The Dynamic Morse Theory of Control Systems

349 Pages·2020·21.38 MB·English

by  Josiney A. Souza

The Dynamic Morse
Theory of Control
Systems

The Dynamic Morse
Theory of Control
Systems
By
Josiney A. Souza
The Dynamic Morse Theory of Control Systems

By Josiney A. Souza

This book first published 2020

Cambridge Scholars Publishing

Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library

Copyright © 2020 by Josiney A. Souza

All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.

ISBN (10): 1-5275-4507-5
ISBN (13): 978-1-5275-4507-6
For Priscila, Larissa, and Henrique.

“The LORD by wisdom hath founded the earth; by understanding hath he
established the heavens. By his knowledge the depths are broken up, and the
clouds drop down the dew. My son, let not them depart from thine eyes; keep
sound wisdom and discretion: So shall they be life unto thy soul, and grace to
thy neck”. Proverbs 3,19-22

Contents
Preface xiii
Introduction 1
1 Fundamental Theory of Control systems 11
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Shift Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Control affine systems . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Control flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5 Group structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.7 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Elementary Concepts 41
2.1 Invariant sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Orbits and semi-orbits . . . . . . . . . . . . . . . . . . . . . . . 44
2.3 Critical points. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Periodic points . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Minimal sets and transitivity 69
3.1 Minimal sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Weak control sets . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Dynamic order . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . 80
ix