Titles in This Series

Volume 2 Phili p D. Loewen

Optimal control via nonsmooth analysis 1993

1 M . Ram Murthy, Editor Theta function s 1993

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Optimal Contro l vi a Nonsmooth Analysi s

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Volume 2

CRM PROCEEDINGS & LECTURE NOTES Centre d e Recherches Mathematique s Universite d e Montrea l

Optimal Contro l vi a Nonsmooth Analysi s Philip D . Loewe n

The Centr e d e Recherche s Mathematique s (CRM ) of l'Universite d e Montrea l wa s create d i n 196 8 t o promot e research i n pur e an d applie d mathematic s an d relate d disciplines. Amon g it s activitie s ar e specia l them e years , summer schools , workshops , postdoctora l an d publica -tion programs . CR M i s supporte d b y rUniversit e d e Montreal, th e Provinc e o f Quebe c (FCAR) , and th e Natural Science s an d Engineerin g Researc h Counci l o f Canada. I t i s affiliate d wit h l'lnstitu t de s Science s Mathematiques d e Montrea l (ISM) , whose constituen t members ar e Concordi a University , McGil l University , l'Universite d e Montreal , l'Universit e d u Quebe c a Montreal, an d l'Ecol e Polytechnique .

American Mathematical Society Providence, Rhode Island US A

https://doi.org/10.1090/crmp/002

T h e product io n o f this volum e wa s suppor te d i n pa r t b y th e Fond s pou r l a Format io n de Chercheur s e t l 'Aid e a l a Recherch e (Fond s F C A R ) an d th e Na tu ra l Science s an d Engineering Researc h Counci l o f C a n a d a (NSERC) .

1991 Mathematics Subject Classification Primar y 49-01 , 49-02 , 49J52; Secondar y 49J15 , 49K15 , 49L05 .

Library o f Congres s Cataloging-in-Publicatio n D a t a

Loewen, Phili p D . (Phili p Daniel) , 1960 -Optimal contro l vi a nonsmoot h analysis/Phili p D . Loewen . p. cm . - (CR M proceeding s & lectur e notes , ISS N 1065-8580 ; v . 2 ) Includes bibliographica l references . ISBN 0-8218-6996- 5 (acid-free ) 1. Contro l Theory—Congresses . 2 . Nonsmoot h optimization—Congresses . I . Title .

II. Series . QA402.3.L64 199 3 93-414 3 515'.64-dc20 CI P

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and submitte d t o th e America n Mathematica l Societ y i n camera-read y form b y th e Centr e d e Recherche s Mathematiques .

10 9 8 7 6 5 4 3 2 1 9 8 9 7 9 6 9 5 9 4 9 3

Contents

Preface i x

Chapter 1 . Motivatio n 1 A. Th e Calculu s o f Variations 2 B. Optima l Contro l 1 2 C. Nonsmoot h Analysi s 2 5 D. Recommende d Readin g 2 8 E. Exercise s 2 9

References 3 1

Chapter 2 . Existenc e o f Solution s 3 3 A. Revie w o f Measure Theor y 3 4 B. Measurabl e Multifunction s an d Selection s 3 4 C. Differentia l Inclusion s 3 8 D. Th e Se t o f Trajectories 4 2 E. Existenc e o f Solution s 4 9 F. Relaxatio n 5 6 G. Exercise s 5 7 References 6 0

Chapter 3 . Variationa l Principle s 6 1 A. Introductio n 6 1 B. Th e Smoot h Variationa l Principl e o f Borwein an d Preis s 6 3 C. Application s 6 5 D. Exercise s 7 0 References 7 2

Chapter 4 . Th e Geometr y o f Nonsmoot h Analysi s 7 3 A. Proxima l Normal s an d Subgradient s 7 4 B. Th e Wea k Topolog y 7 8 C. Limitin g Normal s an d Subgradient s 8 0 D. Dualit y 8 6 E. Exercise s 9 1

References 9 4

Chapter 5 . Subgradien t Calculu s 9 5 A. Th e Mai n Result s 9 5 B. Suggeste d Readin g 10 6 C. Exercise s 10 6

References 10 7

V l l l CONTENTS

Chapter 6 . Necessar y Condition s i n Dynami c Optimi -zation 10 9

A. Th e Generalize d Proble m o f Bolza 10 9 B. Th e Lipschit z Proble m o f Bolz a 11 1 C. Th e Free-Endpoin t Differentia l Inclusio n Proble m 11 3 D. Endpoin t Constraint s 12 0 E. Th e Pontryagi n Maximu m Principl e 12 9 F. A Bolza Proble m 13 1 G. Exercise s 13 4 References 13 7

Chapter 7 . Dynami c Programmin g 13 9 A. Th e Principl e o f Optimalit y 13 9 B. Th e Verificatio n Theore m 14 1 C. Feedbac k Optima l Contro l 14 5 D. Differentia l Characterizatio n o f the Valu e Functio n 14 7 E. Exercise s 15 1

References 15 3

Preface

For fou r wonderfu l week s during Jul y an d Augus t 1992 , the Centr e d e Recher -ches Mathematiques o f the Universit e d e Montrea l sponsore d a Summe r Schoo l o n Control Theory . Th e Schoo l offered fou r courses—thes e note s grow out o f the one I was fortunate enoug h to teach. Mos t o f these note s were written befor e th e lecture s began, an d hande d ou t t o th e student s t o supplemen t th e materia l w e discussed i n class. I n polishing the m fo r publication , I have tried t o kee p the student s an d thei r needs constantl y i n mind . M y goa l i n thi s writeup , a s i n th e lectures , i s t o buil d an accessibl e an d thoroug h foundatio n describin g th e theory' s mai n results , fro m which newcomer s t o th e fiel d ca n undertak e furthe r exploration s wit h confidence .

Many peopl e hav e influence d th e content s an d presentatio n o f thes e notes , and I a m gratefu l t o al l o f them . Bu t I woul d lik e t o singl e ou t thre e fo r specia l thanks. First , I than k Franci s Clarke , t o who m man y o f th e bes t idea s betwee n these cover s ow e thei r beginnings , an d whos e leadershi p bot h i n organizin g th e school and in shaping its content mad e the whole event suc h a profitable experience . Many o f th e exercise s i n thes e note s com e fro m assignment s Professo r Clark e se t for a graduat e cours e I too k i n 1982-83 . Second , I than k Joh n D.L . Rowland , whose activ e engagemen t wit h th e materia l befor e th e course , durin g th e lectures , and no w during th e fina l stage s o f the writeu p ha s don e muc h t o improv e th e fina l product. Third , an d most important , I thank my beloved wife, Kimberley T . Ponic h Loewen, wh o wen t beyon d bein g patien t wit h thi s projec t t o th e poin t o f activel y encouraging it .

Philip D . Loewe n Vancouver 199 2

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