Nonlinear Analysis, Differential Equations and Control

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Series C: Mathematical and Physical Sciences - Vol. 528

Nonlinear Analysis, Differential Equations and Control

edited by

F. H. Clarke Institut Gerard Desargues, Universite Claude Bernard (Lyon 1), Villeurbanne, France

and

R. J. Stern Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada

Technical Editor

G. Sabidussi Departement de Mathematiques et de Statistique, Universite de Montreal, Montreal, Quebec, Canada

~.

" Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Study Institute and Seminaire de mathematiques superieures on Nonlinear Analysis, Differential Equations and Control Montreal, Canada July 27 - August 7, 1998

A C.I.P. Catalogue record lor this book is available lrom the Library 01 Congress.

ISBN 978-0-7923-5666-0 ISBN 978-94-011-4560-2 (eBook) DOI 10.1007/978-94-011-4560-2

Printed an acid-free paper

AII Rights Reserved

@ 1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1 st edition 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Table of Contents

Preface

Key to group picture

Participants

Contributors

E. N. BARRON

Viscosity solutions and analysis in LOO

Jonathan M. BORWEIN, Qiji J. ZHU

Multifunctional and functional analytic techniques in nonsmooth analysis

Giuseppe BUTTAZZO

Relaxed optimal control problems and applications to shape optimization

F.H.CLARKE,Yu. S.LEDYAEV,R.J.STERN

Invariance, monotonicity, and applications

Jean-Michel CORON

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1

61

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207

On the stabilization of some nonlinear control systems: results, tools, and applications 307

Robert DEVILLE

Smooth variational principles and non-smooth analysis in Banach spaces 369

Wendell H. FLEMING

Controlled Markov processes and mathematical finance 407

Alexander IOFFE

Variational methods in local and global non-smooth analysis 447

Etienne PARDOUX

BSDEs, weak convergence and homogenization of semilinear PDEs 503

Eduardo D. SONTAG

Stability and stabilization: discontinuities and the effect of disturbances 551

Index 599

Preface

Recent years have witnessed important developments in those areas of the mathematical sciences where the basic model under study is a dynamical system such as a differential equation or control process. Many of these recent advances were made possible by parallel developments in nonlinear and nonsmooth analysis. The latter subjects, in general terms, encompass differential analysis and optimization theory in the absence of traditional linearity, convexity or smoothness assumptions. In the last three decades it has become increasingly recognized that nonlinear and nonsmooth behavior is naturally present and prevalent in dynamical models, and is therefore significant theoretically. This point of view has guided us in the organizational aspects of this ASI. Our goals were twofold: We intended to achieve "cross fertilization" between mathematicians who were working in a diverse range of problem areas, but who all shared an interest in nonlinear and nonsmooth analysis. More importantly, it was our goal to expose a young international audience (mainly graduate students and recent Ph.D.'s) to these important subjects. In that regard, there were heavy pedagogical demands placed upon the twelve speakers of the ASI, in meeting the needs of such a gathering. The talks, while exposing current areas of research activity, were required to be as introductory and comprehensive as possible. It is our belief that these goals were achieved, and that these proceedings bear this out.

Each of the twelve speakers presented a mini-course of four or five hours duration. Three of the lectures (F. H. Clarke, Yu. S. Ledyaev, R. J. Stern) dealt with the application to non­smooth calculus techniques to general issues arising in control theory, including stabilization, optimal control, invariance and equilibria. Two (J.-M. Coron, E. Sontag) provided a compre­hensive overview of recent results on feedback stabilization in a nonsmooth setting. Three of the speakers (J. Borwein, R. Deville, A. Ioffe) took a functional analytic perspective, and presented results on nonsmooth analysis and optimization in an infinite dimensional setting. G. Buttazzo focused on relaxed control methods in problems of shape optimization, and E. N. Barron provided an exposition of the theory of viscosity solutions, with applications to various optimal control and variational problems. Two speakers (W. H. Fleming, E. Pardoux) dealt with control problems where probabalistic techniques come to the fore. The summaries of all of these minicourses are included in these proceedings.

In addition to the speakers, the ASI was attended by 78 participants from 18 countries. We owe them a debt of gratitude for their attentiveness, their critical and insightful contributions, and for helping to set the tone for this lively forum. We also wish to express our deep gratitude to A. Daigneault, G. David and G. Sabidussi for their hard work and success on the organizational side of the ASI; G. Sabidussi is also thanked for his editorial work, and the contributors for their exemplary cooperation during the creation process of these proceedings.

Finally, we thank NATO for supporting this workshop through its Advanced Study In­stitute program, and the Universite de Montreal, NSERC and FCAR for additional support.

