Dynamical systems: Stability, controllability and chaotic behavior

Dynamical Systems Werner Krabs • Stefan Pickl Dynamical Systems Stability, Controllability and Chaotic Behavior Pr...

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Dynamical Systems

Werner Krabs • Stefan Pickl

Dynamical Systems Stability, Controllability and Chaotic Behavior

Prof. Dr. Werner Krabs Department of Mathematics Technical University of Darmstadt Schlossgartenstr. 7 64289 Darmstadt Germany

Prof. Dr. Stefan Pickl Universität der Bundeswehr München Department of Computer Science Werner Heisenberg Weg 39 85577 Neubiberg-München Germany [email protected]

ISBN 978-3-642-13721-1 e-ISBN 978-3-642-13722-8 DOI 10.1007/978-3-642-13722-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010932600 © Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

Reflecting modelling dynamical systems by mathematical methods can be enriched by philosophical categories. The following introduction might catch the reader’s interest concerning some interdisciplinary dimensions and completes the holistic approach. “It has been said– and I was among those saying it– that any theory of explanation worth its salt should be able to make good predictions. If good predictions could not be made, the explanation could hardly count as serious. (. . .) I now want to move on to my main point. There is, I claim, no major conceptual difference between the problems of explaining the unpredictable in human affairs and in non-human affairs. There are, it is true, many remarkable successes of prediction in the physical sciences of which we are all aware, but these few successes of principled science making principled predictions are, in many ways, misleading. (. . .) The point I want to emphasize is that instability is as present in purely physical systems as it is in those we think of as characteristically human. Our ability to explain but not predict human behavior is in the same general category as our ability to explain but not predict many physical phenomena. The underlying reasons for the inability to predict are the same. (. . .) The concept of instability which accounts for many of these failures is one of the most neglected concepts in philosophy. We philosophers have as a matter of practice put too much emphasis on the contrast between deterministic and probabilistic phenomena. We have not emphasized enough the salient differences between stable and unstable phenomena. One can argue that the main sources of probabilistic or random behavior lie in instability. We might even want to hold the speculative thesis that the random behavior of quantum systems will itself in turn be explained by unstable behavior of classical dynamical systems. (. . .)

vi

Foreword

Chaos, the original confusion in which all the elements were mixed together, was personified by the Greeks as the most ancient of the gods. Now in the twentieth century, chaos has returned in force to attack that citadel of order and harmony, classical mechanics. We have come to recognize how rare and special are those physical systems whose behavior can be predicted in detail. The naivet and hopes of earlier years will not return. For many phenomena in many domains there are principled reasons to believe that we shall never be able to move from good explanations to good predictions.”

PATRICK SUPPES “EXPLAINING THE UNPREDICTABLE” Lucie Stern Professor of Philosophy, Stanford University Director and Faculty Advisor, Education Program for Gifted Youth, Stanford University

Preface

At the end of the nineteenth century Lyapunov and Poincaré developed the so called qualitative theory of differential equations and introduced geometrictopological considerations which have led to the concept of dynamical systems. In its present abstract form this concept goes back to G.D. Birkhoff. This is also the starting point of Chapter 1 of this book in which uncontrolled and controlled time-continuous and time-discrete systems are investigated under the aspect of stability and controllability. Chapter 1 starts with time-continuous dynamical systems. After the description of elementary properties of such systems it focusses on stability in the sense of Lyapunov and gives applications to systems in the plane such as the mathematical pendulum, to general predator-prey models, and to evolution matrix games. The time-discrete case is divided into the autonomous and the non-autonomous part where the latter is no more a dynamical system in the strong sense. It is the counter part of the time-continuous case where the right-hand side of the system of differential equations which describes the dynamics of the system depends explicitly on the time. Controlled dynamical systems could be considered as dynamical systems in the strong sense, if the controls were incorporated into the state space. We, however, adopt the conventional treatment of controlled systems as in control theory. We are mainly interested in the question of controllability of dynamical systems into equilibrium states. In the non-autonomous time-discrete case we also consider the problem of stabilization. Chapter 3 is concerned with chaotic behaviour of autonomous time discrete systems. We consider three different types of chaos: chaos in the sense of Devaney, disorder chaos and chaos in the sense of Li and Yorke. The chapter ends with two examples of strange (or chaotic) attractors.

viii

Preface

The Appendix A is concerned with a dynamical method for the calculation of Nash equilibria in non-cooperative n-person games. The method is based on the fact that Nash equilibria are fixed points of certain continuous mappings of the Cartesian product of the strategy sets of the players into itself. This gives rise to an iteration method for the calculation of Nash equilibria the set of which can be considered as the Ω-limit set of a time-discrete dynamical system. In Appendix B we consider two optimal control problems in chemotherapeutic treatment of cancer. These two problems are somehow dual to each other and are shown to have solutions of the same type. The authors want to thank Korcan Görg¨ ul¨ u for excellent typesetting. He solved every TEX- problem which occurred in minimal time.

Werner Krabs, Stefan Pickl,

Munich, May 2010

Contents

1

2

Uncontrolled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Abstract Definition of Dynamical Systems . . . . . . . . . . . . . . . . . . 1.2 Time-Continuous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Systems in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Stability: The Direct Method of Lyapunov . . . . . . . . . . . . 1.2.4 Application to a General Predator-Prey Model . . . . . . . . 1.2.5 Application to Evolution Matrix Games . . . . . . . . . . . . . . 1.3 Time-Discrete Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Autonomous Case: Definitions and Elementary Properties . . . . . . . . . . . . . . . 1.3.2 Localization of Limit Sets with the Aid of Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Stability Based on Lyapunov’s Method . . . . . . . . . . . . . . . 1.3.4 Stability of Fixed Points via Linearisation . . . . . . . . . . . . 1.3.5 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Discretization of Time-Continuous Dynamical Systems . 1.3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.8 The Non-Autonomous Case: Definitions and Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.9 Stability Based on Lyapunov’s Method . . . . . . . . . . . . . . . 1.3.10 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.11 Application to a Model for the Process of Hemo-Dialysis

1 1 4 4 8 14 21 32 36

62 64 67 73

Controlled Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Time-Continuous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Problem of Controllability . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Controllability of Linear Systems . . . . . . . . . . . . . . . . . . . . 2.1.3 Restricted Null-Controllability of Linear Systems . . . . . . 2.1.4 Controllability of Nonlinear Systems into Rest Points . .

77 77 77 78 82 85

36 40 42 47 50 55 57

x

Contents

2.1.5 Approximate Solution of the Problem of Restricted Null-Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.1.6 Time-Minimal Restricted Null-Controllability of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.2 The Time-Discrete Autonomous Case . . . . . . . . . . . . . . . . . . . . . . 101 2.2.1 The Problem of Fixed Point Controllability . . . . . . . . . . . 101 2.2.2 Null-Controllability of Linear Systems . . . . . . . . . . . . . . . . 111 2.2.3 A Method for Solving the Problem of Null-Controllability118 2.2.4 Stabilization of Controlled Systems . . . . . . . . . . . . . . . . . . 124 2.2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 2.3 The Time-Discrete Non-Autonomous Case . . . . . . . . . . . . . . . . . . 134 2.3.1 The Problem of Fixed Point Controllability . . . . . . . . . . . 134 2.3.2 The General Problem of Controllability . . . . . . . . . . . . . . 138 2.3.3 Stabilization of Controlled Systems . . . . . . . . . . . . . . . . . . 141 2.3.4 The Problem of Reachability . . . . . . . . . . . . . . . . . . . . . . . . 143 3

Chaotic Behavior of Autonomous Time-Discrete Systems . . 149 3.1 Chaos in the Sense of Devaney . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.2 Topological Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.3 The Topological Entropy as a Measure for Chaos . . . . . . . . . . . . 163 3.3.1 Definition and Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.3.2 The Topological Entropy of the Shift-Mapping . . . . . . . . 165 3.3.3 Disorder-Chaos for One-Dimensional Mappings . . . . . . . . 167 3.4 Chaos in the Sense of Li and Yorke . . . . . . . . . . . . . . . . . . . . . . . . 172 3.5 Strange (or Chaotic) Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

A

A Dynamical Method for the Calculation of NashEquilibria in n−Person Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 A.1 The Fixed Point Theorems of Brouwer and Kakutani . . . . . . . . 195 A.2 Nash Equilibria as Fixed Points of Mappings . . . . . . . . . . . . . . . . 197 A.3 Bi-Matrix Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.4 Evolution Matrix Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

B

Optimal Control in Chemotherapy of Cancer . . . . . . . . . . . . . . 217 B.1 The Mathematical Model and Two Control Problems . . . . . . . . 217 B.2 Solution of the First Control Problem . . . . . . . . . . . . . . . . . . . . . . 220 B.3 Solution of the Second Control Problem . . . . . . . . . . . . . . . . . . . . 224 B.4 Pontryagin’s Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 229

C

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

1 Uncontrolled Systems

1.1 Abstract Definition of Dynamical Systems The concept of dynamical systems has developed out of the qualitative theory of differential equations which was established by Lyapunov and Poincaré in the course of the two last decades of the nineteenth century. As final result of a development which lasted more than half a century the following abstract definition of a dynamical system has grown out: Let X be a metric space with metric d. Further let I be an additive semigroup of real numbers, i.e. a subset I of IR with 0 ∈ I, t, s ∈ I t, s, r ∈ I

=⇒

t + s = s + t ∈ I,

=⇒

(t + s) + r = t + (s + r).

A dynamical system on X, sometimes also called a flow, is then defined by a continuous mapping π : X × I → X with the following properties: (identity property), (a) π(x, 0) = x for all x ∈ X (b) π(π(x, t), s) = π(x, t + s) for all x ∈ X (semigroup property). The metric space X is the space of states of the system and the mapping π describes the temporal change of the system where the time t proceeds in a semigroup I ⊆ IR. As a rule we have I = IR, I = IR+ = {t ∈ IR | t ≥ 0} or I = IN0 = {0, 1, 2, . . .}. In the latter case the dynamical system is called time-discrete. Historically the first example of a dynamical system is the following: S. Pickl and W. Krabs, Dynamical Systems: Stability, Controllability and Chaotic Behavior, DOI 10.1007/978-3-642-13722-8_1, © Springer-Verlag Berlin Heidelberg 2010

1

2


Let W be a non-empty, open and connected subset of IRr and let f : W → IRr be a Lipschitz-continuous mapping, i.e., there exists a constant K > 0 such that for all x, y ∈ W ||f (x) − f (y)||2 ≤ K||x − y||2 where || · ||2 denotes the Euclidian norm in IRr . Then, for every x0 ∈ W , the initial value problem x(t) ˙ = f (x(t)),

x(0) = x0

has a unique solution x(t) = ϕ (x0 , t) which is defined on an open interval −α < t < α for some α > 0 and which belongs to C 1 ((−α, α), W ). We assume α = ∞ and put I = IR, X = W . We equip X with the metric d(x, y) = ||x − y||2 ,

x, y ∈ I.

Then we define π : X × I → X by π(x, t) = ϕ(x, t)

for all x ∈ W

and t ∈ I.

Obviously we have π(x, 0) = ϕ(x, 0) = x

for all

x ∈ X,

i.e., π has the identity property. If t ∈ I is given, then we define, for every x ∈ X: ϕ(x, e s) = ϕ(x, t + s)

for all

s∈I

and conclude ϕ(x, e˙ s) = ϕ(x, ˙ t + s) = f (x(t + s)) = f (ϕ(x, e s)) and ϕ(x, e 0) = ϕ(x, t). This implies π(π(x, t), s) = π(ϕ(x, t), s) = ϕ(x, e s) = ϕ(x, t + s) = π(x, t + s), i.e., π also has the semigroup property. The continuity of the mapping π : X × I → X is a consequence of a well known theorem about the continuous dependence of the solutions of initial value problems on the initial values. Let us return to the general definition of a dynamical system and define an orbit or a trajectory through x ∈ X by the point set [ γI (x) = {π(x, t) | t ∈ I} = {π(x, t)}. (1.1) t∈I

1.1 Abstract Definition of Dynamical Systems

3

Obviously we have x = π(x, 0) ∈ γI (x). For every x ∈ X we then define the limit set by \ LI (x) = γI (π(x, t))

(1.2)

t∈I

where for an arbitrary subset A of X the set A denotes the closure of A.

4


1.2 Time-Continuous Dynamical Systems 1.2.1 Elementary Properties Let I = IR. Then the dynamical system is called time-continuous. For every x ∈ X the positive and negative halftrajectory is defined by γ+ (x) = {π(x, t) | t ≥ 0}

and

γ− (x) = {π(x, t) | t ≤ 0},

(1.3)

respectively. Further the omega and alpha limit set is defined by \ \ Ω(x) = γ+ (π(x, t)) and A(x) = γ− (π(x, t)).

(1.4)

t≥0

t≤0

The omega limit set is of primary interest, because it describes the asymptotic behavior of the dynamical system for t → ∞. In this case we can prove the Lemma 1.1. For every x ∈ X we have n Ω(x) = y ∈ X | y = lim π (x, tn ) for a sequence n→∞ ª (tn )n∈IN0 in IR+ with tn → ∞ .

(1.5)

Proof. 1) Let y ∈ X be such that y = lim π (x, tn ) for a sequence (tn )n∈IN0 in IR+ n→∞

with tn → ∞. Then it follows that y ∈ γ+ (x). However, for every t ≥ 0 we also have y = lim π (x, tn ) = lim π (π(x, t), tn − t) , n→∞

n→∞

since tn − t ≥ 0 for sufficiently large n ∈ IN0 and tn − t → ∞. Hence it follows that y ∈ γ+ (π(x, t)) for every t ≥ 0 which implies y ∈ Ω(x) defined by (1.4). 2) Conversely let y ∈ Ω(x) be defined by (1.4). Then y ∈ γ+ (π(x, t))

for every

t≥0

and there exists a sequence (xn )n∈IN0 in γ+ (π(x, t)) with y = lim xn . n→∞

For every n ∈ IN0 we can also write xn = π (x, t + tn )

for some

t≥0

such that y = lim π (x, t + tn ) n→∞

for all

t ≥ 0.

For every k ∈ IN therefore there exists some nk ∈ IN with d (y, π (x, k + tn )) ≤

1 k

for all

n ≥ nk .

This implies y = lim π (x, k + tnk ) , k→∞

i.e., y belongs to Ω(x) defined by (1.5).

u t

1.2 Time-Continuous Dynamical Systems

5

Definition. A subset A ⊆ X is called invariant, if γIR (x) ⊆ A

for all

x∈A

where γIR (x), for every x ∈ A, is the trajectory defined by (1.1). In words: A ⊆ X is called invariant, if every trajectory through a point x ∈ A is contained in A. Lemma 1.2. A ⊆ X is invariant, if and only if [ γIR (x). A=

(1.6)

x∈A

Proof. 1) Let A be invariant. Then [ [ γIR (x) ⊆ A ⊆ γIR (x), i.e. (1.6) holds true. x∈A

x∈A

2) Conversely it follows from (1.6) that γIR (x) ⊆ A

for all

x ∈ A, i.e., A is invariant.

u t

Proposition 1.3. The closure of an invariant subset of X is also invariant. Proof. Let A ⊆ X be invariant and let y ∈ A. Then we have to show that γIR (y) ⊆ A. Since y ∈ A there is a sequence (yn )n∈IN in A with y = lim yn . n→∞ For every t ∈ IR it therefore follows because of the continuity of the mapping π : X × IR → X that π(y, t) = lim π (yn , t) . n→∞

Because of γIR (yn ) ⊆ A we have π (yn , t) ∈ A for all n and therefore π(y, t) ∈ A for every t ∈ IR which implies that γIR (x) ⊆ A. u t Proposition 1.4. The omega limit set Ω(x) is closed and invariant for every x ∈ X. Proof. For every x ∈ X the set Ω(x) is closed as the intersection of closed sets. In order to prove invariance of Ω(x) we choose some y ∈ Ω(x). Then because of Lemma 1.1 there is a sequence (tn )n∈IN0 in IR+ with tn → ∞ such that y = lim π (x, tn ). If we put n→∞

xn = π (x, tn )

for every n ∈ IN0 ,

then it follows y = lim xn and the continuity of π : X × IR → X implies n→∞

π(y, t) = lim π (xn , t) n→∞

for all

t ∈ IR.

