically Improbable Phrases (SIPs):
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proximal calculus, generalized gradients, integral functionals, banach space, mean value inequality, density theorem, solvability theorem, proximal subgradient inequality, derivate calculus, proximal normal inequality, proximal subgradients, proximal subdifferential, weak invariance, strong invariance, linear growth condition, proximal normals, polygonal arcs, lower semicontinuous, multiplier rule, nonautonomous case Key Phrases - Capitalized Phrases (CAPs): (learn more)
Short Course, Standing Hypotheses, Control Theory, Special Topics, Hilbert Space, Mean Value Theorem, Chain Rule, Sum Rule, Solving Equations, Gronwall's Lemma, Trajectories of Differential Inclusions, Prove Corollary, Brouwer Fixed Point Theorem, The Gradient Formula, Fubini's Theorem, Prove Proposition, Growth Hypothesis Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
proximal calculus, generalized gradients, integral functionals, banach space, mean value inequality, density theorem, solvability theorem, proximal subgradient inequality, derivate calculus, proximal normal inequality, proximal subgradients, proximal subdifferential, weak invariance, strong invariance, linear growth condition, proximal normals, polygonal arcs, lower semicontinuous, multiplier rule, nonautonomous case Key Phrases - Capitalized Phrases (CAPs): (learn more)
Short Course, Standing Hypotheses, Control Theory, Special Topics, Hilbert Space, Mean Value Theorem, Chain Rule, Sum Rule, Solving Equations, Gronwall's Lemma, Trajectories of Differential Inclusions, Prove Corollary, Brouwer Fixed Point Theorem, The Gradient Formula, Fubini's Theorem, Prove Proposition, Growth Hypothesis Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
Citations (learn more)
100
books
cite this book:
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- Optimal Control, Stabilization and Nonsmooth Analysis (Lecture Notes in Control and Information Sciences) by Marcio S. de Queiroz on 5 pages
- Handbook of Differential Equations: Ordinary Differential Equations, Volume 2 by A. Canada on page 236
- An Introduction to Markov Processes (Graduate Texts in Mathematics) by Daniel W. Stroock in Back Matter
- Algebraic Geometry (Graduate Texts in Mathematics) by Robin Hartshorne in Front Matter
- Optima and Equilibria: An Introduction to Nonlinear Analysis (Graduate Texts in Mathematics) by Jean-Pierre Aubin in Back Matter
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