The Editors

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Key to group picture

1 M. Lassonde 41 C. Imbert 2 M. Frigon 42 M.-N. Dietsch 3 D. Aussel 43 L. Rifford 4 F. Picard 44 T. Champion 5 M. Ferreira 45 B. Sirakov 6 K. Cam libel 46 A. Briani 7 D. Bessis 47 A. Biglio 8 F. Fontes 48 F. da Lio 9 N. Sengor 49 C. Pignotti 10 R. Choksi 50 P. Albano 11 1. Nikolaev 51 L de Pascale 12 J. Sousa 52 H. Bauschke 13 X. Wang 53 F. Bagagiolo 11 P. Wolenski 54 O. Ley 15 B. Piccoli 55 P. Gauthier 16 F. Peireira 56 R. Lanard 17 F. Schuricht 57 P. Legault 18 Y. Chitour 58 1. Taralova-Roux 19 J. Guinez 59 N. Chung Siong Fah 20 J. Ye 60 N. El Khattabi 21 Y. Lucet 61 N. Harraj 22 P. Man~chal 62 N. Ra"issi 23 G. Gabor 63 G. Sabidussi 24 F. Jules G4 G. David 25 W. Kryszewski G5 L. Freddi 26 M. de Pinho 66 J. Avelin 27 A. Cwiszewski 67 1. Zelenko 28 I. Fleischer 68 A. Camyshev 29 E. Rocha 69 M. Krichman 30 C. Troestler 70 A. Guesima 31 Z. Wu 71 A. Daigneault 32 D. Marado Torres 72 J. Borwein 33 F. Ceragioli 73 A. Ioffe 34 K. Eisenhart 74 Yu. Ledyaev 35 M. Ghazi 75 J.-M. Co ron 3G M. Czarnecki 76 E. Pardoux 37 A. Matos 77 R. Stern 38 J. Zhu 78 E. Sontag 39 S. Koike 79 G. Buttazzo 40 G. Pop 80 F. Clarke

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Participants

ALBANO, Paolo Dpto. di Matematica Univ. di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma Italy albano@axp.mat.uniroma2.it

AUSSEL, Didier Dep. de Mathematiques Univ. de Perpignan 52 Ave de Villeneuve 66860 Perpignan Cedex France aussel@univ-perp.fr

AVELIN, Jonas Dept. of Mathematics Univ. Coll. Gavle-Sandviken 80176 Gavle Sweden jai@hgs.se

BACHIR, Mohammed Centre de Mathematiques Univ. de Bordeaux I 351 cours de la Liberation 33405 Talence Cedex France bachir@math.u-bordeaux.fr

BAGAGIOLO, Fabio Dpto. di Matematica Univ. di Trento Via Sommarive 14 38050 Povo (TN) Italy bagagiol@science.unitn.it

BAUSCHKE, Heinz Dept. of Combinatorics & Optimization Univ. of Waterloo Waterloo (ON) N2L 3G1 Canada hbauschke@math.uwaterloo.ca

BESSIS, Dorothee Ctr. Process Systems Engineering Imperial College SW7 2AZ London United Kingdom d.bessis@ic.ac.uk

BIGLIO, Alessandra Dpto. di Matematica Politecnico di Torino Corso Duca degJi A bruzzi 24 10129 Torino Italy biglio@dm.unito.it

BRIANI, Ariela Dpto. di Matematica Univ. di Pisa Via F. Buonarroti 2 56127 Pisa Italy briani@dm.unipi.it

CAMLIBEL, Kanat Stichting Mathematisch Centrum CWI, Post bus 94079 1090 GB Amsterdam The Netherlands kanat.carnlibel@cwi.nl

CAMYSHEV, Andrei lnst. of Mathematics Latvian Acad. Sci. & Univ. of Latvia Akademijas Paukums 1 1524 Riga Latvia camysh@lanet.lv

CERAGIOLI, Francesca Dpto. di Matematica "U. Dini" Univ. di Firenze Viale Morgagni 67/ A 50134 Firenze Italy ceragio@udini.math.unifi.it

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CHAMPION, Thierry Dep. de Sciences Mathematiques - CC 051 Univ. de Montpellier II PI. Eugene Bataillon 34095 Montpellier Cedex 05 France thierry@num.math.univ-montp2.fr

CHITOUR, Yadne Dep. de Mathematiques Bat. 425 Univ. Paris-Sud (Paris XI) 91405 Orsay Cedex France yacine.chitour@math.u-psud.fr

CHOKSI, Rustum Dept. of Mathematics & Statistics Simon Fraser Univ. Burnaby (BC) V5A IS6 Canada choksi@math.sfu.edu