6


Now we have π (xn , t) = π (π (x, tn ) , t) = π (x, tn + t)

and

tn + t → ∞

for every t ∈ IR. Therefore it follows that π(y, t) ∈ Ω(x)

for all

t ∈ IR and hence

γIR (y) ⊆ Ω(x)

which shows that Ω(x) is invariant.

u t

For every x ∈ X we call the function t 7→ π(x, t), t ∈ IR, a movement through x. Definition. A movement π(x, t), t ∈ IR, through x ∈ X is called positively compact, if the closure of the positive halftrajectory γ+ (x) defined by (1.3) is compact which is equivalent to the existence of a compact subset K of X with π(x, t) ∈ K for all t ≥ 0. With this definition we can prove the following Proposition 1.5. Let π(x, t), t ∈ IR, for some x ∈ X, be a positively compact movement. Then the omega limit set Ω(x) defined by (1.4) is non-empty, compact, invariant and connected. Proof. By Proposition 1.4 the set Ω(x) is invariant. By assumption γ+ (x) with γ+ (x) = {π(x, t) | t ≥ 0} is compact. Further we have γ+ (π(x, t)) ⊆ γ+ (x)

for all

t ≥ 0.

Hence γ+ (π(x, t))

is compact for all t ≥ 0.

Finally we have γ+ (π (x, t2 )) ⊆ γ+ (π (x, t1 ))

for

t2 ≥ t1 ≥ 0

which implies that Ω(x) is non-empty and compact. Let us assume that Ω(x) is not connected. Then there exist two disjoint, nonempty, open subsets A and B of X with Ω(x) = Ω(x)∩(A∪B) where A∩Ω(x) and B ∩ Ω(x) are non-empty and we can find sequences (sn )n∈IN and (tn )n∈IN in IR+ such that 0 < s1 < t1 < . . . < sn < tn < . . . , sn → ∞, tn → ∞, π (x, sn ) ∈ A and π (x, tn ) ∈ B for all n ∈ IN. Since, for every n ∈ IN, the path π (x, [sn , tn ]) is connected, there is, for every n ∈ IN, some t∗n ∈ [sn , tn ] with π (x, t∗n ) ∈ A ∪ B. From π (x, [sn , tn ]) ⊆ γ ¡ + (x) ¡ and¢¢the compactness of γ+ (x) it follows that there exists a subsequence π x, t∗ni i∈IN and a point y ∈ X \ (A ∪ B) with y = lim π (x, tni ), i.e. i→∞

y ∈ Ω(x) by Lemma 1.1. However, this is impossible because of Ω(x) = Ω(x) ∩ (A ∪ B). This contradiction shows that Ω(x) is connected. u t


7

Definition. A subset M ⊆ X is called minimal, if it is non-empty, closed and invariant and none of its genuine subsets has these properties. Proposition 1.6. Every non-empty, compact and invariant subset A of X contains a minimal subset M of X. Proof. Let A be the set of all non-empty, compact and invariant subsets of A. The set A is semi-ordered with respectTto inclusion. Now let B a well ordered B is non-empty, in A and a lower subset of A. Then the intersection B∈B

bound of B. By Zorn’s Lemma it follows that A possesses minimal elements. u t As an immediate consequence of Proposition 1.5 and Proposition 1.6 we obtain the Corollary. If π(x, t), t ∈ IR, for some x ∈ X, is a positively compact movement through x, then the omega limit set Ω(x) contains a minimal subset. Definition. A point x ∈ X is called a rest point or equilibrium of a flow π : X × IR → X, if π(x, t) = x for all t ∈ IR. If x ∈ X is a rest point of a flow, then M = {x} is obviously minimal. Definition. A point x ∈ X and the movement π(x, t), t ∈ IR, through x is called periodic, if there exists some p > 0 with π(x, t + p) = π(x, t)

for all

t ∈ IR.

The number p is called the period of the movement π(x, t), t ∈ IR, through x. Proposition 1.7. If x ∈ X is periodic, then the trajectory γIR (x) is minimal. S π(x, t). Therefore, Proof. The periodicity of x ∈ X implies γIR (x) = t∈[0,p]

γIR (x) is compact, hence closed. Now let y ∈ γIR (x). Then there is some ty ∈ [0, p] with y = π (x, ty ) and for every s ∈ IR it follows that π(y, s) = π (π (x, ty ) , s) = π (x, s + ty ) ∈ γIR (x), hence, γIR (y) ⊆ γIR (x) which proves the invariance of γIR (x). Now let z ∈ γIR (x) be chosen arbitrarily. Then there is some tz ∈ [0, p] with z = π (x, tz ). This implies that z = π (x, tz + ty − ty ) = π (π (x, ty ) , tz − ty ) = π (y, tz − ty ) ∈ γIR (y), hence, γIR (x) ⊆ γIR (y), which implies γIR (x) = γIR (y)

for all

y ∈ γIR (x).

Therefore, there cannot exist a genuine non-empty, closed subset of γIR (x) u t which is invariant. This shows the minimality of γIR (x).

8


1.2.2 Systems in the Plane We start with the historically first example of a dynamical system for the case r = 2 (see Section 1.1). This is defined by two differential equations of the form ¾ x˙ 1 = f1 (x1 , x2 ) , (1.7) (x1 , x2 ) ∈ W ⊆ IR2 , x˙ 2 = f2 (x1 , x2 ) , where W is non-empty, open and connected. We assume f1 , f2 ∈ C 1 (W ) which implies that for every point (x10 , x20 ) ∈ W there is exactly one solution x(t) = ϕ (t, x0 ) , t ∈ (−α, α) for some α > 0, of (1.7) with ϕ (·, x0 ) ∈ C 1 ((−α, α), W ) and ϕ (0, x0 ) = x0 = (x10 , x20 ) . (1.8) If X = W is equipped with the metric d(x, y) = ||x − y||2 ,

x, y ∈ W,

where || · ||2 denotes the Euclidian norm in IR2 and if we assume α = ∞, then (as been shown in Section 1.1) by π(x, t) = ϕ(t, x),

t ∈ IR, x ∈ X,

(1.9)

a (time-continuous) dynamical system π : X × IR → X is defined. A point x ∈ X is a rest point of the flow π, if f1 (x) = 0

and

f2 (x) = 0.

We already observed that every rest point of π as a one point subset of X is minimal. The question arises then whether there exist further minimal sets. For the case that these are compact the following proposition gives a complete answer. Proposition 1.8. Let M ⊆ X be compact and minimal. Then either M is a rest point of the flow (1.9) or a periodic movement π(x, t), t ∈ IR+ , through some x ∈ M . In the proof of this proposition we need the following Proposition 1.9. If, for some x ∈ X, the positive orbit γ+ (x) of (1.3) and the omega limit set Ω(x) have non-rest points in common, then γ+ (x) is a periodic orbit, i.e., there exists some p > 0 such that π(x, t + p) = π(x, t)

for all

t ≥ 0.

For the proof of this proposition we refer to the book “Ordinary Differential Equations” by J.K. Hale (Wiley-Interscience 1969).


9

Proof. Choose x ∈ M arbitrarily. Then γ+ (x) ∈ M . By Proposition 1.4 and Proposition 1.5 the omega limit set Ω(x) is in M , non-empty, closed and invariant which implies, by the minimality of M , that M = Ω(x). If M contains a rest point y, then it follows that M = {y} = {x}. If M does not contain a rest point, then it follows from γ+ (x) ⊆ M = Ω(x) that γ+ (x) and Ω(x) have common non-rest points so that according to Proposition 1.9 γ+ (x) is a periodic orbit. u t Further we have Proposition 1.10. Let γIR (x), for some x ∈ X, be a positively compact orbit, i.e., the movement π(x, t), t ∈ IR, through x is positively compact. Further, let the omega limit set Ω(x) contain no rest point. Then Ω(x) = γ+ (x) is a periodic orbit. For the proof of this proposition we need the following Proposition 1.11. Does Ω(x), for some x ∈ X, contain non-rest points and a periodic orbit γ+ (x) then Ω(x) = γ+ (x). For the proof of this proposition we again refer to the book by J.K. Hale mentioned above. Proof. By Proposition 1.5 Ω(x) is non-empty, compact and invariant. Therefore, Ω(x) contains a minimal subset M of X and M does not contain rest points. By Proposition 1.8 M is a periodic movement π(x, t), t ∈ IR+ , through u t x. By Proposition 1.11, therefore, Ω(x) = γ+ (x) is a periodic orbit. Let us demonstrate Proposition 1.10 by two examples

10


1) The Mathematical Pendulum We consider the movement of a nonlinear plane pendulum with a fixed suspension point as being depicted in the following picture: S

S S ϕ S S l S S

l

z m

S Szm S S S S

mg sin ϕ

By Newton’s law f orce = mass × acceleration we obtain the following equation of movement m · l · ϕ(t) ¨ = −mg sin ϕ(t),

t ∈ IR,

(1.10)

where m and l is the mass and the length of the pendulum, respectively, g is the gravity constant and ϕ(t) the vibration angle of the time t. The expression l · ϕ(t) ¨ is the acceleration along the orbit and −mg sin ϕ(t) the force upon the mass of the pendulum at the time t. If we define x1 (t) = ϕ(t)

and x2 (t) = ϕ(t), ˙

t ∈ IR,

(1.11)

then (1.10) turns out to be equivalent to x˙ 1 (t) = x2 (t), g x˙ 2 (t) = − sin x1 (t), t ∈ IR. (1.12) l g With f1 (x1 , x2 ) = x2 , f2 (x1 , x2 ) = − · sin x1 , (x1 , x2 ) ∈ IR2 , the system l (1.12) takes the form of (1.7) with W = IR2 . Obviously f1 , f2 ∈ C 1 (W ) such that for every point (x10 , x20 ) = (ϕ0 , ϕ˙ 0 ) ∈ IR2 there exists exactly one solution (x1 (·), x2 (·)) ∈ C 1 ((−α, α), W ) of (1.12) for some α > 0 with x1 (0) = x10 ,

x2 (0) = x20 .

(1.13)


11

With (1.11) it follows from (1.10) that g x ¨1 = − sin x1 l which implies that x˙ 21 =

2g cos x1 + C l

for some

C ∈ IR

such that with (1.13) we obtain x220 =

2g cos x10 + C. l

This leads to x22 = x˙ 21 = x220 +

2g 4g (cos x1 − cos x10 ) ≤ x220 + l l

and finally to

r

4g . l If we choose X = (−π, π) × IR, then for every point (x10 , x20 ) ∈ X the unique solution (x1 (·), x2 (·)) ∈ C 1 ((−α, α), X) of (1.12) with (1.13) satisfies r 4g |x2 | ≤ x220 + |x1 (·)| ≤ π, l |x2 | ≤

x220 +

so that we can choose α = ∞. If one defines, for every (x10 , x20 ) ∈ X and every t ∈ IR π (x10 , x20 , t) = (x1 (t), x2 (t)) ,

t ∈ IR,

where (x1 (·), x2 (·)) is the unique solution of (1.12), (1.13) in C 1 (IR, X), then one obtains a flow such that for every x ∈ X the movement π(x, t), t ∈ IR, through x is positively compact. Obviously (0, 0) is the only rest point of (1.12) in X. Now let y ∈ Ω(x) be given for some x ∈ X. By Lemma 1.1 there exists a sequence (tn )n∈IN0 in IR+ with tn → ∞ and y = lim π (x, tn ). This implies n→∞

y22 = x22 +

2g (cos y1 − cos x1 ) . l

If y were (0, 0), then it would follow that 0 = x22 +

2g (1 − cos x1 ) l

which is only possible in the case x1 = x2 = 0. Therefore, if x = (0, 0), no y ∈ Ω(x) can be y = (0, 0), i.e., Ω(x) does not contain rest points. By Proposition 1.10 it therefore follows that Ω(x) = γ+ (x) is a periodic orbit.

12


2) A Predator-Prey Model This model is described by the differential equations x˙ 1 = −c x1 + a x1 x2 , x˙ 2 = r x2 − p x1 x2 ,

x1 > 0, x2 > 0,

(1.14)

where a, c, p, r ∈ IR are given positive constants. By x1 (t) and x2 (t) the size of a predator and prey population, respectively, at time t within a given habitat are denoted. The first equation of (1.14) describes the temporal change of the size of the predator and the second equation that of the prey population. In this case we have ª © X = W = (x1 , x2 ) ∈ IR2 | x1 > 0 and x2 > 0 and all the assumptions we have made with respect to (1.7) are satisfied. We shall show that every solution of (1.14) is bounded and hence defined on all of IR such that (1.14) defines a flow π : X × IR → X. The only rest point of this flow is given by r c x1 = , x2 = . p a In order to apply Proposition 1.10 we at first show that for every x = (x1 , x2 ) ∈ X the orbit γIR (x) is positively compact. For that purpose we consider the homeomorphic mapping T : X → IR2 which is defined by µ ¶ x1 x2 T (x1 , x2 ) = (u1 , u2 ) = ln , ln , (x1 , x2 ) ∈ X, x1 x2 and whose inverse mapping T −1 : IR2 → X is given by T −1 (u1 , u2 ) = (x1 , x2 ) = (x1 eu1 , x2 eu2 ) ,

(u1 , u2 ) ∈ IR2 .

The system (1.14) is then transformed into u˙ 1 = −c (1 − eu2 ) , u˙ 2 = r (1 − eu1 ) ,

(u1 , u2 ) ∈ IR2 .

(1.15)

Further it follows that T (x1 , x2 ) = (0, 0). If (x1 (t), x2 (t)) , t ∈ IR, is a solution of (1.14), then x1 (t) x2 (t) , u2 (t) = ln u1 (t) = ln x1 x2 is a solution of (1.15). If (u1 (t), u2 (t)) is a solution of (1.15), then x1 (t) = x1 eu1 (t) , is a solution of (1.14).

x2 (t) = x2 eu2 (t)


13

If γIR (x1 , x2 ) is an orbit of the system (1.14) through (x1 , x2 ) ∈ X, then T (γIR (x1 , x2 )) is an orbit of the system (1.15) through (u1 , u2 ) = T (x1 , x2 ). If we can show that every orbit of the system (1.15) is positively compact, then this is also true for every orbit of the system (1.14). We consider at first a so called Lyapunov function of the form (u1 , u2 ) ∈ IR2 .

V (u1 , u2 ) = r (eu1 − u1 ) + c (eu2 − u2 ) − r − c, For this we have V (0, 0) = 0

and

V (u1 , u2 ) > 0

for all

(u1 , u2 ) 6= (0, 0)

and for every (u1 (t), u2 (t)) of (1.15) it follows that ³ ´ ³ ´ d V (u1 (t), u2 (t)) = r eu1 (t) − 1 u˙ 1 + c eu2 (t) − 1 u˙ 2 = 0 dt for all t ∈ (−α, α) = definition interval of (u1 (t), u2 (t)). This further implies that ³ ´ ³ ´ V (u1 (t), u2 (t)) = r eu1 (t) − u1 (t) + c eu2 (t) − u2 (t) − r − c = k (1.16) for all t ∈ (−α, α) where k ≥ 0 is a constant. This equality shows that the orbit γIR (u1 (0), u2 (0)) is bounded and hence positively compact. This also shows α = ¡∞ (see above!). Hence also the ¢ inverse image T −1 (γIR (u1 (0), u2 (0))) = γIR x1 eu1 (0) , x2 eu2 (0) is positively compact. In order to apply Proposition 1.10 we have to show that in case (x1 , x2 ) 6= (x1 , x2 ) the limit set Ω (x1 , x2 ) which belongs to γIR (x1 , x2 ) does not contain the only rest point (x1 , x2 ) of the system (1.14). For this it suffices to see that in the case (u1 (0), u2 (0)) 6= (0, 0) the limit set Ω (u1 (0), u2 (0)) which belongs to γIR (u1 (0), u2 (0)) does not contain the point (0, 0). For that purpose we consider a limit point (u1 , u2 ) ∈ Ω (u1 (0), u2 (0)). Then by Lemma 1.1 there exists a sequence (tn )n∈IN0 with tn → ∞ and u1 = lim u1 (tn ) , n→∞

u2 = lim u2 (tn ) n→∞

where (u1 (t), u2 (t)) is a solution of (1.15). From (1.16) it then follows that V (u1 , u2 ) = k = V (u1 (0), u2 (0)) > 0 which implies (u1 , u2 ) 6= (0, 0). By Proposition 1.10 we therefore obtain the Result. The omega limit set Ω (x1 , x2 ) of an arbitrary orbit γIR (x1 , x2 ) through (x1 , x2 ) ∈ X with (x1 , x2 ) 6= (x1 , x2 ) is equal to γ+ (x1 , x2 ) and simultaneously a periodic orbit.

