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http://dx.doi.org/10.1090/pspum/064 Selected Titles in This Series

64 G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Editors, Differential and control (University of Colorado, Boulder, 1997) 63 Alejandro Adem, Jon Carlson, Stewart Priddy, and Peter Webb, Editors, representations: Cohomology, group actions and (University of Washington, Seattle, 1996) 62 Janos Kollar, Robert Lazarsfeld, and David R. Morrison, Editors, —Santa Cruz 1995 (University of California, Santa Cruz, July 1995) 61 T. N. Bailey and A. W. Knapp, Editors, and automorphic forms (International Centre for Mathematical Sciences, Edinburgh, Scotland, March 1996) 60 David Jerison, I. M. Singer, and Daniel W. Stroock, Editors, The legacy of Norbert Wiener: A centennial symposium (Massachusetts Institute of Technology, Cambridge, October 1994) 59 William Arveson, Thomas Branson, and Irving Segal, Editors, Quantization, nonlinear partial differential , and operator (Massachusetts Institute of Technology, Cambridge, June 1994) 58 Bill Jacob and Alex Rosenberg, Editors, X-theory and algebraic geometry: Connections with quadratic forms and division (University of California, Santa Barbara, July 1992) 57 Michael C. Cranston and Mark A. Pinsky, Editors, Stochastic analysis (Cornell University, Ithaca, July 1993) 56 William J. Haboush and Brian J. Parshall, Editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, Editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, Editors, (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, Editors, and (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, Editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, Editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, Editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, Editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., Editor, The mathematical heritage of (Duke University, Durham, May 1987) 47 Paul Fong, Editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, Editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E. Browder, Editor, Nonlinear and its applications (University of California, Berkeley, July 1983) 44 William K. Allard and Frederick J. Almgren, Jr., Editors, and the of variations (Humboldt State University, Areata, California, July/ August 1984) 43 Frangois Treves, Editor, Pseudodifferential operators and applications (University of Notre Dame, Notre Dame, Indiana, April 1984) (Continued in the back of this publication) (Photograph courtesy of Harolyn Gardner)

Robert Brown Gardner February 27, 1939-May 5, 1998 Proceedings of Symposia in PURE

Volume 64

Differential Geometry and Control

Summer Research Institute on Differential Geometry and Control June 29-July 19, 1997 University of Colorado, Boulder

G. Ferreyra R. Gardner H. Hermes H. Sussmann Editors

^y^TPHTOI MH^V ^^ ^ American Mathematical Society Providence, Rhode Island PROCEEDINGS OF A SUMMER RESEARCH INSTITUTE ON DIFFERENTIAL GEOMETRY AND CONTROL HELD AT THE UNIVERSITY OF COLORADO, BOULDER JUNE 29-JULY 19, 1997

with support from the National Science Foundation, Grant DMS-9713413

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the view of the National Science Foundation.

1991 Mathematics Subject Classification. Primary 49-XX, 53-XX, 93-XX; Secondary 22-XX, 60-XX, 35-XX.

Library of Congress Cataloging-in-Publication Data Summer Research Institute on Differential Geometry and Control (1997 : University of Colorado, Boulder) Differential geometry and control : Summer Research Institute on Differential Geometry and Control, June 29-July 19, 1997, University of Colorado, Boulder / G. Ferreyra... [et al.], editors. p. cm. — (Proceedings of symposia in , ISSN 0082-0717 ; v. 64) Includes bibliographical references. ISBN 0-8218-0887-7 (alk. paper) 1. —Congresses. 2. Geometry, Differential—Congresses. 3. Exterior differen• tial systems—Congresses. I. Ferreyra, Guillermo Segundo, 1953- . II. Title. III. Series. QA402.3.S887 1997 629.8'312—dc21 98-38711 CIP

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1999 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 04 03 02 01 00 99 Contents

Preface vii Remembering the mathematics of Robert Brown Gardner GEORGE R. WILKENS 1 A Hamiltonian approach to strong minima in optimal control ANDREI AGRACHEV, GIANNA STEFANI, AND PIERLUIGI ZEZZA 11 The Heisenberg group H% in terms of the orthonormal frame bundle 0(R2) and the control of variation of the Heisenberg diffusion HELENE AIRAULT 23 Linear control systems on Lie groups and controllability VICTOR AYALA AND JUAN TIRAO 47 Real and local observability ZBIGNIEW BARTOSIEWIGZ 65 External dynamic feedback equivalence of observable discrete-time control systems EWA PAWLUSZEWICZ AND ZBIGNIEW BARTOSIEWICZ 73 Good needle-like variations R. M. BIANCHINI 91 Representations of Dirac structures on vector spaces and nonlinear L-C circuits ANTHONY M. BLOCH AND PETER E. CROUCH 103 Non integrable geodesies in SR-Martinet geometry BERNARD BONNARD, MONIQUE CHYBA, AND IVAN KUPKA 119 Singular extremals of order 3 and chattering V. F. BORISOV 135 Some open problems in control theory JEAN-MICHEL CORON 149 Quadratic order conditions of a local minimum for singular extremals in a general optimal control problem A. V. DMITRUK 163 vi CONTENTS