CHUNG SIONG FAH, Nicolas Ecole Normale Superieure de Paris 45 rue d'Ulm 75005 Paris France chung@clipper.ens.fr

CHYBA, Monique Robotics Laboratory Harvard Univ. Pierce G12E Cambridge MA 01238 USA chyba@sunny-unige.ch

CWISZEWSKI, Aleksander Fac. of Mathematics & Informatics Nicholas Copernicus Univ. Chopin a 12/18 87100 Tomil Poland aleks@mat.uni.tomn.pl

CZARNECKI, Marco Dep. de Sciences Mathematiques - CC 051 Univ. de Montpellier II PI. Eugene BatailJon 34095 Montpellier Cedex 05 France marco@math.univ-montp2.fr

DA LIO, Francesca Dpto. di Matematica Pur a ed Appl. Univ. di Padova Via Belzoni 7 35131 Padova Italy dalio@math.unipd.it

DANIlLIDIS, Aris Dept. of Mathematics Univ. of the Aegean 83200 Karlouassi-Samos Greece arisd@karkis.math.aegean.gr

DELFOUR, Michel

Participants

Dcp. de mathCmatiques & de statistique Univ. de Montreal CP 6128-Centre-vilJe Montreal (QC) H3C 3J7 Canada delfour@dms.umontreal.ca

DE PASCALE, Luigi Dpto. di Matematica Univ. di Pisa Via Buonarroti 2 56127 Pisa Italy depascal@dm.unipi.it

DIETSCH, Marie-Noelle Dep. de Mathematiques Univ. de Metz BP 80794 - lie de Sauley 57012 Metz Cedex 01 France dietsch@poncelet.univ-metz.fr

EISENHART, Kirsty Dept. of Mathematics & Statistics Western Michigan Univ. Kalamazoo, MI 49008-5152 USA kirsty.eisenhart@wmich.edu

EL KHATTABI, Noha Dep. de Mathematiques Univ. Mohamed V Rabat, Maroc elkhatab@fsr.ac.ma

Participants

FERREIRA, Maria DEEC/FEUP, Univ. do Porto Rua dos Bragas 4099 Porto Codex Portugal mmf@fe.up.pt

FLEISCHER, Isidore Dept. of Mathematics Univ. of Windsor Windsor (ON) N9B 3P4 Canada fieischi@crm.umontreal.ca

FONTES, Fernando Ctr. Process Systems Engineering Imperial College London SW7 2AZ United Kingdom f.fontes@ic.ac.uk

FREDDI, Lorenzo Dpto. di Matematica e Informatica Univ. degli Studi di Udine Via delle Science 206 33100 Udine Italy freddi@dimLuniud.it

FRIGON, Marlene Dep. de mathematiques & de statistique Univ. de Montreal CP 6128-Centre-ville Montreal (QC) H3C 3J7 Canada frigon@dms.umontreai.ca

GABOR, Grzegorz Fac. of Mathematics & Informatics Nicholas Copernicus Univ. Chopina 12/18 87100 Toruli Poland ggabor@mat.uni.torun.pl

GAUTHIER, Paul Dep. de mathematiques & de statistique Univ. de Montreal CP 6128-Centre-ville Montreal (QC) H3C 3J7 Canada gauthier@dms.umontreal.ca

GEOFFROY, Michel Dep. de MatMmatiques Univ. Antilles-Guyane 97159 Pointe-a.-Pitre France michel.geoffroy@univ-ag.fr

GUESMIA, Aissa IRMA - UFR MatMmatiques Univ. Louis Pasteur 7 rue Rene Descartes 67084 Strasbourg Cedex France guesmia@math.u-strasbg.fr

GUINEZ, Jorge Dpto. de Matematicas Fac. Ingenieria Univ. del Zulia Maracaibo, Venezuela jguinez@vibora.ica.luz.ve

HARRAJ, Najoua Dep. de Mathematiques Univ. Mohamed V Rabat Maroc

IMBERT, Cyril

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Lab. Approximation & Optimisation, UFR-MIG Univ. Paul Sabatier 118 route de Narbonne 31062 Toulouse Cedex 4 France cimbert@cict.fr

JULES, Florence Dep. de MatMmatiques Univ. Antilles-Guyane 97159 Pointe-a.-Pitre France fiorence.jules@univ-ag.fr

KOIKE, Shigeaki Dept. of Mathematics Saitama Univ. 556-1-54 Horisakincho 330 Omiya-Saitama Japan skoike@rimath.saitama-u.ac.jp