14


1.2.3 Stability: The Direct Method of Lyapunov As in Section 1.2.1 we at first consider a dynamical system π on a metric space (X, d) with I = IR. Definition. A rest point x ∈ X of the flow π : X × IR → X is called stable, if for every ε > 0 there exists a δ = δ(ε) such that d(x, y) ≤ δ

=⇒

d(x, π(x, t)) ≤ ε

for all

t ≥ 0,

and asymptotically stable, if x is stable and there exists a δ0 > 0 such that lim π(y, t) = x

t→∞

for all

y∈X

with

d(x, y) ≤ δ0 .

In words stability means that a positive semi-orbit γ+ (y) stays arbitrarily close to x, if y is sufficiently close to x. Asymptotic stability means that in addition there exists a neighborhood {y ∈ X | d(x, y) ≤ δ0 } = Vδ0 (x) such that for every y ∈ Vδ0 (x) the positive semi-orbit γ+ (y) gets arbitrarily close to the rest point x. After this definition we turn to the first example of a time-continuous dynamical system in Section 1.1 and consider again an autonomous system of the form x˙ = f (x) (1.17) where f ∈ C 1 (W, IRn ) , W ⊆ IRn is open and connected. For every x ∈ W then there exists exactly one solution ϕ = ϕ(x, t), t ∈ (−α, α) for some α > 0 of (1.17) with ϕ ∈ C 1 ((−α, α), W ) and ϕ(x, 0) = x. We assume that α = ∞. If we define π : X × IR → X, X = W , by π(x, t) = ϕ(x, t),

x ∈ X, t ∈ IR,

then we obtain a flow as being shown in Section 1.1. A point x ∈ X is a rest point of this flow, if and only if f (x) = On = nullvector of IRn . In order to derive sufficient conditions for the stability and asymptotic stability of such a rest point we will apply the so called direct method of Lyapunov. For that purpose we start with the following Definition. A function V ∈ C 1 (X) is called a Lyapunov function with respect to f , if for all x ∈ X. V˙ (x) = grad V (x)T f (x) ≤ 0 With this definition we formulate the


15

Proposition 1.12. If there exists a Lyapunov function V ∈ C 1 (X) with respect to f which is positive definite with respect to some rest point x ∈ X, i.e., which satisfies the condition V (x) = 0

and

V (x) > 0

for all

x ∈ X, x 6= x,

and

V˙ (x) < 0

for all

x ∈ X, x 6= x,

then x is stable. If in addition V˙ (x) = 0

then x is asymptotically stable. Proof. Since X is open, there is some r > 0 such that Br (x) ⊆ X where Br (x) denotes the ball around x with radius r. Now let ε ∈ (0, r) be chosen arbitrarily. Then kε = min {V (x) | ||x − x||2 = ε} > 0, and because of the continuity of V there is some δ = δ(ε) with 0 < δ < ε and for ||x − x||2 ≤ δ.

V (x) < kε

Now let y ∈ Bδ (x). Then it follows for the solution x = x(t) of (1.17) with x(0) = y dV (x(t)) ˙ = V˙ (x(t)) = grad V (x(t))T f (x(t)) = grad V (x(t))T x(t) dt and hence Z V (x(t)) − V (y) = 0

t

dV (x(τ )) dτ = dt

Z

t

V˙ (x(τ )) dτ ≤ 0

0

for every t ≥ 0. Therefore we have V (x(t)) ≤ V (y) < kε

for all t ≥ 0

which implies x(t) ∈ Bε (x) for all t ≥ 0. If this were not the case, i.e., x(t) 6∈ Bε (x) or ||x − x(t)||2 > ε for some t > 0, then by the intermediate value theorem for continuous functions there exists some t∗ ∈ (0, t) with ||x−x (t∗ ) ||2 = ε which implies V (x (t∗ )) ≥ kε , a contradiction to V (x (t∗ )) < kε . Therefore ||x − π(y, t)||2 ≤ ε which shows the stability of x.

for all

t ≥ 0 and all

y ∈ Bδ (x)

16


The stability of x implies in particular the existence of b0 > 0 and h > 0 such that for the solution x = x(t) of (1.17) with x(0) = x0 ∈ X it follows that ||x − x(t)||2 < h

for all

t ≥ 0, if

||x0 − x||2 < b0 .

Also, for every sufficiently small ε > 0, there exists some δ = δ(ε) with ||x − x(t)||2 < ε

for all

t ≥ 0, if

||x0 − x||2 < δ.

In order to show asymptotic stability it suffices to show the existence of some Tε > 0 with ||x − x(t)||2 < ε

for all

t ≥ Tε , if ||x0 − x||2 < b0 .

Assumption: There exists some x0 ∈ X with ||x0 − x||2 < b0

and

||x − x(t)||2 ≥ δ(ε) for some t ≥ 0.

Let γ > 0 be chosen such that V˙ (x(t)) < − γ,

if δ(ε) ≤ ||x − x(t)||2 < h.

Then it follows that dV /dt (x(t)) < − γ and hence V (x(t)) < V (x0 ) − γ t,

if δ(ε) ≤ ||x − x(t)||2 < h.

Let β, k > 0 be chosen such that β ≤ V (x) ≤ k,

if δ(ε) ≤ ||x − x||2 < h.

Then we put Tε = (k − β)/γ and conclude for t > Tε that V (x(t)) < V (x0 ) − γ Tε ≤ k − (k − β) = β. Therefore there must exist some t0 ∈ [0, Tε ] with ||x − x (t0 ) ||2 < δ(ε). The stability of x then implies ||x − x(t)||2 < ε

for all

t ≥ t0 ,

This completes the proof.

hence for all

t ≥ Tε . u t

Let us demonstrate Proposition 1.12 by two examples. At first we consider the system ´ ³ u˙ 1 (t) = −c 1 − eu2 (t) , ´ ³ t ∈ IR. (1.15) u˙ 2 (t) = r 1 − eu1 (t) , The only rest point of this system is (0, 0).


17

If we define V : X = IR2 → IR by V (u1 , u2 ) = r (eu1 − u1 ) + c (eu2 − u2 ) − r − c for (u1 , u2 ) ∈ IR2 , then it follows that V (u1 , u2 ) > 0 for all (u1 , u2 ) 6= (0, 0). ¡ ¢ The function V = V (u1 , u2 ) is obviously in C 1 IR2 and positive definite with respect to (0, 0). Further it follows that V (0, 0) = 0

and

V˙ (u1 , u2 ) = −rc (eu1 − 1) (1 − eu2 ) + rc (eu2 − 1) (1 − eu1 ) = 0 for all (u1 , u2 ) ∈ X, i.e., V is a Lyapunov function. By Proposition 1.12 therefore (0, 0) is a stable rest point. As a second example we choose a modification of the system (1.14) and consider the system x˙ 1 (t) = −c x1 (t) − b x21 (t) + a x1 (t)x2 (t), x˙ 2 (t) = r x2 (t) − k x22 (t) − p x1 (t)x2 (t),

t ∈ IR,

(1.18)

where a, b, c, k, p, r ∈ IR are positive constants. We again choose ª © X = (x1 , x2 ) ∈ IR2 | x1 > 0, x2 > 0 . If we require that ra − kc > 0 then the only rest point (x1 , x2 ) ∈ X of the system (1.18) is given by x1 =

ra − kc , bk + ap

x2 =

br + pc . bk + ap

If we again apply the homeomorphic mapping T : X → IR2 which is given by µ ¶ x1 x2 T (x1 , x2 ) = (u1 , u2 ) = ln , ln , (x1 , x2 ) ∈ X, x1 x2 then the system (1.18) is transformed into the system ³ ´ ³ ´ u˙ 1 (t) = b x1 1 − eu1 (t) − a x2 1 − eu2 (t) , ³ ´ ³ ´ u˙ 2 (t) = p x1 1 − eu1 (t) + k x2 1 − eu2 (t) ,

t ∈ IR,

(1.19)

and the only rest point of system (1.19) is given by (0, 0) = T (x1 , x2 ). If (u1 (t), u2 (t)) ,¡ t ∈ (−α, α) for some α > 0, is a solution of (1.19), then ¢ (x1 (t), x2 (t)) = x1 eu1 (t) , x2 eu2 (t) , t ∈ (−α, α), is a solution of (1.18) and if this is the case, then (u1 (t), u2 (t)) = (ln (x1 (t)/x1 ) , ln (x2 (t)/x2 )) is a solution of (1.19).

18


If we define a function V : IR2 → IR by V (u1 , u2 ) = p x1 (eu1 − u1 ) + a x2 (eu2 − u2 ) − p x1 − a x2 ¡ ¢ for (u1 , u2 ) ∈ IR2 , then V ∈ C 1 IR2 and V (0, 0) = 0

and

V (u1 , u2 ) > 0 for all

(u1 , u2 ) 6= (0, 0).

Hence V is positive definite with respect to (0, 0). Further it follows that 2 2 V˙ (u1 , u2 ) = −pb x21 (1 − eu1 ) − ak x22 (1 − eu2 ) ≤ 0

(1.20)

for all (u1 , u2 ) ∈ IR2 . Now let (u1 (t), u2 (t)) , t ∈ (−α, α) for some α > 0 a solution of (1.19). Then for every t ∈ IR it follows that µ ¶ d u˙ 1 (t) T ˙ V (u1 (t), u2 (t)) ≤ 0, V (u1 (t), u2 (t)) = grad V (u1 (t), u2 (t)) = u˙ 2 (t) dt hence V (u1 (t), u2 (t)) ≤ V (u1 (0), u2 (0))

for all

t ∈ IR

and {(u1 (t), u2 (t)) , t ∈ IR} ⊆ V −1 [0, V (u1 (0), u2 (0))] which shows that (u1 (t), u2 (t)) is defined on all of IR. The inequality (1.20) shows that V : IR2 → IR is a Lyapunov function. From Proposition 1.12 it therefore follows that (0, 0) is a stable rest point. In addition it follows from (1.20) that V˙ (0, 0) = 0

and

V˙ (u1 , u2 ) < 0

for all

(u1 , u2 ) 6= (0, 0)

which implies that (0, 0) is an asymptotically stable rest point. After these two examples we consider a linear system x˙ = A x,

x ∈ IRn ,

(1.21)

where A is a real n × n−matrix. Then for every x ∈ IRn there exists a unique solution ϕ ∈ C 1 (IR, IRn ) with ϕ(0) = x and by the definition π(x, t) = ϕ(x, t)

for all x ∈ X = IRn , t ∈ I = IR,

we obtain a dynamical system π : X × I → X. We assume that all the eigenvalues of A have negative real parts which implies that A is non-singular and x = Θn is the only rest point of the system (1.21).


19

Now let C be an arbitrary symmetric and positive definite matrix. Then we define the symmetric and positive definite matrix Z ∞ T e tA C e tA dt B= 0

(B is well defined, since there exist positive constants K and α with ||e tA || ≤ K e−αt , t ≥ 0). If we define V (x) = xT B x,

x ∈ IRn ,

it follows that V : IRn → IR+ is positive definite with respect to Θn . Further it follows that ¡ ¢ V˙ (x) = 2 xT BA x = xT BA x + xT AT B x = xT BA + AT B x < 0 for all x ∈ IRn and V˙ (Θn ) = 0 because of Z ∞ ´ d ³ tAT AT B + BA = e C e tA dt = −C. dt 0 Hence V is a Lyapunov function and by Proposition 1.12 it follows that Θn is an asymptotically stable rest point of the system (1.21). The asymptotic stability of Θn also implies that the eigenvalues of the matrix A have only negative real parts (see, for instance, H.W. Knobloch and F. Kappel: Gewönliche Differentialgleichungen, Verlag B.G. Teubner, Stuttgart 1974, Kap. III, Satz 7.5). Let us return to the system x˙ = f (x)

(1.17)

with f ∈ C 1 (W, IRn ) where W ⊆ IRn is open and connected. Now let x ∈ W be a rest point of this system, i.e., f (x) = Θn . Then for every x ∈ W there is a representation f (x) = A (x − x) + r (x − x) µ ¶ ∂fi (x) is the Jacobi matrix of f in x and where A = ∂xj 1≤i,j≤n r : (W − x) → IRn a vector function with lim

||x−x||2 →0

||r (x − x) ||2 = 0. ||x − x||2

We assume that A only possesses eigenvalues with negative real parts.

20


According to the considerations above then there exists a symmetric and positive definite matrix B such that AT B + BA = −I

where I = n × n − unit matrix.

If we define T

V (x) = (x − x) B (x − x)

for

x ∈ W,

then V is positive definite with respect to x. Further it follows that T V˙ (x) = (x − x) B f (x) + f (x)T B (x − x) ¢ T ¡ T = (x − x) BA + AT B (x − x) + 2 (x − x) B r (x − x)

≤ −||x − x||22 + 2||x − x||2 ||B|| ||r (x − x) ||2 = −||x − x||22 (1 − 2 ||B|| E (||x − x||2 )) where E (||x − x||2 ) =

||r (x − x) ||2 →0 ||x − x||2

for

||x − x||2 → 0.

Let r > 0 be such that E (||x − x||2 ) ≤

1 4 ||B||

for all

x ∈ IRn

with

||x − x||2 ≤ r

and Br (x) = {x ∈ IRn | ||x − x||2 ≤ r} ⊆ W . Then it follows that V˙ (x) = 0

and

V˙ (x) < 0

for all

x ∈ Br (x) .

As a consequence of the proof of Proposition 1.12 we therefore obtain Proposition 1.13.µLet x ∈ ¶ W be a rest point of the system (1.17) such that ∂fi the Jacobi matrix (x) has only eigenvalues with negative real ∂xj 1≤i,j≤n parts. Then x is asymptotically stable. Let us demonstrate this result by the system u˙ 1 = −a x2 (1 − eu2 ) , u˙ 2 = p x1 (1 − eu1 ) + k x2 (1 − eu2 ) , where x1 =

c´ 1³ r−k , p a

x2 =

(u1 , u2 ) ∈ IR2 c , a

a, c, k, p, r > 0 and ra − kc > 0. The point (0, 0) is the only rest point of this system.

(1.22)


21

The Jacobi matrix of the right-hand side of (1.22) is given by µ

0 a x 2 e u2 u1 −p x1 e −k x2 e u2

¶ ,

(u1 , u2 ) ∈ IR2 .

For u1 = 0, u2 = 0 we obtain the matrix µ A=

0 a x2 −p x1 −k x2

¶

with the eigenvalues λ1, 2

k x2 ± =− 2

r

k 2 x22 − ap x1 x2 4

which are either both real and negative or have a negative real part. By Proposition 1.13 the point (0, 0) is an asymptotically stable rest point of the system (1.22). 1.2.4 Application to a General Predator-Prey Model We consider n populations Xi , i = 1, . . . , n, of animals or plants that are living in mutual predator-prey relations or are pairwise neutral to each other. Let us denote by xi (t) the size of population Xi at the time t. We assume the temporal development of these sizes to be described by the following system of differential equations  x˙ i (t) = ci +

n X

 cij xj (t) xi (t)

(1.23)

j=1

for

i = 1, . . . , n

cii ≤ 0

for

where i = 1, . . . , n

and, for i 6= j, cij > 0,

if Xi = predator and Xj = prey,

cij < 0, if Xi = prey and Xj = predator, cij = cj i = 0, if Xi and Xj are neutral to each other. This model is a generalization of the predator-prey models for two populations which were considered in Sections 1.2.2 and 1.2.3.

22


Let us further assume that there is a rest point or an equilibrium state x = (x1 , . . . , xn ) of (1.23), i.e. a solution of the linear system n X

cij xj = −ci ,

i = 1, . . . , n,

(1.24)

j=1

with xj > 0,

j = 1, . . . , n.

(1.25)

Then (1.23) can be written in the form   n X cij (xj (t) − xj ) xi (t) x˙ i (t) = 

(1.26)

j=1

for

i = 1, . . . , n.

We assume that, for every initial state x0 ∈ IRn with x0i > 0 for all i = 1, . . . , n, there is exactly one solution x(t) = (x1 (t), . . . , xn (t)) of (1.26) with xi (0) = x0i ,

xi (t) > 0

for all

t > 0 and

i = 1, . . . , n.

