Tangency, Flow Invariance for Differential Equations, and Optimization Problems

by Pavel; Nicolae H.

• ISBN13: 9780824773410
• ISBN10: 0824773411
• Format: Hardcover
• Copyright: 1999-04-14
• Publisher: CRC Press

Summary

Discusses basic results of the flow invariance of closed or locally closed set S of a Banach space (or manifold) with respect to differential equations.

Table of Contents

Preface	v
Tangent Vectors to Closed Sets	1	(24)
Tangency in normed spaces	1	(12)
Tangency in Banach manifolds	13	(12)
Flow-invariant Sets	25	(48)
Flow-invariant sets with respect to differential equations. Nagumo-Brezis' theorem	25	(18)
Generation of semigroups on closed subsets	43	(11)
Flow-invariance with respect to vector fields on differentiable manifolds	54	(5)
Applications	59	(14)
Integral surfaces of the first order PDE	59	(5)
Invariant parallelepipeds and quadrants. Component-wise positive solutions	64	(3)
Theorems of Brouwer and Miranda in terms of Bouligand-Nagumo fields. Zeros of tangential vector field	67	(6)
Second Order differential Equations and Flow-invariance	73	(48)
Flow-invariant sets with respect to a second order differential equation in Banach spaces	73	(13)
Extension to Banach manifolds	86	(13)
Applications to flight mechanics (orbital motions)	99	(22)
Flow-invariant sets with respect to Semilinear Differential Equation y' = Ay + f (t, y)	121	(86)
S-fields on a closed set	121	(4)
Flow-invariant sets with respect to semilinear evolution equation u' = Au + f(t, u)	125	(17)
Semilinear equations with dissipative perturbations	142	(23)
The extension of the mild solutions. Global existence	165	(6)
Differentiable semigroups and evolution operators on Banach manifolds and quasi-flow-invariant sets	171	(22)
Semigroups	171	(15)
Evolution operators	186	(7)
Applications of S-fields to PDE	193	(14)
A Transversality approach to Flow-invariance	207	(20)
Transversality theory. Parametric transversality theorem	207	(12)
Applications of transversality to flow-invariance	219	(8)
Optimization and Optimal Control Via Tangential Cones	227	(54)
Ekeland's variational Principle	227	(8)
Geometric optimization	235	(24)
Nonlinear programming and optimal control via tangential cones and closed range operators	259	(22)
Critical point Theory on Flow-Invariant Sets	281	(48)
Elements of Critical point theory	281	(13)
Minimax principles on closed sets	294	(8)
Applications to the theory of geodesics	302	(17)
Nonlinear elliptic boundary value problems	319	(10)
Appendix 1. Elements of Nonlinear Analysis	329	(94)
Strictly (and uniformly) convex spaces. The projection on a closed set	329	(3)
Duality mapping and weak derivatives	332	(18)
Definitions and basic properties	332	(11)
Duality mapping of some concrete spaces	343	(5)
Weak derivatives and duality mappings. Kato's Lemma	348	(2)
Dissipative (accretive, monotone) sets	350	(13)
Basic properties of dissipative sets	350	(6)
Maximal dissipative and m-dissipative sets	356	(4)
Quasi-dissipative sets	360	(3)
Monotone and hemicontinuous Operators	363	(17)
Local boundedness	363	(2)
Maximal monotone sets	365	(15)
Subgradients (subdifferentials)	380	(5)
Cyclically monotone sets	385	(3)
Examples of maximal monotone operators	388	(11)
Subdifferentiability of convex functions. Convex regularizations	399	(9)
On the equation y ∈ dx -- Ax -- Bx	408	(5)
Some characterizations of convex functions	413	(6)
The extendibility of continuous and dissipative operators on closed sets	419	(4)
Appendix 2. The Approximate difference Scheme and Nonlinear Semigroups	423	(28)
Approximate difference schemes (DS)	423	(7)
DS-limit solutions	430	(2)
Generation of nonlinear Semigroups. The Exponential formula of Crandall and Liggett	432	(6)
Existence of DS-approximate solutions and flow-invariance	438	(13)
Kobayashi's theorem	438	(7)
Flow-invariance theorems via DS-approximate solutions	445	(6)
Appendix 3. Banach Manifolds and Vector Fields	451	(6)
Appendix 4. Generalized Gradients	457	(4)
References	461	(16)
Index	477