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KRICHMAN, Mikhail Dept. of Mathematics Rutgers Univ. 110 Frelinghuisen Rd. Piscataway, NJ 08854-8019 USA krichman@math.rutgers.edu

KRYSZEWSKI, Wojciech Fac. of Mathematics & Informatics Nicholas Copernicus Univ. Chopina 12/18 87100 TOruli Poland wkrysz@mat.uni.torun.pl

LANARD, Renaud Inst. Girard Desargues Bat. 101 Univ. Claude Bernard (Lyon I) 43 bd du 11 Novembre 1918 69622 Villeurbanne Cedex France lanard@desargues.univ-lyonl.fr

LASSONDE, Marc Dep. de MatMmatiques Univ. Antilles-Guyane 97159 Pointe-a.-Pitre France lassonde@univ-ag.fr

LEY, Olivier Dep. de Mathematiques Univ. de Tours Parc de Grandmont 37200 Tours France ley@math.univ-tours.fr

LUCET, Yves CECM, Dept. of Mathematics & Statistics Simon Fraser Univ. Burnaby (HC) V5A IS6 Canada lucet@cccm.sfu.ca

MARECHAL, Pierre CBCM, Dept. of Mathematics & Statistics Simon Fraser Univ. Burnaby (BC) V5A 1S6 Canada marechal@cecm.sfu.ca

MATOS, Anibal DEEC/FEUP Univ. do Porto Rua dos Bragas 4099 Porto Codex Portugal anibal@fe.up.pt

MONTOKI, Emmanuel

Participants

Dep. de mathematiques & de statistique Univ. de Montreal CP 6128-Centre-ville Montreal (QC) H3C 3J7 Canada montoki@dms.umontreal.ca

NOBAKHTIAN, Soghra Dept. of Mathematics & Statistics McGill Univ., Burnside Hall 805 Sherbrooke St. West Montreal (QC) H3A 2K6 Canada nobakht@math.mcgill.ca

PEREIRA, Fernando DEEC/FEUP Univ. do Porto Rua dos Bragas 4099 Porto Codex Portugal flp@fe.up.pt

PICARD, Frederic Dep. de mathematiques & de statistique Univ. de Montreal CP 6128-Centre-ville Montreal (QC) 1I3C 3J7 Canada picard@dms.umontreal.ca

PICCOLI, Benedetto SISSA-ISAS Via Beirut 2-4 34013 Trieste Italy piccoli@sissa.it

PIGNOTTI, Cristina Dpto. di Matematica Univ. di Roma Tor Vergata Via della Ricerca Scientifica 00133 Roma Italy pignotti@axp.mat.uniroma2.it

Participants

PINHO, Maria DEEC/FEUP Univ. do Porto Rua dos Bragas 4099 Porto Codex Portugal mrpinho@fe.up.pt

POLOTSKI, Vladimir Dep. de genie electrique Ecole Poly technique de Montreal CP 6079-Centre-ville Montreal (QC) H3C 3A7 Canada polotski@ai.polymtl.ca

POP, Gabriela Fac. of Mathematics Univ. of Bucharest Academiei 14 70109 Bucharest Romania pop@math.math.unibuc.ro

RAISSI, Nadia Fac. des Sciences Univ. Ibn Tofai1 BP 133, Kenitra Maroc raissi@emi.ac.ma

RENOU, Stephane Dep. de genie chimique Ecole Poly technique de Montreal CP 6079-Centre-ville Montreal (QC) H3C 3A7 Canada renou@gch.polymtl.ca

RIFFORD, Ludovic lnst. Gerard Desargues Biit. 101 Univ. Claude Bernard (Lyon I) 43 bd du 11 Novembre 1918 69622 Villeurbanne Cedex France rifford@desargues.univ-lyonl.fr

ROCHA, Eugenio Dpto. Matematica Univ. de Aveiro 3810 Aveiro Portugal eugenio@mat.ua.pt

ROSIER, Lionel Dep. de Mathematique Biit. 425 Univ. Paris-Sud (Paris XI) 91405 Orsay Cedex France lionel.rosier@math.u-psud.fr

ROUSSEAU, Christiane Dep. de mathematiques & de statistique Univ. de Montreal CP 6128-Centre-ville Montreal (QC) H3C 3J7 Canada rousseac@dms.umontreal.ca

SCHURICHT, Friedemann Max-Planck-Inst. f. Mathematik Inselstr. 22-26 04103 Leipzig Germany schur@mis.mpg.de

SENGOR, Neslihan Dept. of Electronics & Communiation Istanbul Technical Univ. 80626 Maslak-lstanbul Turkey neslihan@ehb.itu.edu.tr