The system (1.26) has the form (1.17) with   P n c1j (xj − xj ) x1   j=1     .. f (x) =  , .   n  P cnj (xj − xj ) xn

x ∈ IRn ,

j=1

and the Jacobi matrix of f in x is given by   c11 x1 . . . c1n x1  ..  . Jf (x) =  ... .  cn1 xn . . . cnn xn

(1.27)

By Proposition 1.13 the equilibrium state x is asymptotically stable, if the Jacobi matrix (1.27) has only eigenvalues with negative real parts. Let us consider the case n = 2 where X1 is the population of predators and X2 that of preys. The system (1.23) reads x˙ 1 (t) = (c1 + c11 x1 (t) + c12 x2 (t)) x1 (t), x2 (t) = (c2 + c21 x1 (t) + c22 x2 (t)) x2 (t)

(1.23)

with c11 ≤ 0, c22 ≤ 0, c12 > 0, c21 < 0. It is reasonable to assume that c1 < 0 and c2 > 0.


23

The system (1.24) reads c11 x1 + c12 x2 = −c1 , c21 x1 + c22 x2 = −c2 and has the unique solution x1 =

−c1 c22 + c2 c12 , c11 c22 − c12 c21

x2 =

c1 c21 − c2 c11 . c11 c22 − c12 c21

(1.28)

This implies that x1 > 0 and x2 > 0, if and only if the condition −c1 c22 + c2 c12 > 0

(1.29)

is satisfied. This is for instance the case, if c22 = 0. The Jacobi matrix

µ Jf (x) =

c11 x1 c12 x1 c21 x2 c22 x2

¶

of f at x = (x1 , x2 ) has the eigenvalues s 2 c11 x1 + c22 x2 (c11 x1 + c22 x2 ) ± − (c11 c22 − c12 c21 ) x1 x2 . λ1, 2 = 2 4 This implies < (λ1, 2 ) < 0, if c11 + c22 < 0. Proposition 1.13 therefore leads to the Result. If the conditions c11 + c22 < 0 and (1.29) are satisfied, then the system (1.23) has exactly one equilibrium state x = (x1 , x2 ) with x1 > 0 and x2 > 0 which is given by (1.28) and which is asymptotically stable. In Section 1.2.3 we have obtained the same result under the assumption c11 < 0 and c22 < 0 by applying Lyapunov’s method. We also considered the case c11 = c22 = 0 in which it only follows that x = (x1 , x2 ) is a stable equilibrium state. Next we consider the case n = 3 where we assume that X1 and X2 are predator populations that have the X3 −population as prey and are neutral to each other. This leads to c12 = 0, c21 = 0,

c13 > 0,

c23 > 0,

c31 < 0,

c32 < 0.

c11 < 0,

c22 < 0,

c33 < 0.

In addition we assume

24


Finally it is reasonable to assume that c1 < 0,

c2 < 0,

c3 > 0.

The system (1.24) reads c11 x1 + c12 x2 + c13 x3 = −c1 , c21 x1 + c22 x2 + c23 x3 = −c2 , c31 x1 + c32 x2 + c33 x3 = −c3 . and has the unique solution (−c23 c32 + c22 c33 ) c1 + c13 c32 c2 − c22 c13 c3 , c11 c23 c32 − c22 c33 c11 + c22 c13 c31 (c33 c11 − c13 c31 ) c2 − c23 c11 c3 + c23 c31 c1 x2 = , c11 c23 c32 − c22 c33 c11 + c22 c13 c31 −c2 c32 c11 + c3 c22 c11 − c1 c22 c31 x3 = > 0. c11 c23 c32 − c22 c33 c11 + c22 c13 c31 x1 =

(1.30)

This implies that xi > 0 for i = 1, 2, 3, if and only if the conditions (c22 c33 − c23 c32 ) c1 > c22 c13 c3 − c13 c32 c2 , (c11 c33 − c13 c31 ) c2 > c11 c23 c3 − c23 c31 c1

(1.31)

are satisfied. The characteristic polynomial of Jf (x1 , x2 , x3 ) is given by P (λ) = −λ3 − a1 λ2 − a2 λ − a3 ,

λ ∈ C,

where a1 = −c11 x1 − c22 x2 − c33 x3 > 0, a2 = c11 c22 x1 x2 + (c22 c33 − c23 c31 ) x2 x3 + (c11 c33 − c13 c31 ) x1 x3 , a3 = (−c11 c22 c33 + c11 c23 c32 + c22 c13 c31 ) x1 x2 x3 > 0. | {z } − det Jf (x1 , x2 , x3 )

By a theorem of Hurwitz (see [9]) all the eigenvalues of Jf (x1 , x2 , x3 ) which are zeros of P = P (λ) have a negative real part, if and only if a1 a2 − a3 > 0 which can be verified in this case. Result. There exists exactly one equilibrium state (x1 , x2 , x3 ) of (1.23) for n = 3 with xi > 0 for i = 1, 2, 3 which is given by (1.30) and is asymptotically stable, if and only if the conditions (1.31) are satisfied.


25

The method above for deriving sufficient conditions for asymptotic stability of equilibrium states becomes rather complicated, if n > 3. Therefore we will apply Lyapunov’s method in order to find sufficient conditions. For this purpose we define a homeomorphic mapping T : X = {x ∈ IRn | x1 > 0 for i = 1, . . . , n} → IRn by T (x1 , . . . , xn ) = (u1 , . . . , un ) =

µ ¶ x1 xn ln , . . . , ln , x1 xn

(x1 , . . . , xn ) ∈ X,

whose inverse mapping T −1 : IRn → X is given by T −1 (u1 , . . . , un ) = (x1 , . . . , xn ) = (x1 e u1 , . . . , xn e un ) ,

(u1 , . . . , un ) ∈ IRn .

By this mapping the system (1.26) is transformed into the system u˙ i (t) =

n X

cij xj

³ ´ e uj (t) − 1

(1.32)

j=1

for

i = 1, . . . , n.

This system has (0, . . . , 0) = T (x1 , . . . , xn ) as equilibrium state. If x ∈ C 1 (IR+ , X) is a solution of the system (1.23), then u = T x ∈ C 1 (IR+ , IRn ) is a solution of (1.32) and if u ∈ C 1 (IR+ , IRn ) is a solution of (1.32), then x = T −1 (u) is a solution of (1.23). This implies (according to the assumption above) that for every u0 ∈ IRn there is exactly one solution u ∈ C 1 (IR+ , IRn ) of (1.32) with u(0) = u0 . Further it follows that x ∈ X is an asymptotically stable equilibrium state of the system (1.23), if and only if Θn = T (x) is an asymptotically stable equilibrium state of the system (1.32). Now let us define a function V : IRn → IR by V (u1 , . . . , un ) =

n X n X

cj i sgn (−cj ) xi (e ui − ui ) −

i=1 j=1

n X

|cj |.

j=1

Then it follows that V (0, . . . , 0) = 0 and V (u1 , . . . , un ) > 0 for all

(u1 , . . . , un ) 6= (0, . . . , 0),

if n X

cj i sgn (−cj ) > 0

for all i = 1, . . . , n,

j=1

i.e., V is positive definite with respect to Θn .

(1.33)

26


Further it follows that V˙ (u1 , . . . , un )  Ã ! n n n X X X ui uk   cj i sgn (−cj ) xi (e − 1) cik xk (e − 1) = i=1

=

n X

j=1



cii 

i=1

n X j=1

+

n X

2

cj i sgn (−cj ) x2i (e ui − 1)





cik 

n X



+ ck i 

=

n X

cii 

i=1

 cj i sgn (−cj )

j=1

i,k=1



k=1



n X j=1

n X

 cjk sgn (−cj ) xi xk (e ui − 1) (e uk − 1)  2

cj i sgn (−cj ) x2i (e ui − 1) < 0

j=1

for (u1 , . . . , un ) 6= (0, . . . , 0), if cii < 0

for all

i = 1, . . . , n

(1.34)

and  cik 

n X





cj i sgn (−cj ) + ck i 

j=1

n X

 cjk sgn (−cj ) = 0

(1.35)

j=1

for all

i 6= k.

Finally we have V˙ (0, . . . , 0) = 0. By Proposition 1.12 it therefore follows that Θn is an asymptotically stable equilibrium state of the system (1.32) and hence x = T −1 (Θn ) an asymptotically stable equilibrium state of the system (1.26), (1.23), if the conditions (1.33), (1.34), (1.35) are satisfied. If cii = 0 for at least one i = 1, . . . , n, then Θn is only a stable equilibrium state of the system (1.32). Next we consider the special cases n = 2 and n = 3. For n = 2 we assume that X1 is a predator population which has the population X2 as prey. This leads to and c21 < 0. c12 > 0 We further assume that c11 = c22 = 0.


27

This implies

c2 c1 and x2 = − c21 c12 which requires c1 < 0, c2 > 0 in order to make x1 and x2 positive. The conditions (1.33) and (1.35) are satisfied which implies that (0, 0) is a stable equilibrium state of the system (1.32) for n = 2. x1 = −

For n = 3 we consider at first the case that X1 is a predator population that has the populations X2 and X3 as prey and would decay exponentially in the absence of X2 and X3 . We further assume that X2 and X3 would grow logistically in the absence of X1 and are neutral to each other. All this leads to the conditions c1 < 0, c11 = 0, c12 > 0, c21 < 0,

c2 > 0, c22 < 0, c13 > 0, c31 < 0,

c3 > 0, c33 < 0, c23 = 0, c32 = 0.

The condition (1.33) read −c21 − c31 > 0, c12 − c22 > 0, c13 − c33 > 0 and are satisfied. The conditions (1.35) turn out to be equivalent to c12 c31 + c21 c22 = 0,

c13 c21 + c31 c33 = 0.

(1.36)

By Proposition 1.12 we conclude that (0, 0, 0) is a stable equilibrium state of the system (1.32) for n = 3, if the conditions (1.36) are satisfied. The unique solution of the system (1.24) for n = 3 is given by c22 c33 c1 − c12 c33 c2 − c13 c22 c3 , c12 c21 c33 + c13 c31 c22 −c13 c31 c2 − c21 c33 c1 + c21 c13 c3 x2 = , c12 c21 c33 + c13 c31 c22 −c12 c21 c3 − c31 c22 c1 + c31 c12 c2 x3 = , c12 c21 c33 + c13 c31 c22 x1 =

and xi > 0 for i = 1, 2, 3 is satisfied, if and only if c22 c33 c1 − c12 c33 c2 − c13 c22 c3 > 0, −c13 c31 c2 − c21 c33 c1 + c21 c13 c3 > 0, −c12 c21 c3 − c31 c22 c1 + c31 c12 c2 > 0.

28


Next we consider the case that X1 and X2 are predator populations which have the population X3 as prey and are neutral to each other. Further we assume that X1 and X2 would decay exponentially in the absence of X3 and that X3 would increase logistically in the absence of X1 and X2 . This leads to the conditions c1 < 0, c11 = 0, c12 = 0, c21 = 0,

c2 < 0, c22 = 0, c13 > 0, c31 < 0,

c3 > 0, c33 < 0, c23 > 0, c32 < 0.

The above choice of a Lyapunov function V : IR3 → IR does not lead to sufficient conditions for stability of (0, 0, 0) as we shall see later. We therefore define V (u1 , u2 , u3 ) = −c31 x1 (e u1 − u1 ) − c32 x2 (e u2 − u2 ) −c33 x3 (e u3 − u3 ) − c3 for (u1 , u2 , u3 ) ∈ IR3 and conclude that V (0, 0, 0) = 0,

V (u1 , u2 , u3 ) > 0 for all

(u1 , u2 , u3 ) 6= (0, 0, 0).

Further we obtain V˙ (u1 , u2 , u3 ) = −c31 c13 x1 x3 (e u1 − 1) (e u3 − 1) − c32 c23 x2 x3 (e u2 − 1) (e u3 − 1) −c33 x3 (e u3 − 1) (c3 + c31 x1 e u1 + c32 x2 e u2 + c33 x3 e u3 ) {z } | c31 x1 (e u1 − 1) + c32 x2 (e u2 − 1) + c33 x3 (e u3 − 1)

³ = −x3 (e u3 − 1) c31 (c13 + c33 ) x1 (e u1 − 1) ´ 2 +c32 (c23 + c33 ) x2 (e u2 − 1) − c233 x23 (e u3 − 1) ≤ 0 for all

(u1 , u2 , u3 ) ∈ IR3 ,

if c13 = c23 = −c33 .

(1.37)

By Proposition 1.12 it follows that (0, 0, 0) is a stable equilibrium state of the system (1.32) for n = 3, if the condition (1.37) is satisfied.


29

Further the system (1.24) for n = 3 has infinitely many solutions (x1 , x2 , x3 ) with xi > 0, i = 1, 2, 3, which are stable equilibrium states of the system (1.26), (1.23), if the conditions (1.37) and c23 c1 + c13 c2 = 0, c33 c1 − c13 c3 < 0,

(1.38)

c33 c2 − c23 c3 < 0 are satisfied. The first condition in (1.38) implies that x3 = −

c1 c2 =− . c13 c23

From this it follows that the first two equations of (1.23) can be written in the form x˙ 1 (t) = c13 (x3 (t) − x3 ) x1 (t), x˙ 2 (t) = c23 (x3 (t) − x3 ) x2 (t). On using (1.37) we therefore conclude that x2 (t) =

x2 (0) x1 (t) x1 (0)

(1.39)

and the system (1.23) can be replaced by (1.39) and x˙ 1 (t) = (c1 + c13 x3 (t)) x1 (t), c31 x1 (t) + c33 x3 (t)) x3 (t). x˙ 3 (t) = (c3 + e where e c31 = c31 + c32

x2 (0) < 0. x1 (0)

Finally, we get from c31 x1 + c33 x3 = 0 c3 + e that x1 = −

1 e c31

µ ¶ c1 c3 − c33 > 0, c13

if c3 − c33

c1 > 0. c13

The last two cases are special cases of the following general situation: Let 1 ≤ m < n and let X1 , . . . , Xm be predator populations that have the populations Xm+1 , . . . , Xn as prey and are neutral to each other.

30


Further we assume that X1 , . . . , Xm would decay exponentially in the absence of Xm+1 , . . . , Xn and Xm+1 , . . . , Xn would increase logistically in the absence of X1 , . . . , Xm . Finally Xm+1 , . . . , Xn are neutral to each other. All these assumptions lead to the following conditions: c1 < 0, . . . , cm < 0, cm+1 > 0, . . . , cn > 0, c11 = 0, . . . , cmm = 0, cm+1 m+1 < 0, . . . , cnn < 0, for i, j = 1, . . . , m, i 6= j. cij = 0 cij > 0 cij < 0 cij = 0

for

i = 1, . . . , m

for for

i = m + 1, . . . , n and j = 1, . . . , m, i, j = m + 1, . . . , n, i 6= j.

and j = m + 1, . . . , n,

The conditions (1.33) then read n X

(−cj i ) > 0

for

i = 1, . . . , m

j=m+1

and

m X

cj i − cii > 0

for

i = m + 1, . . . , n

j=1

and are satisfied. The conditions (1.35) turn out to be equivalent to   n m X X (−cj i ) + cki  cj k − ckk  = 0, cik j=m+1

 cik 

m X

j=1

i = 1, . . . , m, k = m + 1, . . . , n,



n X

cj i − cii  + cki

j=1

(−cj k ) = 0,

j=m+1

i = m + 1, . . . , n, k = 1, . . . , m. For n = 3 and m = 1 these conditions are equivalent to (1.36) and can be satisfied. For n = 3 and m = 2 they are equivalent to c13 − c33 = 0

and

c23 − c33 = 0

and are not satisfied. We therefore define as Lyapunov function V : IRn → IR V (u1 , . . . , un ) =

n n X X

(−cj i ) xi (e ui − ui ) −

i=1 j=m+1

for

(u1 , . . . , un ) ∈ IRn

n X j=m+1

cj


31

and obtain V (0, . . . , 0) = 0 and V (u1 , . . . , un ) > 0 because of

n X

for all

(−cj i ) > 0

(u1 , . . . , un ) 6= (0, . . . , 0) for

i = 1, . . . , n.

j=m+1

The system (1.32) reads u˙ i (t) = u˙ i (t) =

n X

cij xj

j=m+1 m X

cij xj

³ ´ e uj (t) − 1 ,

i = 1, . . . , m,

³ ³ ´ ´ e uj (t) − 1 + cii xi e ui (t) − 1 ,

i = m + 1, . . . , n,

j=1

and we obtain V˙ (u1 , . . . , un ) =

m X i=1

+



n X



(−ck i ) xi (e ui − 1)

Ã

i=m+1

cij xj (e uj − 1)

j=m+1

k=m+1 n X



n X

!Ã

n X

(−ck i ) xi (e

k=m+1

ui

− 1)

m X

cij xj (e uj − 1)

j=1

!

