Optimal Periodic Control
This research monograph deals with optimal periodic control problems for systems governed by ordinary and functional differential equations of retarded type. Particular attention is given to the problem of local properness, i.e. whether system performance
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Fritz Colonius
Optimal Periodic Control
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Fritz Colonius Institut fur Dynamische Systeme, Universitat Bremen Fb 3 Postfach 330440, 2800 Bremen 33, Federal Republic of Germany
Mathematics Subject Classification (1980): 49-02, 49B 10, 49B27, 93-02, 34K35 ISBN 3-540-19249-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-19249-2 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
To Didi Hinrichsen, who taught me how to do mathematics
CONTENTS
Chapter
II
Page
INTRODUCTION
1
OPTIMIZATION THEORY
8
1. First Order Optimality Conditions "2. Second Order Optimality Conditions 3. Further Results III
IV
V
VI
VII
8 18
28
RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
31
1. Structure Theory of Linear Equations 2. Extendability to the Product 3. Nonlinear Equations
31 39 44
STRONG LOCAL MINIMA
48
1. Problem Formulation 2. A Global Maximum Principle
48
WEAK LOCAL MINIMA
65
1. Problem Formulation 2. First Order Necessary Optimality Conditions 3. Second Order Necessary Optimality Conditions
65
LOCAL RELAXED MINIMA
86
1. 2. 3. 4.
86 92
Problem Formulation Relations between Ordinary and Relaxed Problems First Order Necessary Optimality Conditions Second Order Necessary Optimality Conditions
51
69
79
96
101
TESTS FOR LOCAL PROPERNESS
104
1. Problem Formulation 2. Analysis of First Order Conditions
104 106
VIII
IX
3. The n-Test 4. The High-Frequency n-Test 5. Strong Tests
115 122 127
A SCENARIO FOR LOCAL PROPERNESS
129
1. Problem Formulation 2. Controlled Hopf Bifurcations 3. Example: Periodic Control of Retarded Lienard Equations
129 131 136
OPTIMAL PERIODIC CONTROL OF ORDINARY DIFFERENTIAL EQUATIONS 145 1. 2. 3. 4.
Problem Formulation Necessary Optimality Conditions Local Properness under State Constraints Example: Controlled Hopf Bifurcation in a Continuous Flow Stirred Tank Reactor (CSTR)
REFERENCES
145 146 149 151 167
CHAPTER I INTRODUCTION
1. These notes are concerned with optimal periodic control for ordinary and functional differential equations of retarded type. In its simplest version this problem can be stated as follows:
Consider a controlled system
x( t)
f(x(t),u(t)),
t E R+:= [O,co)
( 1)
where x(t) ERn, u(t) E Rm. Look for a T-periodic control function u and a corresponding T-periodic traje
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