SIRAKOV, Boyan Analyse Numerique Univ. Pierre & Marie Curie (Pavis VI) Tour 55-65 - 5e etage 4 pI. J ussieu 75230 Paris Cedex 05 France sirakov@ann.jussieuJr

SOUSA, Joao DEEC/FEUP Univ. do Porto Rua dos Bragas 4099 Porto Codex Portugal jtasso@alf.fe.up.pt

SUBBOTINA, Nina Inst. of Mathematics & Mechanics Ural Branch of Russian Acad. Sci. S. Kovalevskoi St. 16 620919 Ekaterinburg Russia subb@uran.ru

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SZPIRO, Adam Div. of Applied Mathematics Brown Univ. Providence, RT 02912-0001 USA aas(cj)dam.hrown.edu

TARALOVA-ROUX, Ina Dep. de Genie Electriqne INS A Toulouse Complexe Scientifique de Rangueil 31077 Toulouse Cedex 04 France taralova@dge.insa-tlse.fr

TORRES, Delfim Dpto. Matematica Univ. de Aveiro 3810 Aveiro Portugal delfim@mat.ua.pt

TROESTLER, Christophe Inst. Mathematiques et Informatique Univ. de Mons-Hainaut "Le Pentagone" 6 Ave du Champ de Mars 7000 Mons Belgique trch@sun1.umh.ac.he

WANG, Xianfu CECM, Dept. of Mathematics & Statistics Simon Fraser Univ. Burnaby (HC) V5A ]S6 Canada xwang@cecm.sfn.ca

WOLENSKI, Peter Dept. of Y1athematics Louisiana State Univ. Baton Rouge LA 70803 eSA wolenski@math.lsu.edu

WU, Zili Dept. of Mathematics & Statistics Univ. of Victoria P.O. Box 304S Victoria (HC) V8W 3P4 Canada wzl@math.uvic.ca

YE, Jane Dept. of Y1athematics ,'\,~ Statistics UIIiv. of Victoria P.O. Box 3045 Victoria (BC) V8W 3P4 Canada janeye@math.uvic.ca

ZELENKO, Igor Dept. of Mathematics

Participants

Technion - Israel Inst. of Technology 32000 Haifa Israel zigorrgtx. technion .ac.il

Contributors

BARRON, E. N. Dept. of Mathematical Sciences Loyola Univ. of Chicago Chicago, IL 60626-5311 USA enb@math.luc.edu

BORWEIN, Jonathan Dept. of Mathematics & Statistics Simon Fraser Univ.

Burnaby (BC) V5A IS6 Canada jborwein@cecm.sfu.ca

BUTTAZZO, Giuseppe Dpto. di Matematica Univ. di Pisa Via Buonarroti 2 56127 Pisa Italy buttazzo@dm.unipi.it

CLARKE, Francis Inst. Gerard Desargues BAt. 10 1, U niv. Claude Bernard (Lyon I) 43 bd du 11 Novembre 1918 69622 Villeurbanne Cedcx France clarke@jonas.univ-lyonl.fr

CORON, Jean-Michel Dep. de Mathematique Anal. Numerique et EDP Bat. 425 Univ. Paris-Sud (Paris XI) 91405 Orsay Cedex France jean-michel coron@math.u-psud.fr

DEVILLE, Robert Centre de Mathematiques U niv. de Bordeaux I 351 cours de la Liberation 33405 Talence Cedex France deville@math.u-bordeaux.fr

FLEMING, Wendell H. Div. of Applied Mathematics Brown Univ. Providence, RI 02912-0001 USA whf@cfm.brown.edu

TOFFE, Alexander Dept. of Mathematics Technion - Israel Inst. of Technology 32000 Haifa Israel ioffe@math.technion.ac.il

LEDYAEV, Yu. S. Dept. of Mathematics & Statistics Western Michigan Univ. Kalamazoo, MI 49008-5152 USA ledyaev@math-stat.wmich.edu

PARDOUX, Etienne LATP-CMI Universite de Provence 39 rue F. Joliot Curie 13453 Marseille Ccdex 13 France pardoux@gyptis.univ-mrs.fr

SONTAG, Eduardo Dept. of Mathematics Rutgers Univ. 110 Frelinghuisen Rd. Piscataway, NJ 08854-8019 USA esontag@hilbert.rutgers.edu

STERN, Ronald Dept. of Mathematics & Statistics Concordia Univ. 7141 Sherbrooke West Montreal (QC) H4B 1R6 Canada stern@vax2.concordia.ca

ZIIU, Jim Dept. of Mathematics & Statistics Western Michigan Univ. Kalama7.0o, MI 49008-5152 USA zhu@math-stat.wmich.edu

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