+cii xi (e ui − 1) =

Ã

m n X X i=1 j=m+1

n X

(−ck i ) cij

k=m+1

+

n X

! (−ck j ) cj i

xi xj (e ui − 1) (e uj − 1)

k=m+1

+

n X

n X

2

(−ck i ) cii x2i (e ui − 1) ≤ 0,

i=m+1 k=m+1

if Ã

n X k=m+1

zÃ

! ck i

cij +

cjj

}|

n X

!{ ck j

k=m+1

and j = m + 1, . . . , n.

cj i = 0

for i = 1, . . . , m

(1.40)

32


In the case n = 3, m = 2 these conditions are equivalent to (1.37). In the case n = 3, m = 1 they are equivalent to (c21 + c31 ) c12 + c22 c21 = 0,

(c21 + c31 ) c13 + c33 c31 = 0

and can be satisfied. The same holds true for the conditions (1.40). 1.2.5 Application to Evolution Matrix Games In the following we give a mathematical definition of an evolution matrix game. As to a definition within a biological context we refer to [26] (see also Appendix A). We start with a bi-matrix game that is played with mixed strategies so that the strategy sets of the two players are given by two simplices ) ( m X m xi = 1 X = x ∈ IR | xi ≥ 0 for i = 1, . . . , m and i=1

and

  Y =



y ∈ IRn | yj ≥ 0 for j = 1, . . . , n and

 

n X

yj = 1

j=1



.

The corresponding payoff functions are given by Φ1 (x, y) = xT Ay

and Φ2 (x, y) = xT By

with

x ∈ X and y ∈ Y

for player 1 and 2, respectively, where A and B are m × n−matrices. A pair (b x, yb) ∈ X × Y is called a Nash equilibrium, if

and

y ≥ xT Ab y x bT Ab

for all

x∈X

y≥x bT Ay x bT Bb

for all

y ∈ Y.

Now we assume that the game is symmetric, i.e., m = n =⇒ X = Y

and

B = AT .

Then the two conditions for a Nash equilibrium are equivalent to y ≥ xT Ab y and x bT Ab

ybT Ab x ≥ y T Ab x

for all

x, y ∈ X.

If x b = yb, these two conditions are equivalent to x ≥ xT Ab x x bT Ab

for all x ∈ X.

In symmetric games the two players cannot be distinguished and only symmetric strategy pairs (x, x) are of interest. So there is only one player who plays against himself. But then the game has to be given another interpretation.


33

One interpretation could be that the player is represented by a population whose individuals have n strategies at their disposal in the struggle of life. Then the (mixed) strategies of the game can be considered as a probability distribution of the strategies of the individuals. These probability distributions are called population states and with this interpretation the game is called an evolution matrix game. The last condition for a Nash equilibrium of the form (b x, x b) , x b ∈ X then leads to the following. Definition. A population state x b ∈ X is called a Nash equilibrium, if x≤x bT Ab x xT Ab

for all

x ∈ X.

In words that means that a deviation from x b does not lead to a higher payoff. For rational behavior this would suffice to maintain the population state x b. However, animals do not behave rationally so that the stability of a Nash equilibrium is not guaranteed. This leads to the concept of an evolutionarily stable Nash equilibrium given by the Definition. A Nash equilibrium x b ∈ X is called evolutionarily stable, if x=x bT Ab x for some x ∈ X with x 6= x b implies that xT Ax < x bT Ax. xT Ab In words this means that, if a change from x b to x leads to the same payoff, x cannot be a Nash equilibrium. Evolutionary stability can be characterized by a condition which is needed in a dynamical treatment of the game. In this connection we also need a necessary condition for a Nash equilibrium. in order to derive this we at first define for every x ∈ X a support by S(x) = {i ∈ {1, . . . , n} | xi > 0} . Then we can prove the Lemma 1.14. If x b ∈ X is a Nash equilibrium, then it follows that eTi Ab x=x bT Ab x

for all

i ∈ S (b x)

where eTi = (0, . . . , 0, 1, 0, . . . , 0). i

Proof. At first we have x= x bT Ab

n X i=1

x bi eTi Ab x=

X i∈S(b x)

x bi eTi Ab x ≤ max eTi Ab x. i∈S(b x)

34


Since x b is a Nash equilibrium, it follows that eTi Ab x≤x bT Ab x hence

for all

i ∈ S (b x) ,

x bT Ab x = max eTi Ab x. i∈S(b x)

Let

eTi0 Ab x = max eTi Ab x. i∈S(b x)

Then we obtain X

0=x bT Ab x−

x bi eTi Ab x=

i∈S(b x)

X

¡ ¢ x bi eTi0 Ab x − eTi Ab x

i∈S(b x)

which implies x bT Ab x = eTi0 Ab x = eTi Ab x

for all

i ∈ S (b x)

and concludes the proof. The characterization of evolutionary stability is given by Lemma 1.15. A population state x b ∈ X is an evolutionarily stable Nash equilibrium, if and only if there exists some ε > 0 such that xT Ax < x bT Ax

for all

x ∈ X with x 6= x b and ||x − x b||2 < ε.

For the proof we refer to [22]. The game is stable by its very nature. For its application to problems of evolution, however, it is desirable to embed it into a dynamics in the course of time. In [22] we have chosen a time-discrete dynamical model that was first introduced in [26]. Here we choose a time-continuous dynamical model that has been introduced independently by Taylor and Jonker, Zuman and Hofbauer et. al. It is based on the following system of differential equations ¡ ¢ (∗) x˙ i (t) = xi (t) (Ax(t))i − x(t)T Ax(t) , t ∈ IR, for i = 1, . . . , n. x(t) denotes the state of the population at the time t, i.e., x(t) ∈ X. x(t)T Ax(t) can be considered as the average payoff in the population and (Ax(t))i the payoff to the i−th strategy. So the terms (Ax(t))i − x(t)T Ax(t) can be interpreted as the growth rate of xi (t). Now let x b ∈ X be a Nash equilibrium. Then by Lemma 1.14 x b is a solution of (Ab x)i − x bT Ab x=0

for all

i ∈ {1, . . . , n} with x bi > 0.

This implies that x b is a rest point of the system (∗).


35

The question now arises under which condition this rest point is asymptotically stable. We shall show that this is the case, if x b is evolutionarily stable. For the proof we will make use of Proposition 1.12. So the first step consists of defining a Lyapunov function. For this purpose we define the set X (b x) = {x ∈ X | S(x) ⊇ S (b x)} and the function

X

W (x) =

x bi ln xi ,

x ∈ X (b x) .

i∈S(b x)

x) be such that Now let x∗ ∈ X (b W (x∗ ) ≥ W (x)

for all

x ∈ X (b x) .

Since W = W (x), x ∈ X (b x), is differentiable and strictly concave, this is equivalent with the existence of some λ ∈ IR, λ 6= 0, such that ¶ X µx bi + λ (x∗i − xi ) ≥ 0 for all x ∈ IRn with xi ≥ 0 for i ∈ S (b x) . x∗i i∈S(b x)

This again is equivalent with x bi + λ = 0 ⇐⇒ x bi = −λx∗i x∗i

for all

i ∈ S (b x)

which implies λ = −1 and x b = x∗ . Therefore, it follows that W (b x) ≥ W (x)

for all

x ∈ X (b x) .

If we define the function V (x) = W (b x) − W (x),

x ∈ X (b x) ,

then it follows that V (x) ≥ 0

for all

x ∈ X (b x)

and

V (x) = 0 ⇐⇒ x = x b.

Further it follows that Vxi (x) = − If we put

x bi , xi

i ∈ S (b x) ,

for all

¡ ¢ fi (x) = xi (Ax)i − xT Ax ,

x ∈ X (b x) .

i ∈ S (b x) ,

then it follows that ∇V (x)T f (x) = −b xT Ax + xT Ax

for all

x ∈ X (b x) .

If we now assume that x b is evolutionarily stable, then it follows by Lemma 1.15 that there exists some ε > 0 such that ∇V (x)T f (x) < 0

for all x ∈ X (b x) with x 6= x b and ||x − x b||2 < ε.

b Hence Proposition 1.12 with X = {x ∈ X (b x) | ||x − x b||2 < ε} implies that x is asymptotically stable.

36


1.3 Time-Discrete Dynamical Systems 1.3.1 The Autonomous Case: Definitions and Elementary Properties Let X be a metric space with metric d : X × X → IR+ and let f : X → X be a continuous mapping. If we define a mapping π : X × IN0 → X, by π(x, k) = f k (x) = f ◦ . . . ◦ f (x), | {z }

x ∈ X,

k ∈ IN0 ,

(1.41)

k−times

then π is continuous and it follows π(x, 0) = x

for all

x∈X

and ¡ ¢ π(π(x, k), l) = f l f k (x) = f l+k (x) = π(x, k + l)

for all

x ∈ X.

Hence by the definition (1.41) we obtain a time-discrete dynamical system. Conversely, if π : X × IN0 → X is a time-discrete flow and if we define a continuous mapping f : X → X by f (x) = π(x, 1)

for all

x ∈ X,

then it follows that f k (x) = π(x, k)

for all

x∈X

and k ∈ IN0 .

Without loss of generality we therefore can consider a time-discrete dynamical system as a pair (X, f ) where X is a metric space and f : X → X is a continuous mapping. The flow is then defined by (1.41) and the system is called autonomous in contrast to non-autonomous systems which will be defined in Section 1.3.8. For every x ∈ X we define an orbit starting with x by [ {f n (x)} γf (x) =

(1.42)

n∈IN0

(see (1.1)). Further we define as limit set of γf (x) the set Lf (x) =

\

[

{f n (x)}

(1.43)

n∈IN0 m≥n

(see (1.2)). This limit set can be given by an equivalent definition which is the content of

1.3 Time-Discrete Dynamical Systems

37

Proposition 1.16. For every x ∈ X the limit set Lf (x) being defined by (1.43) consists of all accumulation points of the sequence (f n (x))n∈IN0 . Proof. 1) Let y ∈ X be an accumulation point of (f n (x))n∈IN0 . Then there exists a subsequence (f ni (x))i∈IN0 with f ni (x) → y. This implies that, for every n ∈ IN0 , [ y∈ {f m (x)} m≥n

which in turn implies that y ∈ Lf (x). 2) Let y ∈ Lf (x). Then y∈

[

{f m (x)}

for every n ∈ IN0 .

m≥n

¡ ¢ Therefore, for every n ∈ IN0 , there is a sequence f ki +n (x) i∈IN with 0 f ki +n (x) → y as i → ∞. Hence, for every n ∈ IN0 , there exists an in ∈ IN0 such that ¢ ¡ 1 for all i ≥ in d f ki +n (x), y ≤ n ¡ ¢ and we can assume that in+1 > in . This implies that f kin +n (x) n∈IN is a subsequence of (f n (x))n∈IN0 with f kin +n (x) → y as n → ∞. This means that u t y is an accumulation point of (f n (x))n∈IN0 which completes the proof. Definition. A non-empty subset H ⊆ X is called positively (negatively) invariant (with respect to a mapping f : X → X), if f (H) ⊆ H

(H ⊆ f (H)),

and invariant, if f (H) = H. Exercise 1.1. Show that for a continuous mapping f : X → X the following holds true: (a) The closure of a positively invariant subset of X is also positively invariant (with respect to f ). (b) The closure of a relatively compact invariant subset of X is also invariant (with respect to f ). According to Proposition 1.16 the limit set Lf (x) for some x ∈ X given by (1.43) can be empty, if the sequence (f n (x))n∈IN0 does not have accumulation points. If this is not the case, then we have the Proposition 1.17. If, for some x ∈ X, the limit set Lf (x) given by (1.43) is non-empty, then it is closed and positively invariant.

38


Proof. The closedness of Lf (x) is an immediate consequence of the definition (1.43). Let y ∈ Lf (x) be given. Then, by Proposition 1.16, there exists a subsequence (f ni (x))i∈IN0 of the sequence (f n (x))n∈IN0 with f ni (x) → y as i → ∞. This implies, due to the continuity of f , that f ni +1 (x) → f (y), hence f (y) ∈ Lf (x). This shows f (Lf (x)) ⊆ Lf (x), i.e. that Lf (x) is positively invariant. u t If X is compact, then, for every x ∈ X, the limit set Lf (x) given by (1.43) is non-empty and we can even prove Proposition 1.18. If X is compact, then, for every x ∈ X, the limit set Lf (x) given by (1.43) is compact, invariant and the smallest closed subset S ⊆ X with lim % (f n (x), S) = 0

where

n→∞

%(y, S) = min{d(y, z) | z ∈ S}.

(1.44)

Proof. As a non-empty closed subset of the compact metric space X the limit set Lf (x) is also compact. In order to show the invariance of Lf (x) it suffices to show that Lf (x) ⊆ f (Lf (x)). Let y ∈ Lf (x) be given. Then there exists a subsequence (f ni (x))i∈IN0 of the sequence (f n (x))n∈IN0 with f ni (x) → y and with no loss generality we can assume that there is some z ∈ Lf (x) with f ni −1 (x) → z. By the continuity of f this implies f ni (x) → f (z), hence y = f (z) ∈ Lf (x) and therefore Lf (x) ⊆ f (Lf (x)). In order o show (1.44) we at first show that lim % (f n (x), Lf (x)) = 0.

n→∞

For that purpose we assume that % (f n (x), Lf (x)) 6→ 0

as

n → ∞.

Then there is a subsequence (f ni (x))i∈IN0 of (f n (x))n∈IN0 with % (f ni (x), Lf (x)) 6→ 0

as

i → ∞,

and some y ∈ Lf (x) with lim d (f ni (x), y) = 0 which implies i→∞

lim % (f ni (x), Lf (x)) = 0

i→∞

and leads to a contradiction. Now let S ⊆ X be any closed subset with lim % (f n (x), S) = 0.

n→∞

Then we choose any y ∈ Lf (x) and conclude by Proposition 1.16 the existence of a subsequence (f ni (x))i∈IN0 of (f n (x))n∈IN0 with f ni (x) → y.


Further it follows that

39

lim % (f ni (x), S) = 0,

i→∞

which implies y ∈ S, since S is closed. This completes the proof of Proposition 1.18. u t Definition. A closed invariant subset of X is called invariantly connected, if it is not representable as a disjoint union of two non-empty, invariant and closed subsets of X. Definition. A sequence (f n (x))n∈IN0 , x ∈ X, is called periodic or cyclic, if there exists a number k ∈ IN with f k (x) = x. The smallest number with this property is called the period of the sequence. If k = 1, then x ∈ X is called a fixed point of f : X → X. Exercise 1.2. Show that a finite subset H ⊆ X is invariantly connected, if and only if for every x ∈ H the sequence (f n (x))n∈IN0 is periodic and its period is equal to the cardinality of H. Proposition 1.18 can be supplemented by Proposition 1.19. If X is compact, then, for every x ∈ X, the limit set Lf (x) given by (1.43) is invariantly connected. Proof. Let us assume that, for some x ∈ X, there exist two non-empty, invariant and closed subsets A1 and A2 of Lf (x) which are disjoint and satisfy A1 ∪ A2 = Lf (x). Since Lf (x) is compact, A1 and A2 are also compact and there exist disjoint open subsets U1 and U2 of X with A1 ⊆ U1 and A2 ⊆ U2 . Since f is uniformly continuous on A1 , there is an open subset V1 of X with A1 ⊆ V1 and f (V1 ) ⊆ U1 . Since Lf (x) is the smallest closed set S ⊆ X with (1.44), the sequence (f n (x))n∈IN0 must intersect V1 as well as U2 infinitely many times. This implies the existence of a subsequence (f ni (x))i∈IN0 which is neither contained in V1 nor U2 and which can be assumed to be convergent. This, however, is impossible and leads to a contradiction to the assumption that Lf (x) is not invariantly connected. u t Given a compact and positively invariant subset K ⊆ X. Show Exercise T 1.3. f n (K) with f n (K) = {f n (x) | x ∈ K} for all n ∈ IN0 is non-empty, that n∈IN0

compact and the largest invariant subset of K.

40


1.3.2 Localization of Limit Sets with the Aid of Lyapunov Functions Let X be a metric space and let f : X → X be a continuous mapping. Further let G ⊆ X be a non-empty subset. Definition. A function V : X → IR is called a Lyapunov function with respect to f on G, if (1) (2)

V is continuous on X, V (f (x)) − V (x) ≤ 0 for all x ∈ G.

