Calculus of Variations, Classical and Modern: Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in ... 10-18, 1966 (C.I.M.E. Summer Schools, 39)

Cover
C.I.M.E. Summer Schools, 39
Calculus of Variations, Classical
and Modern
ISBN 9783642110412
Contents
Quelques Aspects Geometriques Des Processus Optimaux
	I Definition Des Surfaces Limits, Proprietes Globales De Ces Surfaces
		1. Introduction
		2. Trajectoires Dans l\'espace des etats augmente
		3. Surface limits et surfaces isocôut optimales
		4. Quelques propriete des surfaces limites
		5. Equations d\'etat
		6. Critère de coût integral
		7. Proprietes d\'une transformation lineaire
		8. Transformation du plan tangent
		9. Points interieurs fortement reguliers d\'une surface limite
		10. Trajectoires optimales fortement regulières
		11. Condition de transversalite terminale
		12. Condition de transversalite initiale
		13. Le Principe du Maximum pour les trajectoires optimales fortement regulières
		14. Equation de la Programmation Dynamique
	II Proprietes Locales Des Surfaces Limites
		1. Introduction
		2. Une Hypothèse de Base
		3. Defnition des Cones Locaux (Image)
		4. Points Interieurs de (Image) et (Image), une autre Hypothèse de Base
		5. Cone Local (Image)
		6. Une Partition de (Image)
		7. Lemmes 3 et 4
		8. Cone des Vecteurs (Image)
		9. Hyperplan Separant d\'un Cone Local
		10. Cone des Normales
		11. Points Interieurs Faiblement Reguliers, et Nonreguliers, d\'une Surface Limite
		12. Transformation Lineaire (Rappel)
		13. Lemmes 6 et 7
		14. Theorèmes de Separabilite
		15. Corollaires 4 et 5
		16. Sous-Ensembles Attractifs et Repulsifs d\'une Surface Limite
		17. Sous-Ensemble Reguliers et Antireguliers d\'une Surface Limite
		18. Sous-Ensembles Symmetriques du Cone Local (Image)
		19. Cas Degenere
		20. Principe du Maximum (Points interieurs de (Image), reguliers ou non - reguliers)
		21. Principe du Maximum Trivial
	III Exemples Illustrant La Theorie
		1. Introduction
		2. Problème du Regulateur à Une-Dimension
		3. Problème du Rocket de Puissance Limitee
		4. Un Problème de Navigation
Quelques Problemes De Mesurabilite Lies A La Theorie Des Commandes
Existence Theorems For Langrange And Pontryagin Problems Of The Calculus Of Variations And Optimal Control. More Dimensional Extensions In Sobolev Spaces
	Lecture 1. Usual and generalized solutions in Optimal control and the calculus of variations
		1. Usual solutions
		2. Generalized solutions
		3. The distance function (Image)
	Lecture 2. Upper semicontinuity of variable sets generalizations
		4. Upper semicontinuity of variable sets
		6. Properties (U) and (Q) of variable sets
	Lecture 3. Closure Theorems
		5. Closure Theorems 1
		6. Another closure theorem
	Lecture 4. Existence theorems for usual solutions of Langrage problems
		7. Notations
		8. Statement of the first existence theorem
		9. Another Existence Theorem for Lagrange Problems with Unilateral Constraints. Existence Theorem II (L. Cesari (Image) )
		10. A few corollaries
		11. Examples
		12. Further existence theorems for Langrage problems
	Lecture 5. Existence theorems for generalized solutions of Langrange problems
		13. Notations
		14. Property (P)
		15. Existence theorem
		16. An exemple
	Lecture 6. A system of partial differential equations in Sobolev spaces
		17. Notations
	Lecture 7. A closure theorem in Sobolev spaces
		18. Closure Theorem III (in Sobolev spaces)
	Lecture 8. Existence theorems for Pontryagin\'s problems in Sobolev speces
		19. More notations for the existence theorems
		20. An existence theorem for multidimensional problems of optimal control
		21. Examples
	References
Optimal Control As Programming In Infinite Dimensional Spaces
	Introduction
		Section I. A Mathematical Programming Problem in Infinite Dimensional Spaces
		Section II. Optimal Control Problem
		Acknowledgement
	References
The range of integrals of a certain class of vector-valued functions
	Introduction
		Concerning the proofs
	References
Weak Topology And Calculus Of Variations
	1. Introduction
	2. General theorems on lower semi-continuous and on convex functions
	3. Weak topology in Banach spaces
	4. On the existence of extrema
	5. On the relation between f and its Gâteaux differential (Image)
	6. Applications to Sobolev spaces
	Bibliography
Problems About The Set Of Attainability
	I. Control systems. Attainable set
		1. Control systems
		2. Admissible controls
		3. Contingent and paratingent equations
		4. Attainable set
	II. Properties of the attainable set
		1. Notation
		2. Some examples
		3. The closedness of the attainable set
		4. Chattering, sliding regime and quasitrajectories
	III. Control systems defined abstractly
		1. Generalized dynamical systems
		2. Properties of the generalized dynamical systems. Motions
		3. Relation with contingent equations
	IV. Control systems and calculus of variations
		1. The optimal control problem
		2. Pontryagin\'s Maximum Principle
		3. Relation with the classical Weierstrass-function
	V. Boundary controls
		1. Boundary motions and boundary controls
		2. Extremal solutions for linear systems
	VI. Controllability of linear systems
		1. Some examples
		2. Controllability and observability
		3. Conditions for complete controllability
		4. Controllablity under boundedness conditions
	VII. Controllability of nonlinear systems
		1. Attainablity and controllability
		2. Nonlinear system with linear control. Definition of complete controllablity
		3. General nonlinear systems
	VIII. Singular problems. Not controllable systems
		1. Systems linear in the control
		2. Finite stability
	References