Some solvable infinite time horizon stochastic control problems in hyperbolic three T. E. DUNCAN 199 Geometric control and Lie semigroup theory JIMMIE D. LAWSON 207 Analysis of abnormal extremals in optimal control URSZULA LEDZEWICZ AND HEINZ SCHATTLER 223 Rolling bodies with regular : The holonomic case ALESSIA MARIGO AND ANTONIO BICCHI 241 On degenerate Monge-Ampere equations VISWANATH RAMAKRISHNA 257 Lie brackets and impulsive controls: An unavoidable FRANCO RAMPAZZO 279 Survey on controllability of systems on solvable Lie groups YURI L. SACHKOV 297 Centro- in the and feedback invariants of two-state scalar control systems GEORGE R. WILKENS 319 On hypersurfaces satisfying a basic equality JIE YANG 335 Preface

The 1997 American Mathematical Society Summer Research Institute on Dif• ferential Geometry and Control was held in June-July 1997 at the University of Colorado in Boulder. It was an exciting and scientifically rewarding summer insti• tute with the participation of more than 90 and mathematically- oriented engineers from Bulgaria, Chile, Colombia, France, Germany, Israel, Italy, Mexico, Poland, Portugal, Russia, Switzerland and the United States. The organizer of the conference was Hector J. Sussmann with Henry Hermes as co-organizer. Guillermo S. Ferreyra and the late Robert B. Gardner were additional members of the organizing committee. This edited represents a cross of the presentations made at the conference. The papers included here were refereed to the highest standards and contain complete results, as well as interesting open problems, in the of Geo• Control Theory. The reader interested in this topic will find here an excellent source of challenging research problems and results. The objective of this collection of papers is to provide a coherent global perspective of recent develop• ments and important open problems in the area of Geometric Control Theory. Control theory is a very large and active area of contemporary science and . In the United States, most work in control takes place in industry, and academic activity in this occurs primarily in departments of engineer• ing, especially electrical, mechanical and aerospace. The main professional society of control theorists and engineers is the IEEE Control Systems Society, which has about 10,000 members, i.e. a size comparable to about half that of the entire Amer• ican Mathematical Society. There is also significant activity by mathematicians, working in engineering or mathematics departments. Within the Society for Indus• trial and (SIAM) there is an activity group in control with about 300 members, which sponsors a triennial conference with about 500 partici• pants. There are many large control conferences, such as the IEEE Conference on Decision and Control, held every year in December, where about 1,000 papers are presented each year. As counterpart to the wide breadth of the large conferences held yearly in control theory, the 1997 NSF/AMS Summer Research Institute fo• cused mainly on the differential geometric aspects of control theory. Participants exchanged ideas and presented recent work to colleagues and graduate students. The field of "Differential-geometric nonlinear control theory" was born in the 1960's, with the pioneering work of R. Hermann, H. Hermes, and C. Lobry, who were the first to adopt the perspective that a control system should be regarded primarily as a collection of vector fields. Starting from this vector-field of view it is reasonable to expect that the natural algebraic structure of the space of vector vii Vlll PREFACE