If V : X → IR is a Lyapunov function with respect to f on G, then we define the set ª © E = x ∈ X | V (f (x)) = V (x), x ∈ G where G is the closure of G. Further we put, for every c ∈ IR, V −1 (c) = {x ∈ X | V (x) = c}. Then we prove Proposition 1.20. Let G ⊆ X be non-empty and relatively compact. Further let V be a Lyapunov function with respect to f on G and finally let x0 ∈ G be such that for all n ∈ IN. f n (x0 ) ∈ G Then there exists some c ∈ IR such that Lf (x0 ) ⊆ M ∩ V −1 (c) where M is the largest invariant subset of E. ¢ ¡ Proof. If we define xn = f n (x0 ) , n ∈ IN0 , then the sequence V (xn )n∈IN0 ¡ ¢ is contained in V G and therefore bounded. Further it follows that V (xn+1 ) ≤ V (xn )

for all

n ∈ IN0 .

Therefore there exists some c ∈ IR with c = lim V (xn ). n→∞

Now let p ∈ Lf (x0 ). Then there exists a subsequence (xni )i∈IN0 of (xn )n∈IN0 with xni → p which implies c = lim V (xni ) = V (p) due to the continuity of i→∞

V . Hence p ∈ V −1 (c) and thus Lf (x0 ) ⊆ V −1 (c). Since, by Proposition 1.18, Lf (x0 ) is invariant, it follows that V (f (p)) = c for all p ∈ Lf (x0 ) and hence V (f (p)) = V (p) for all p ∈ Lf (x0 ) which implies Lf (x0 ) ⊆ E and in turn u t Lf (x0 ) ⊆ M . This completes the proof.


41

Let us demonstrate this result by an example: We take X = IR2 equipped with the Euclidian norm and define f : X → X by T

f (x, y) = (f1 (x, y), f2 (x, y)) , with f1 (x, y) =

y , 1 + x2

(x, y) ∈ X,

f2 (x, y) =

x . 1 + y2

(1.45)

Further we choose V (x, y) = x2 + y 2 ,

(x, y) ∈ X.

(1.46)

Then it follows that Ã V (f (x, y)) − V (x, y) =

!

1 2

(1 + y 2 )

for all

−1

Ã x2 +

!

1 2

(1 + x2 )

−1

y 2 ≤ 0,

(x, y) ∈ X.

This shows that V is a Lyapunov function with respect to f on X = IR2 . Further we see that E = {(x, 0) | x ∈ IR} ∪ {(0, y) | y ∈ IR}. From f (x, 0) = (0, x) for all x ∈ IR and f (0, y) = (y, 0) for all y ∈ IR, it follows that E is invariant and hence M = E. Further we conclude f 2 (x, 0) = f (0, x) = (x, 0)

for all

x ∈ IR

and f 2 (0, y) = f (y, 0) = (0, y) for all y ∈ IR. ª © Now let G = (x, y) ∈ IR2 | x2 + y 2 < r for any r > 0. Then it follows that f (G) ⊆ G and for every (x0 , y0 ) ∈ G there exists some c ∈ IR such that ª © Lf (x0 , y0 ) ⊆ E ∩ (x, y) | x2 + y 2 = c2 = {(c, 0), (0, c)}. Since Lf (x0 , y0 ) is invariantly connected by Proposition 1.19 it follows from Exercise 1.2 that Lf (x0 , y0 ) = {(c, 0), (0, c)}.

42


1.3.3 Stability Based on Lyapunov’s Method Let f : X → X be a continuous mapping where X is a metric space. Definition. A relatively compact set H ⊆ X is called stable with respect to f , if for every relatively compact open set U ⊆ X with U ⊇ H = closure of H there exists an open set W ⊆ X with H ⊆ W ⊆ U such that f n (W ) ⊆ U where

for all

n ∈ IN0

f n (W ) = {f n (x) | x ∈ W } .

Theorem 1.1. Let H ⊆ X be relatively compact and such that for every relatively compact open set U ⊆ X with U ⊇ H there exists an open subset BU of U with BU ⊇ H and f (BU ) ⊆ U . Further let G ⊆ X be an open set with G ⊇ H such that there exists a Lyapunov function V with respect to f on G which is positive definite with respect to H, i.e., ¡ ¢ V (x) ≥ 0 for all x ∈ G and V (x) = 0 ⇐⇒ x ∈ H . Then H is stable with respect to f . Proof. Let U ⊆ X be an arbitrary relatively compact open set with U ⊇ H. Then U ∗ = U ∩ G is also a relatively compact open set with U ∗ ⊇ H and there exists an open set BU ∗ ⊆ U ∗ with BU ∗ ⊇ H and f (BU ∗ ) ⊆ U ∗ . Let us put

ª © m = min V (x) | x ∈ U ∗ \ BU ∗ . ¢ ¡ Since H ∩ U ∗ \ BU ∗ is empty, it follows that m > 0. If we define W = {x ∈ U ∗ | V (x) < m} , then W is open and H ⊆ W ⊆ BU ∗ . Now let x ∈ W be chosen arbitrarily. Then x ∈ BU ∗ and therefore f (x) ∈ U ∗ . Further we have V (f (x)) ≤ V (x) < m, hence f (x) ∈ W ⊆ BU ∗ . This implies f 2 (x) = f (f (x)) ∈ U ∗ and ¡ ¢ V f 2 (x) ≤ V (f (x)) < m, hence f 2 (x) ∈ W. By induction it therefore follows that f n (x) ∈ W ⊆ U ∗ ⊆ U

for all

this shows that H is stable with respect to f .

n ∈ IN0 . u t


43

Exercise 1.4. (a) Give an explicit definition of the instability of a relatively compact set H ⊆ X with respect to f as logical negation of the notion of stability. (b) Show: If a relatively compact set H ⊆ ¡X is ¢ stable with respect to f , then its closure H is positively invariant, i.e. f H ⊆ H. Definition. A set H ⊂ X is called an attractor with respect to f , if there exists an open set U ⊆ X with U ⊇ H such that ¡ ¢ ¡ ¢ lim % f n (x), H = 0 in short: f n (x) → H for all x ∈ U n→∞

where

¡ ¢ © ª % y, H = inf d(y, z) | z ∈ H .

If H ⊆ X is stable and an attractor with respect to f , then H is called asymptotically stable with respect to f . Theorem 1.2. Let H ⊆ X be such that there exists a relatively compact open set U ⊇ H with f (U ) ⊆ U . Further let V : X → IR be a Lyapunov function with respect to f on U which is positive definite with respect to H, i.e. ¡ ¢ V (x) ≥ 0 for all x ∈ U and V (x) = 0 ⇐⇒ x ∈ H . Finally let lim V (f n (x)) = 0

n→∞

for all

x ∈ U.

Then H is an attractor with respect to f . Proof. Let x ∈ U be chosen arbitrarily. Since the sequence (f n (x))n∈IN0 is contained in U , for every subsequence (f ni (x))i∈IN0 of (f n (x))n∈IN0 there exists a subsequence (f nij (x))j∈IN0 and some q ∈ U with lim f nij (x) = q.

j→∞

This implies lim V (f nij (x)) = V (q) = 0,

j→∞

hence q ∈ H, and therefore f nij (x) → H as j → ∞. From this it follows that u t f n (x) → H which shows that H is an attractor with respect to f . Corollary. Under the assumptions of Theorem 1.1 and 1.2 it follows that H ⊆ X is asymptotically stable. As an important special case we prove the following

44


Theorem 1.3. Let x∗ ∈ X be a fixed point of f , i.e. f (x∗ ) = x∗ . Further let there exist an open set G ⊆ X with x∗ ∈ G and a Lyapunov function V with respect to f on G which is positive definite with respect to x∗ , i.e. V (x) ≥ 0

for all

x∈G

(V (x) = 0 ⇐⇒ x = x∗ ) .

and

Then {x∗ } is stable with respect to f . If in addition V (f (x)) < V (x)

for all

x∈G

x 6= x∗ ,

with

(1.47)

then {x∗ } is asymptotically stable with respect to f . Proof. Let U ⊆ X be a relatively compact open set with x∗ ∈ U . Then there exists some r > 0 such that Br (x∗ ) = {x ∈ X | d (x, x∗ ) < r} ⊆ U. Since f is continuous in x∗ , there exists some s ∈ (0, r) such that f (Bs (x∗ )) ⊆ Br (x∗ ). Hence, if we put BU = Bs (x∗ ), then f (BU ) ⊆ U , BU is open and {x∗ } ⊆ BU . By Theorem 1.1 therefore {x∗ } is stable with respect to f . From the stability part of the proof of Theorem 1.1 we deduce for every relatively compact open subset U of X the existence of an open subset W of U ∩ G with x∗ ∈ W, W ⊆ G and f n (x) ∈ W

for all x ∈ W

and all

n ∈ IN0 .

Now let x ∈ W be chosen arbitrarily. Assumption: f n (x) 6→ x∗ . Then there exists a subsequence (f nk (x))k∈IN0 and some x ∈ W with x 6= x∗ and lim f nk (x) = x. (∗) k→∞

f

nk

(x) ∈ W for all k ∈ IN0 implies f (f nk (x)) ∈ W for all k ∈ IN0 and hence f (f nk (x)) → f (x) ∈ W ⊆ G.

This implies the existence of a neighborhood B = {z ∈ W | d (z, x) ≤ ε} of x with x∗ 6∈ B and f (B) ⊆ G. For every z ∈ B we therefore have V (f (z)) < V (z) which implies q = sup z∈B

V (f (z)) < 1, V (z)

hence

V (f (z)) ≤ q V (z)

for all

z ∈ B.

Because of (∗) there is some k0 ∈ IN0 with f nk (x) ∈ B for all k ≥ k0 .


45

This implies, for all k ≥ k0 , because of f n (f nk (x)) ∈ W ⊆ B for all n ∈ IN that ¡ ¡ ¢¢ ¡ ¢ V (f nk+1 (x)) = V f nk+1 −nk −1 f nk +1 (x) ≤ V f nk +1 (x) ≤ q V (f nk (x)) and by iteration it follows that V (f nk0 +l (x)) ≤ q l V (f nk0 (x))

for all

l ∈ IN.

This implies lim V (f nk0 +l (x)) = 0

l→∞

and therefore

V (x) = 0

which is impossible because of x 6= x∗ . Thus the above assumption is false and hence for all x ∈ W f n (x) → x∗ which shows that x∗ is an attractor with respect to f .

u t

Let us demonstrate this result by the same example as in Section 1.3.2, i.e. we choose f : IR2 → IR2 as f (x, y) = (f1 (x, y), f2 (x, y)) ,

(x, y) ∈ X = IR2 ,

with f1 = f1 (x, y) and f2 = f2 (x, y) defined by (1.45). If we choose V : X → IR in the form of (1.46), then V is a Lyapunov function with respect to f on G = X = IR2 . If we choose x∗ = (0, 0)T , then V is positive definite with respect to x∗ and x∗ is a fixed point of f . Further (1.47) is satisfied. By Theorem 1.3 it follows that x∗ = (0, 0)T is asymptotically stable with respect to f . With the aid of Proposition 1.20 we can prove Theorem 1.4. Let G ⊆ X be open, relatively compact and positively invariant with respect to f , i.e. f (G) ⊆ G. Further let V be a Lyapunov function with respect to f on G. Then the largest invariant subset M of ª © E = x ∈ G | V (f (x)) = V (x) is an attractor with respect to f . If in addition V is constant on M , then M is asymptotically stable with respect to f . Proof. Let us put U = G. If we choose x0 ∈ U arbitrarily, then the sequence (f n (x0 ))n∈IN0 is contained in G, since G is positively invariant. Proposi¡ ¢ tion 1.20 implies Lf (x0 ) ⊆ M . We have to show that lim % f n (x0 ) , M = 0. n→∞

46


Now let (f ni (x0 ))i∈IN0 be an arbitrary subsequence of (f n (x0 ))n∈IN0 . Then there is a subsequence (f nij (x0 ))j∈IN0 and an element y ∈ Lf (x0 ) ⊆ M ¡ ¢ with f nij (x0 ) → y which implies that lim % f nij (x0 ) , M = 0 and in turn j→∞ ¡ ¢ % f n (x0 ) , M → 0. In order to show that M is asymptotically stable with respect to f we have to show that M is stable with respect to f . For that purpose we apply Theorem 1.1 and verify its assumptions. At first let U ⊆ X be relatively compact, open and such that U ⊇ H. If we define BU = {x ∈ U ∩ G | f (x) ∈ U }, then BU is open, BU ⊇ M , BU ⊆ U , and f (BU ) ⊆ U . Let V (x) = c

for all x ∈ M.

(1.48)

Let us assume that there is some x ∈ G with V (x) < c. Then it follows that V (f n (x)) ≤ V (x) < c

for all n ∈ IN.

Since Lf (x) ⊆ M , there exists some y ∈ Lf (x) ⊆ M with V (y) < c which contradicts (1.48). Hence V (x) ≥ c

for all

x∈G

and further V (x) = c

⇐⇒

x ∈ M.

Therefore, if we define Ve (x) = V (x) − c,

x ∈ X,

then we obtain a Lyapunov function with respect to f on G which is positive u t definite with respect to M . The assertion now follows by Theorem 1.1. Let us again demonstrate this result by the above example (1.45) with Lyapunov function (1.46). Let again ª © for some r > 0. G = (x, y) ∈ IR2 | x2 + y 2 < r2 Then G is open, relatively compact and positively invariant with respect to f . From Theorem 1.4 it therefore follows that ¡ ¢ M = {(x, 0) | x ∈ IR} ∪ {(0, x) | x ∈ IR} ∩ G is an attractor with respect to f . It even follows that for every x0 ∈ G it is true that ¡ ¢ lim % f n (x0 ) , M = 0. n→∞


47

1.3.4 Stability of Fixed Points via Linearisation Let us assume that X is a non-empty open subset of a normed linear space (E, || · ||) and f : X → X is a continuous mapping which is continuously Fréchet differentiable at every x ∈ X. We denote the Fréchet derivative of f at x ∈ X by fx0 which is a continuous linear mapping fx0 : E → E whose norm is given by ||fx0 || = sup {||fx0 (h)|| | h ∈ E with ||h|| = 1} for x ∈ X. Then we can prove Theorem 1.5. Let x∗ ∈ X be a fixed point of f , i.e. f (x∗ ) = x∗ . Then the following two statements are true: a) Let ||fx0 ∗ || < 1. Then there is some ε > 0 and some c ∈ [0, 1) such that x0 ∈ X and ||x0 − x∗ || < ε

=⇒

||f n (x0 ) − x∗ || ≤ cn ||x0 − x∗ || for all n ∈ IN

which implies that x∗ is asymptotically stable with respect to f (Exercise). −1 b) Let fx0 ∗ be continuously invertible and ||fx0 ∗ ||−1 > 1. Then there is some δ > 0 and some d > 1 such that f k (x0 ) ∈ X

and

||f k (x0 ) − x∗ || < δ

for

0≤k ≤n−1

implies that ||f n (x0 ) − x∗ || ≥ dn ||x0 − x∗ || for all n ∈ IN. This implies that x∗ is unstable with respect to f (Exercise, see Exercise 1.4 a)). Proof. a) From the continuity of x → ||fx0 ||, x ∈ X, it follows that there exists some ε > 0 and some c ∈ [0, 1) such that ||fx0 || ≤ c

for all

x ∈ B (x∗ , ε) = {y ∈ E | ||y − x∗ || < ε} .

The mean value theorem implies ||f (x) − f (y)|| ≤ ||fx0 || ||x − y|| for all x, y ∈ B (x∗ , ε) and some z = α x + (1 − α) y with α ∈ (0, 1). This implies ||f (x) − f (y)|| ≤ c ||x − y||

for all x, y ∈ B (x∗ , ε)

and in particular ||f (x) − x∗ || ≤ c ||x − x∗ || < ε

for all

x ∈ B (x∗ , ε) .

48


Therefore it follows that f (B (x∗ , ε)) ⊆ B (x∗ , ε) and hence ||f n (x) − x∗ || = ||f ◦ f n−1 (x0 ) − f (x∗ ) || ≤ c ||f n−1 (x0 ) − x∗ || ≤ . . . ≤ cn ||x0 − x∗ || for all x0 ∈ B (x∗ , ε) and n ∈ IN. b) For every x, y ∈ X we have ||x − y|| = ||fx0 ∗

−1

(fx0 ∗ (x)) − fx0 ∗

−1

(fx0 ∗ (y)) || ≤ ||fx0 ∗

−1

|| ||fx0 ∗ (x − y)||

which implies ||fx0 ∗ (x − y)|| ≥ d 0 ||x − y||

with

d 0 = ||fx0 ∗

−1 −1

||

> 1.

