非光滑分析和控制論

前言

　　Pardon me for writing such a long letter; I had not the time to write a short one.　　——Lord Chesterfield　　Nonsmooth analysis refers to differential analysis in the absence of differentiability. It can be regarded as a subfield of that vast subject known as nonlinear analysis. While nonsmooth analysis has classical roots (we claim to have traced its lineage hack to Dini)， it is only in the last decades that the subject has grown rapidly. To the point， in fact， that further development has sometimes appeared in danger of being stymied， due to the plethora of definitions and unclearly related theories.　　One reason for the growth of the subject has been， without a doubt， the recognition that nondifferentiable phenomena are more widespread， and play a more important role， than had been thought. Philosophically at least， this is in keeping with the coming to the fore of several other types of irregular and nonlinear behavior： catastrophes， fractals， and chaos.　　In recent years， nonsmooth analysis has come to play a role in functional analysis， optimization， optimal design， mechanics and plasticity， differentim equations (as in the theory of viscosity solutions)， control theory， and， increasingly， in analysis generally (critical point theory， inequalities， fixed point theory， variational methods ...). In the long run， we expect its methods and basic constructs to be viewed as a natural part of differential analysis.

內(nèi)容概要

Nonsmooth analysis refers to differential analysis in the absence of differentiability. It can be regarded as a subfield of that vast subject known as nonlinear analysis. While nonsmooth analysis has classical roots (we claim to have traced its lineage hack to Dini), it is only in the last decades that the subject has grown rapidly. To the point, in fact, that further development has sometimes appeared in danger of being stymied, due to the plethora of definitions and unclearly related theories.

書籍目錄

PrefaceList of Figures0 Introduction　1 Analysis Without Lineaxization　2 Flow-Invariant Sets　3 Optimization　4 Control Theory　5 Notation1 Proximal Calculus in Hilbert Space　1 Closest Points and Proximal Normals　2 Proximal Subgradients　3 The Density Theorem　4 Minimization Principles　5 Quadratic Inf-Convolutions　6 The Distance Function　7 Lipschitz Functions　8 The Sum Rule　9 The Chain Rule　10 Limiting Calculus　11 Problems on Chapter 12 Generalized Gradients in Banach Space　1 Definition and Basic Properties　2 Basic Calculus　3 Relation to Derivatives　4 Convex and Regular Functions　5 Tangents and Normals　6 Relationship to Proximal Analysis　7 The Bouligand Tangent Cone and Regular Sets　8 The Gradient Formula in Finite Dimensions　9 Problems on Chapter 23 Special Topics　1 Constrained Optimization and Value Functions　2 The Mean Value Inequality　3 Solving Equations　4 Derivate Calculus and Rademacher's Theorem　5 Sets in L2 and Integral b-~nctionals　6 Tangents and Interiors　7 Problems on Chapter 34 A Short Course in Control Theory　1 Trajectories of DiffercntiM Inclusions　2 Weak Invariance　3 Lipschitz Dependence and Strong Invariance　4 Equilibria　5 Lyapounov Theory and Stabilization　6 Monotonicity and Attainability　7 The Hamilton Jacobi Equation and Viscosity Solutions 　8 Feedback Synthesis from Semisolutions　9 Necessary Conditions for Optimal Control　10 Normality and Controllability　11 Problems on Chapter 4Notes and CommentsList of NotationBibliographyIndex

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