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)

A. Blaquière: Quelques aspects géométriques des processus optimaux.- C. Castaing: Quelques problèmes de mesurabilité liés à la théorie des commandes.- L. Cesari: Existence theorems for Lagrange and Pontryagin problems of the calculus of variations and optimal control of more-dimensional extensions in Sobolev space.- H. Halkin: Optimal control as programming in infinite dimensional spaces.- C. Olech: The range of integrals of a certain class vector-valued functions.- E. Rothe: Weak topology and calculus of variations.- E.O. Roxin: Problems about the set of attainability. 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