From the Fréchet differentiability it follows that f (x) − f (x∗ ) = fx0 ∗ (x − x∗ ) + ε (||x − x∗ ||) where lim∗

x→x

for x ∈ X

||ε (||x − x∗ ||) || = 0. ||x − x∗ ||

If one chooses η > 0 with d = d 0 − η > 1 and δ > 0 such that ||ε (||x − x∗ ||) || ≤η ||x − x∗ ||

for all

B (x∗ , δ) \ {x∗ } ,

then it follows that ||f (x) − f (x∗ ) || ≥ ||fx0 ∗ (x − y)|| − ||ε (||x − x∗ ||) || ≥ d 0 ||x − x∗ || − η ||x − x∗ || = d ||x − x∗ || for all x ∈ B (x∗ , δ). Now let x0 ∈ B (x∗ , δ) be such that ||f k (x0 ) − x∗ || < δ

for 0 ≤ k ≤ n − 1, n ∈ IN.

Then it follows that ||f n (x0 ) − x∗ || = ||f ◦ f n−1 (x0 ) − f (x∗ ) || ≥ d ||f n−1 (x0 ) − x∗ || ≥ . . . ≥ dn ||x0 − x∗ ||. This completes the proof of Theorem 1.5.

u t


49

Let us consider the following important special case: We assume E = IRk equipped with any norm || · ||. Let again X ⊆ E be non-empty and open and let again f : X → X be a continuously Fréchet differentiable (and hence continuous) mapping on X. This is equivalent to the statement that at every x ∈ X there exists the Jacobian matrix   f1 x1 · · · f1 xk   Jf (x) =  ... . . . ...  , x = (x1 , . . . , xk ) , fk x1 · · · fk xk and depends continuously on x. The Fréchet derivative is then given by fx0 (h) = Jf (x) h,

h ∈ IRk ,

and we obtain ||fx0 || = ||Jf (x)|| = sup {||Jf (x) h|| | ||h|| = 1} . Theorem 1.5 now gives rise to the following Corollary. Let x∗ ∈ X be a fixed point of f . Then the following two statements are true: a) Let the spectral radius % (Jf (x∗ )) be smaller than 1. Then x∗ is asymptotically stable with respect to f . b) Let Jf (x∗ ) be invertible and let all the eigenvalues of Jf (x∗ ) be larger than 1 in absolute value. Then x∗ is unstable with respect to f . Proof. By Theorem 3 in Chapter 1 of the book “Analysis of Numerical Methods” by E. Isaacson and H.B. Keller (John Wiley and Sons, New York, London, Sydney 1966) there exists, for every δ > 0, a vector norm in IRk such that ||Jf (x∗ ) || ≤ % (Jf (x∗ )) + δ. a) Let us choose δ=

1 (1 − % (Jf (x∗ ))) 2

(> 0).

Then

1 (1 + δ (Jf (x∗ ))) < 1 2 and the assertion follows from Theorem 1.5 a). ´ ³ −1 < 1. Again by the above b) From the assumption it follows that % Jf (x∗ ) quoted theorem it follows that, for a suitable matrix norm, ||fx0 ∗ || = ||Jf (x∗ ) || ≤

||Jf (x∗ )

−1

|| < 1

=⇒

−1

1 < ||Jf (x∗ )

so that the assertion follows from Theorem 1.5 b).

||−1 = ||fx0 ∗

−1 −1

||

u t

50


Assumption a) in the Corollary is not necessary for the asymptotic stability of x∗ with respect to f as can be seen by the Example (1.45). Here we have f1 x (x, y) = −

2xy 2 x2 )

(1 + 1 f2 x (x, y) = , 1 + y2 hence

µ Jf (0, 0) =

0 1 1 0

1 , 1 + x2 2xy f2 y (x, y) = − 2, (1 + y 2 )

,

f1 y (x, y) =

¶ =⇒

% (Jf (0, 0)) = 1.

However, we have seen in Section 1.3.3 that (0, 0) is asymptotically stable with respect to f . 1.3.5 Linear Systems As at the beginning of Section 1.3.4 we consider a normed linear space (E, ||·||) and a mapping f : E → E which is given by f (x) = A(x) + b,

x ∈ E,

where A : E → E is a continuous linear mapping and b ∈ E a fixed element. Then f is Fréchet differentiable at every x ∈ E and its Fréchet derivative is given by for all x ∈ E. fx0 = A Theorem 1.5 leads immediately to Theorem 1.6. Let x∗ ∈ E be a fixed point of f , i.e. x∗ = A (x∗ ) + b.

(1.49)

Then the following two statements are true: a) If ||A|| < 1, then x∗ is asymptotically stable with respect to f . b) Let A be continuously invertible and ||A−1 ||−1 > 1. Then x∗ is unstable with respect to f . Exercise 1.5. Show that under the assumption ||A|| < 1 there is at most one x∗ ∈ E with (1.49). Further show that, if ||A|| < 1 and there exists x∗ ∈ E with (1.49) (which is then unique), it follows that, for every x0 ∈ E, the sequence (xn )n∈IN0 given by xn+1 = A (xn ) + b, n ∈ IN0 , converges to x∗ . The Corollary of Theorem 1.5 leads to the following


51

Corollary. Let E = IRk equipped with any norm and let x ∈ IRk ,

f (x) = A x + b,

where A is a real k × k−matrix and b ∈ IRk a fixed element. Let x∗ ∈ IRk be a fixed point of f , i.e. x∗ = A x∗ + b. Then the following two statements are true: a) If %(A) < 1, then x∗ is asymptotically stable with respect to f . b) If A is invertible and all eigenvalues of A are larger than 1 in absolute value, then x∗ is unstable with respect to f . In the following we introduce a concept of stability and asymptotic stability for linear systems that differ from the one given in Section 1.3.3 and that we adopt from [23]. For that purpose we consider the sequences (xn )n∈IN0 in E with n ∈ IN0 , x0 ∈ E. (1.50) xn+1 = A xn + b, Definition. A sequence (xn )n∈IN0 in E with (1.50) is called 1. stable, if for every ε > 0 and every N ∈ IN, there exists some δ = δ(ε, N ) such that for every sequence (xn )n∈IN0 with (1.50) for x0 ∈ E and ||xN − xN || < δ it follows that ||xn − xn || < ε

for all

n ≥ N,

2. attractive, if for every N ∈ IN, there exists some δ = δ(N ) such that for every sequence (xn )n∈IN0 with (1.50) for x0 ∈ E and ||xN − xN || < δ it follows that lim ||xn − xn || = 0, n→∞

3. asymptotically stable, if (xn )n∈IN0 is stable and attractive. In order to guarantee the stability of all sequences (xn )n∈IN0 with (1.50) it suffices to guarantee the stability of the zero sequence (xn = ΘE )n∈IN0 which satisfies (1.50) for b = ΘE =zero element of E. This is a consequence of Lemma 1.3. The following statements are equivalent: (1) All sequences (xn )n∈IN0 with (1.50) are stable. (2) One sequence (xn )n∈IN0 with (1.50) is stable. (3) The zero sequence (xn = ΘE )n∈IN0 which satisfies (1.50) for b = ΘE is stable.

52


Proof. (1) =⇒ (2) is trivially true. Now let (2) be true. Then there is a sequence (yn )n∈IN0 with (1.50) which is stable. Let (xn )n∈IN0 be any sequence with (1.50) for b = ΘE . Then (xn + yn )n∈IN0 satisfies (1.50) and for every ε > 0 there exists some δ = δ(ε, N ) such that ||xN − ΘE || = ||xN + yN − yN || < δ ||xn − ΘE || = ||xn + yn − yn || < ε

implies

for all

n ≥ N,

since (yn )n∈IN0 is stable. From this it follows that (xn = ΘE )n∈IN0 is stable which shows (2) =⇒ (3). Now let (yn )n∈IN0 and (y n )n∈IN0 be arbitrary sequences with (1.50). Then we choose any sequence (zn )n∈IN0 in E with (1.50) and define xn = yn − zn

and

xn = y n − z

for n ∈ IN0 .

Since according to (3) the zero sequence (e xn = ΘE )n∈IN0 (which satisfies (1.50) for b = ΘE ) is stable, it follows that for every ε > 0 there exists some δ > δ(ε, N ) such that ||yN − y N || = ||xN − xN − ΘE || < δ ||yn − y n || = ||xn − xn − ΘE || < ε

implies that for all

n≥N

which shows that (1) is true. Hence (1) =⇒ (2) =⇒ (3) =⇒ (1) which completes the proof.

u t

Remark. Lemma 1.3 also holds true, if we replace stable by attractive. Hence it is also true with asymptotically stable instead of stable. Now let us again consider the special case E = IRk equipped with any norm || · ||. According to Lemma 1.3 we consider sequences (xn )n∈IN0 in IRk with xn+1 = A xn ,

n ∈ IN0 , x0 ∈ IRk .

(1.51)

In order to show stability of the zero sequence (xn = Θk )n∈IN0 we assume that A has eigenvalues λ1 , . . . , λk ∈ C such that there exist eigenvectors v1 , . . . , vk ∈ Ck which are linearly independent. Therefore every x0 ∈ IRk has a unique representation x0 =

k X

ci vi ,

c1 , . . . , ck ∈ C.

i=1

This implies that every sequence (xn )n∈IN0 can be represented in the form xn =

k X i=1

ci λni vi ,

n ∈ IN0 .


53

Now let us assume that |λi | ≤ 1

for

i = 1, . . . , k.

(1.52)

If we define, for every z=

k X

ci (z) vi ∈ Ck ,

i=1

a norm by ||z|| =

k X

|ci (z)|,

i=1

it follows that ||xn || =

k X

|λi |n |ci (x0 ) |

i=1

≤

k X

|ci (x0 ) | = ||x0 ||.

i=1

This leads to Theorem 1.7. Let the eigenvalues λ1 , . . . , λk ∈ C of A satisfy (1.52) and be such that the corresponding eigenvectors are linearly independent. Then the zero sequence (xn = Θk )n∈IN0 which satisfies (1.51) is stable and hence every sequence (xn )n∈IN0 that satisfies (1.51) with E = IRk is stable. Proof. Let ε > 0 be chosen. Then we put δ = ε. Now let (xn )n∈IN0 be any sequence with (1.51) for xn = xn and ||x0 || < δ. Then it follows that ||xn − u t Θk || < ε for all n ∈ IN0 which completes the proof. Exercise 1.6. Prove that the zero sequence (xn = Θk )n∈IN0 is asymptotically stable, if all eigenvalues of A are less than 1 in absolute value. Remark. Under the assumption of Theorem 1.7 it is also true that the mapping f : IRk → IRk defined by f (x) = A x,

x ∈ IRk ,

has {Θk } as a stable fixed point in the sense of the definition at the beginning of Section 1.3.3.

54


Proof. Let G = IRk and define V : IRk → IR by V (x) = ||x||,

x ∈ IRk ,

where ||x|| =

k X

x ∈ IRk .

|ci (x)|,

i=1

Then it follows that V (A x) = ||A x|| =

k X

|λi | |ci (x)| ≤

i=1

k X

|ci (x)| = ||x||,

x ∈ IRk .

i=1

Hence V (A x) − V (x) ≤ 0 for all x ∈ G and V is a Lyapunov function on G = IRk . Further we have V (x) ≥ 0 for all

x∈G

and

(V (x) = 0 ⇐⇒ x = Θk ) .

Theorem 1.3 therefore implies that {Θk } is a stable fixed point of f .

u t

We can also use another Lyapunov function in order to show that x b = Θk is a stable fixed point of the system (1.51). For that purpose we choose a symmetric and positive definite real k × k−matrix B and define a function V (x) = xT B x, If

¡ ¢ xT AT B A − B x ≤ 0

x ∈ IRk .

for all

x ∈ IRk ,

then V is a Lyapunov function with respect to f (x) = A x,

x ∈ IRk ,

on G = IRk which is positive definite with respect to {Θk }. Exercise 1.7. a) Show with the aid of Theorem 1.3 that {Θk } is stable with respect to f . b) Show with the aid of Theorem 1.3 that {Θk } is asymptotically stable with respect to f , if ¡ ¢ xT AT B A − B x < 0

for all

x ∈ IRk , x 6= Θk .


55

One can even show that {Θk } is globally asymptotically stable with respect to f , i.e., {Θk } is stable with respect to f and An x0 → Θk

as

n→∞

for all

x0 ∈ IRk

which is equivalent to An → 0 = k × k − zero matrix. Conversely, let {Θk } be globally asymptotically stable with respect to f . Further let C be a symmetric and positive definite real k × k−matrix such that ∞ ¡ ¢k P AT C Ak converges. If we define the series k=0

B=

∞ X ¡

AT

¢k

C Ak ,

k=0

then B is a symmetric and positive definite real k × k−matrix and it follows that AT B A − B = −C which implies ¡ ¢ xT AT B A − B x < 0

x ∈ IRk , x 6= Θk .

for all

1.3.6 Discretization of Time-Continuous Dynamical Systems As in Section 1.1 we consider an autonomous system of the form x˙ = f (x) n

(1.17)

n

n

where f ∈ C 1 (IR , IR ). Then for every x ∈ IR there is exactly one solution ϕ = ϕ(x, t) ∈ (−α, α) for some α > 0 with ϕ ∈ C 1 ((−α, α), IRn ) and ϕ(x, 0) = x. If we assume that α = ∞, then by the definition π(x, t) = ϕ(x, t)

for all x ∈ IRn

and t ∈ IR

a time-continuous dynamical system π : IRn × IR → IRn is defined as being shown in Section 1.1. For a given h > 0 we now replace the derivatives x˙ i = x˙ i (t) on the left-hand side of (1.17) by difference quotients xi (t + h) − xi (t) , h

i = 1, . . . , n,

and obtain a system of difference equations of the form xi (t + h) = xi (t) + h fi (x(t)),

t ∈ IR, i = 1, . . . , n.

56


If one defines a vector function g h : IRn → IRn by gih (x) = xi + h fi (x(t)),

x ∈ IRn , i = 1, . . . , n,

(1.53)

then g h is continuous and with the definition ¡ ¢k πh (x, k) = g h (x) = g h ◦ g h ◦ . . . ◦ g h (x) | {z }

for

x ∈ IRn , k ∈ IN,

k−times

x ∈ IRn ,

πh (x, 0) = x,

(1.54)

we obtain a time-discrete dynamical system (see Section 1.3.1) which is called a discretization of (1.17) (with stepsize h). A point x b ∈ IRn is a rest point of the system (1.17), i.e., a solution of the b is a fixed point of g h , i.e., a solution of equation f (b x ) = Θn , if and only if x h x) = x b. the equation g (b In Section 1.2.3 we have shown (see Proposition 1.13) that a rest point x ∈ IRn of the system (1.17) is asymptotically stable, if the Jacobi matrix ¡ ¢ Jf (b x ) = fixj (b x ) i,j=1,...,n of f at x b has only eigenvalues with negative real parts. The Jacobi matrix of g h in x b ∈ IRn reads ¡ ¢ Jg h (b x ) = δij + h fi xj (b x ) i,j=1,...,n where

½ δij =

0 for i 6= j, 1 for i = j.

Obviously λ ∈ C is an eigenvalue of Jf (b x ), if and only if 1+λ h is an eigenvalue x ). Further we have of Jg h (b 2

|1 + λ h|2 = (1 + Re(λ) h) + (Im(λ) h) = 1 + 2Re(λ) h + h2 |λ|2 ¢ ¡ = 1 + h 2Re(λ) + h |λ|2 .

2

This equation implies that Re(λ) < 0, if and only if |1+λ h| < 1 for sufficiently small h > 0. We can even say that |1 + λ h|2 < 1

⇐⇒

Re(λ) < 0 and h
0, x b is an asymptotically stable fixed point of g h . x ) be invertible and let all the eigenvalues of Jg h (b x ) be larger b) Let Jg h (b than 1 in absolute value. Then x b is an unstable fixed point of g h . 1.3.7 Applications a) Discretization of a Predator-Prey Model We consider the following predator-prey model x˙ 1 (t) = (c1 + c12 x2 (t)) x1 (t), x˙ 2 (t) = (c2 + c21 x1 (t) + c22 x2 (t)) x2 (t),

t ∈ IR+ ,

(1.56)

where c1 < 0, c2 > 0, c12 > 0, c21 < 0, c22 ≤ 0 (see (1.23)). We assume that c2 c12 − c1 c22 > 0. b2 ) , x b1 > 0, x b2 > 0 Then this system has exactly one rest point x b = (b x1 , x which is given by x b1 = Let

c2 c12 − c1 c22 , −c12 c21

x b2 = −

c1 . c12

(1.57)

ª © X = x ∈ IR2 | x1 > 0 and x2 > 0 .