fields —given by the linear operations and the Lie bracket— ought to play a sig• nificant role in the study of control-theoretic problems, and the work of Hermann, Hermes and Lobry provided strong support to this idea, in the form of concrete applications to control problems. Since then, the field has grown and expanded in several directions. The differential-geometric point of view —and in particular the systematic search for properties that are invariant under the appropriate groups of nonlinear transformations, and the representation of control systems as systems of vector fields on — has led to the development of new classification the• ories for nonlinear control systems, and the discovery of new results on feedback stabilization and optimal control. Applications have been devised to numerous problems such as satellite and helicopter control, robot trajectory planning, and optimal drug therapy. In the opposite direction, methods from optimal control have had an impact on the understanding of differential-geometric questions such as the problem of the structure of sub-Riemannian minimizers and various ques• tions about Lie semigroups. The interaction between geometry and control theory involves the use of various differential-geometric tools, in particular Lie algebras of vector fields, free Lie algebras, nilpotent Lie algebras, Lie algebra filtrations, Lie groups and Lie semigroups, , vector bundles, distributions (in the differential-geometric sense of "subbundles of the bundle") and their dual objects (i.e. codistributions), exterior differential systems, Cartan prolonga• tions and equivalence, infinite-dimensional manifolds ("diffieties") of map jets, as well as tools from "analytic geometry" (real-analytic, semianalytic and subanalytic sets, stratifications and desingularization), from control theory (the "Maximum Principle" of Optimal Control and its generalizations, theorems on local and global controllability) and from dynamical systems (Lyapunov techniques, stable and un• stable manifolds, center manifolds). Most of these topics are covered in this volume. With sorrow we learned last month of the untimely death of our friend and great colleague Robert Gardner. Upon our invitation, George Wilkens has written an article for this proceedings in tribute to Robert Gardner's lasting contributions to mathematics. We gratefully acknowledge the financial support of the National Science Foun• dation, and of the University of Colorado in Boulder. We would like to thank Jim Maxwell (American Mathematical Society) for his help with the administration of the NSF grant and Wayne Drady (American Mathematical Society) for the on-site organization of the Summer Institute and associated social events. We also ac• knowledge the superb efforts of the many anonymous referees that were involved with the papers submitted to this publication. Finally, we wish to thank Christine Thivierge of the American Mathematical Society, for her excellent support in the editorial and prepublishing process for this volume.

GUILLERMO FERREYRA

HENRY HERMES

HECTOR J. SUSSMANN Selected Titles in This Series (Continued from the front of this publication)

42 Anil Nerode and Richard A. Shore, Editors, Recursion theory (Cornell University, Ithaca, New York, June/July 1982) 41 Yum-Tong Siu, Editor, of several variables (Madison, Wisconsin, April 1982) 40 Peter Orlik, Editor, Singularities (Humboldt State University, Areata, California, July/August 1981) 39 Felix E. Browder, Editor, The mathematical heritage of Henri Poincare (Indiana University, Bloomington, April 1980) 38 Richard V. Kadison, Editor, Operator algebras and applications (Queens University, Kingston, Ontario, July/August 1980) 37 Bruce Cooperstein and Geoffrey Mason, Editors, The Santa Cruz conference on finite groups (University of California, Santa Cruz, June/July 1979) 36 and Alan Weinstein, Editors, Geometry of the Laplace operator (University of Hawaii, Honolulu, March 1979) 35 Guido Weiss and Stephen Wainger, Editors, in Euclidean spaces (Williams College, Williamstown, Massachusetts, July 1978) 34 D. K. Ray-Chaudhuri, Editor, Relations between and other parts of mathematics (Ohio State University, Columbus, March 1978) 33 A. Borel and W. Casselman, Editors, Automorphic forms, representations and L-functions (Oregon State University, Corvallis, July/August 1977) 32 R. James Milgram, Editor, Algebraic and (Stanford University, Stanford, California, August 1976) 31 Joseph L. Doob, Editor, (University of Illinois at Urbana-Champaign, Urbana, March 1976) 30 R. O. Wells, Jr., Editor, Several complex variables (Williams College, Williamstown, Massachusetts, July/August 1975) 29 Robin Hartshorne, Editor, Algebraic geometry - Areata 1974 (Humboldt State University, Areata, California, July/August 1974) 28 Felix E. Browder, Editor, Mathematical developments arising from Hilbert problems (Northern Illinois University, Dekalb, May 1974) 27 S. S. Chern and R. Osserman, Editors, Differential geometry (Stanford University, Stanford, California, July/August 1973) 26 Calvin C. Moore, Editor, Harmonic analysis on homogeneous spaces (Williams College, Williamstown, Massachusetts, July/August 1972) 25 Leon Henkin, John Addison, C. C. Chang, William Craig, Dana Scott, and Robert Vaught, Editors, Proceedings of the Tarski symposium (University of California, Berkeley, June 1971) 24 Harold G. Diamond, Editor, Analytic (St. Louis University, St. Louis, Missouri, March 1972) 23 D. C. Spencer, Editor, Partial differential equations (University of California, Berkeley, August 1971) 22 Arunas Liulevicius, Editor, (University of Wisconsin, Madison, June/July 1970) 21 Irving Reiner, Editor, Representation theory of finite groups and related topics (University of Wisconsin, Madison, April 1970) 20 Donald J. Lewis, Editor, 1969 Number theory institute (State University of New York at Stony Brook, Stony Brook, July 1969) 19 Theodore S. Motzkin, Editor, Combinatorics (Universit(See they AMof CaliforniaS catalog, foLor searlie Angelesr titles, ) March 1968)