Then it follows (see Section 1.2.4) ¡that x ∈¢ C (IR+ , X) is a solution of the system (1.56), if and only if u ∈ C 1 IR+ , IR2 with µ u1 (t) = ln

x1 (t) x b1

µ

¶ and u2 (t) = ln

x2 (t) x b2

¶ ,

is a solution of the system ³ ´ u˙ 1 (t) = c12 x b2 e u2 (t) − 1 , ³ ³ ´ ´ b1 e u1 (t) − 1 + c22 x b2 e u2 (t) − 1 , u˙ 2 (t) = c21 x

t ∈ IR+ ,

t ∈ IR+ . (1.58)

The only rest point of this system is the point (0, 0). Further it follows that (0, 0) is asymptotically stable, if and only if x b = (b x1 , x b2 ) given by (1.57) is asymptotically stable.

58


If we define a vector function f : IR2 → IR2 by Ã ! ¡ ¢ b2 e u2 (t) − 1 c12 x ¡ ¡ ¢ ¢ , f (u1 , u2 ) = c21 x b1 e u1 (t) − 1 + c22 x b2 e u2 (t) − 1

(u1 , u2 ) ∈ IR2 , (1.59)

then the system (1.58) can be written in the form u(t) ˙ = f (u(t)),

t ∈ IR+ ,

and the Jacobi matrix of f at (0, 0) is given by Ã ! b2 0 c12 x . Jf (0, 0) = c21 x b1 c22 x b2 The eigenvalues of Jf (0, 0) are given by r 1 1 2 2 λ1,2 = c22 x c x b2 ± b + c12 c21 x b1 x b2 . 2 4 22 2 We now distinguish two cases: a) Let c22 < 0. Then it follows that Re (λ1 ) < 0

and

Re (λ2 ) < 0.

Theorem 1.8 then implies that (0, 0) is an asymptotically stable fixed point of the mapping g h : IR2 → IR2 defined by (1.53) with f given by (1.59), if h > 0 is sufficiently small. From the equivalence (1.55) we deduce that this is the case, if µ ¶ 2 Re (λ1 ) 2 Re (λ2 ) , h < min . |λ1 |2 |λ2 |2 b) Let c22 = 0. Then it follows that p λ1,2 = ± i |c12 c21 | x b1 x b2 ,

i=

√

−1.

Hence Re (λ1 ) = Re (λ2 ) = 0 and the absolute values of the eigenvalues of the Jacobi matrix Jg h (0, 0) of g h at (0, 0) are given by p p |1 + λ1 h| = 1 + h2 |λ1 |2 and |1 + λ2 h| = 1 + h2 |λ2 |2 . This implies that they are larger than 1 for every choice of h > 0. Theorem 1.8 then implies that (0, 0) is an unstable fixed point of g h for every choice of h > 0.


59

b) Discretization of a Model of Interacting Logistic Growth of Two Populations The model is described by the system x˙ 1 (t) = (c1 + c11 x1 (t) + c12 x2 (t)) x1 (t), x˙ 2 (t) = (c2 + c21 x1 (t) + c22 x2 (t)) x2 (t),

t ∈ IR+ ,

(1.60)

where c1 > 0, c2 > 0, c11 < 0, c12 < 0, c21 < 0, c22 < 0. If we define

Ã

f (x) =

(c1 + c11 x1 + c12 x2 ) x1 (c2 + c21 x1 + c22 x2 ) x2

! x = (x1 , x2 ) ,

,

(1.61)

then the system (1.60) can be written in the form x(t) ˙ = f (x(t)),

t ∈ IR+ .

b1 > 0 and x b2 > 0 is a rest point of the system (1.60), A point x b ∈ IR2 with x i.e., a solution of the equation f (b x ) = Θ2 , if and only if b1 + c12 x b2 = −c1 , c11 x b1 + c22 x b2 = −c2 . c21 x

(1.62)

c11 c22 − c12 c21 > 0.

(1.63)

Let us assume that b2 ) which is given Then the equation (1.62) has exactly one solution x b = (b x1 , x by −c1 c22 + c2 c12 c1 c21 − c2 c11 x b1 = , x2 = . c11 c22 − c12 c21 c11 c22 − c12 c21 Further it follows that x b1 > 0 and x b2 > 0, if and only if −c1 c22 + c2 c12 > 0

and

c1 c21 − c2 c11 > 0.

(1.64)

The Jacobi matrix of f at x b is given by Ã ! b1 c12 x b1 c11 x Jf (b x) = c21 x b2 c22 x b2 and has the eigenvalues λ1,2 =

1 (c11 x b1 + c22 x b2 ) ± 2

r

1 (c11 x b1 + c22 x b2 ) + (c12 c21 − c11 c22 ) x b1 x b2 4

which have negative real parts. Theorem 1.8 then implies that x b is an asymptotically stable fixed point of the mapping g h : IR2 → IR2 defined by (1.53) with f given by (1.61), if h > 0 is sufficiently small.

60


c) An Emission Reduction Model In [32] a mathematical model for the reduction of carbon dioxide emission is investigated in form of a time-discrete dynamical system which as uncontrolled system is given by the following system of difference equations Ei (t + 1) = Ei (t) +

r X

em ij Mj (t),

j=1

Mi (t + 1) = Mi (t) − λi Mi (t) (Mi∗ − Mi (t)) Ei (t) for i = 1, . . . , r and t ∈ IN0

(1.65)

where Ei (t) denotes the amount of emission reduction and Mi (t) the financial means spent by the i−th actor at the time t, λi > 0 is a growth parameter and Mi∗ > 0 an upper bound for Mi (t) for i = 1, . . . , r and t ∈ IN0 . For t = 0 we assume the system to be in the state E0 i , M0 i , i = 1, . . . , r which leads to the initial conditions Ei (0) = E0 i

and Mi (0) = M0 i

for

i = 1, . . . , r.

¡ ¢T If we define x = E T , M T , E, M ∈ IRr , and functions fi : IRn → IRn , i = 1, . . . , n = 2r by fi (x) = Ei (t) +

r X

em ij Mj ,

i = 1, . . . , r,

j=1

fi (x) = Mi − λi Mi (Mi∗ − Mi ) Ei ,

i = r + 1, . . . , n,

(1.66)

then we can write (1.65) in the form x(t + 1) = f (x(t)),

t ∈ IN0 ,

(1.67)

T

where f (x) = (f1 (x), . . . , fn (x)) . ´T ³ b T , ΘrT For every x b= E , E ∈ IRr , we have x b = f (b x) . Let x b be any such fixed point of f . Then we replace the system (1.67) by the linear system x) x(t), x(t + 1) = Jf (b

t ∈ IN0 ,

x) is given by where the Jacobi matrix Jf (b ! Ã Ir C Jf (b x) = Or D

(1.68)


61

where Ir and Or is the r × r−unit and −zero matrix, respectively, and   em1 1 · · · em1 r  ..  .. C =  ... . .  emr 1 · · · emr r and

  D=

0

d1 1 ..

.

0 with

  

dr r

bi , di i = 1 − λi Mi∗ E

i = 1, . . . , r.

Let us assume that di i 6= 0, di i 6= 1

and |di i | ≤ 1

for all

i = 1, . . . , r.

x) is non-singular. Its eigenvalues are given by Then the matrix Jf (b µi = 1

and

µi+r = di i

for

i = 1, . . . , r,

hence |µi | ≤ 1

for

i = 1, . . . , 2r

and the corresponding eigenvectors Ã ! ei for i = 1, . . . , r Θr and

Ã

−C ej / (1 − dj j ) ej

! for j = 1, . . . , r

(ej = j − th unit vector) are linearly independent. Theorem 1.7 therefore implies that the zero sequence (x(t) = Θn )t∈IN0 which satisfies (1.68) is stable.

62


1.3.8 The Non-Autonomous Case: Definitions and Elementary Properties Let X be a metric space with metric d : X × X → IR+ and let (fn )n∈IN be ¢ of continuous mappings fn : X → X, n ∈ IN. Then the pair ¡ a sequence X, (fn )n∈IN is called a non-autonomous time-discrete dynamical system. The dynamics in this system is defined by the sequence F = (Fn )n∈IN0 of mappings Fn : X → X given by Fn (x) = fn ◦ fn−1 ◦ . . . ◦ f1 (x)

for all

x∈X

and n ∈ IN

and F0 (x) = x

for all

x ∈ X.

If fn = f for all n ∈ IN we obtain an autonomous time-discrete dynamical system (x, f ) as being defined in Section 1.3.1. For every x ∈ X we define an orbit starting with x by [ γF (x) = {Fn (x)}

(1.69)

n∈IN0

and LF (x) =

\

[

{Fm (x)}

(1.70)

n∈IN0 m≥n

where A denotes the closure of A ⊆ X. Then we can prove Proposition 1.19. For every x ∈ X the limit set LF (x) consists of all accumulation points of the sequence (Fn (x))n∈IN0 . The proof is the same as that of Proposition 1.16 and is left as an exercise. Now let X be compact. Then, for every x ∈ X, the sequence (Fn (x))n∈IN0 has at least one accumulation point which implies, by Proposition 1.19, that LF (x) is non-empty. By definition LF (x) is a closed subset of X and hence also compact for every x ∈ X. Further one can show that LF (x) is the smallest closed subset S ⊆ X with lim % (Fn (x), S) = 0

n→∞

with %(y, S) = min{d(y, z) | z ∈ S}.

(1.71)

The proof is left as an exercise (see Proposition 1.18). In addition we can prove the following Proposition 1.20. Let X be compact and let (fn )n∈IN be uniformly convergent to some mapping f0 : X → X. Then it follows that f0 (LF (x)) = LF (x)

for all

x ∈ X.


63

Proof. 1) At first we show that f0 (LF (x)) ⊆ LF (x), x ∈ X. Choose x ∈ X and y ∈ LF (x) arbitrarily. Since f0 is continuous, for every ε > 0, there exists some δ = δ(ε, y) > 0 such that f0 (e x) ∈ Uε (f0 (y))

for all x e ∈ Uδ (y).

Uniform convergence of (fn )n∈IN to f0 implies the existence of some n(ε) ∈ IN such that fn (e x) ∈ Uε (f0 (e x)) ⊆ U2ε (f0 (y))

for all

x e ∈ Uδ (y) and all

n ≥ n(ε).

By Proposition 1.19 there is a subsequence (Fni (x))i∈IN0 of (Fn (x))n∈IN0 with Fni (x) ∈ Uδ (y)

for all

i ∈ IN0 .

This implies Fni +1 (x) = fni +1 ◦ Fni (x) ∈ U2ε (f0 (y))

for all

ni ≥ n(ε).

Since ε > 0 is chosen arbitrarily, it follows that Fni +1 (x) → f0 (y)

and hence, by Proposition 1.19, f0 (y) ∈ LF (x).

2) Next we prove LF (x) ⊆ f0 (LF (x)) , x ∈ X. Choose x b ∈ X and y ∈ LF (b x ) arbitrarily. Then we have to show the existence x ) such that y = f0 (x). of some x ∈ LF (b By Proposition 1.19 there is a subsequence (Fni (b x ))i∈IN0 of (Fn (b x ))n∈IN0 with x ) → y as i → ∞. If we put Fni (b xi = Fni −1 (b x)

for all

i ∈ IN0 ,

then it follows that fni (xi ) → y as i → ∞. We can also assume that xi → x x ). for some x ∈ LF (b Then we have d (f0 (x), y) ≤ d (f0 (x), f0 (xi )) + d (f0 (xi ) , fni (xi )) + d (fni (xi ) , y) ≤ d (f0 (x), f0 (xi )) + sup d (f0 (e x) , fni (e x)) + d (fni (xi ) , y) {z } xe∈X {z } | | {z } | →0 →0 →0

as i → ∞, hence y = f0 (x).

u t

Next we give a more precise localization of the limit sets LF (x), x ∈ X, with the aid of a Lyapunov function which is defined as follows:

64


Let G ⊆ X be non-empty. Then we say that V : X → IR is a Lyapunov function with respect to (fn )n∈IN on G, if (1) (2)

V is continuous on X, V (fn (x)) − V (x) ≤ 0 for all x ∈ G and all n ∈ IN such that fn (x) ∈ G.

For every c ∈ IR we define V −1 (c) = {x ∈ X | V (x) = c}. Then we can prove the following Proposition 1.21. Let V be a Lyapunov function with respect to (fn )n∈IN on G ⊆ X where G is relatively compact. Let further x0 ∈ G be chosen such that fn (x0 ) ∈ G for all n ∈ IN. Then there exists some c ∈ IR such that LF (x0 ) ⊆ V −1 (c) and LF (x0 ) is non-empty. Proof. Since G is relatively compact, it follows that LF (x0 ) is nonempty. For every n ∈ IN we put xn = Fn (x0 ) which implies xn ∈ G and V (xn+1 ) ≤ V (xn )

for all

n ∈ IN.

Since V : X → IR is continuous, V is bounded from below on G which implies the existence of c = lim V (xn ). n→∞

Now let p ∈ LF (x0 ). Then, by Proposition 1.19, there is a subsequence (xni )i∈IN0 of (xn )n∈IN0 with xni → p as i → ∞. This implies V (p) = lim V (xni ) = c, hence p ∈ V −1 (c). i→∞

u t

1.3.9 Stability Based on Lyapunov’s Method Definition. A relatively compact set H ⊆ X is called stable with respect to (fn )n∈IN , if for every relatively compact open set U ⊆ X with U ⊇ H = closure of H there exists an open set W ⊆ X with H ⊆ W ⊆ U such that Fn (W ) ⊆ U

for all

n ∈ IN0

where Fn (W ) = {Fn (x) | x ∈ W } .


65

Theorem 1.9. Let H ⊆ X be relatively compact and such that for every relatively compact open set U ⊆ X with U ⊇ H there exists an open subset BU of U with BU ⊇ H and fn (BU ) ⊆ U

for all

n ∈ IN.

Further let G ⊆ X be an open set with G ⊇ H such that there exists a Lyapunov function V with respect to (fn )n∈N on G which is positive definite with respect to H, i.e. ¡ ¢ V (x) ≥ 0 for all x ∈ G and V (x) = 0 ⇐⇒ x ∈ H Then H is stable with respect to (fn )n∈N . Proof. Let U ⊆ X be an arbitrary relatively compact open set with U ⊇ H. Then U ∗ = U ∩ G is also a relatively compact open set with U ∗ ⊇ H and there exists an open set BU ∗ ⊆ U ∗ with BU ∗ ⊇ H and fn (BU ∗ ) ⊆ U ∗ for all n ∈ IN. Let us put m = min {V (x) | x ∈ U ∗ \ BU ∗ } . Since H ∩ (U ∗ \ BU ∗ ) is empty, it follows that m > 0. If we define

W = {x ∈ U ∗ | V (x) < m} ,

then W is open and H ⊆ W ⊆ BU ∗ . Now let x ∈ W be chosen arbitrarily. Then x ∈ BU ∗ and therefore F1 (x) = f1 (x) ∈ U ∗ . Further we have V (F1 (x)) = V (f1 (x)) ≤ V (x) < m, hence F1 (x) ∈ W ⊆ BU ∗ . This implies F2 (x) = f2 (F1 (x)) ∈ U ∗ and V (F2 (x)) ≤ V (F1 (x)) < m, hence F2 (x) ∈ W . By induction it therefore follows that for all n ∈ IN0 . Fn (x) ∈ W ⊆ U ∗ ⊂ U This shows that H is stable with respect to (fn )n∈IN .

u t

Definition. A set H ⊆ X is called an attractor with respect to (fn )n∈IN , if there exists an open set U ⊆ X with U ⊇ H such that ¡ ¢ ¡ ¢ lim % Fn (x), H = 0 in short: Fn (x) → H for all x ∈ U n→∞

where

¢ © ª ¡ % y, H = inf d(y, z) | z ∈ H .

If H ⊆ X is stable and an attractor with respect to (fn )n∈IN , then H is called asymptotically stable with respect to (fn )n∈IN .

66


Theorem 1.10. Let H ⊆ X be such that there exists a relatively compact open set U ⊇ H with fn (U ) ⊆ U

for all

n ∈ IN.

Further let V : X → IR be a Lyapunov function with respect to (fn )n∈IN on U which is positive definite with respect to H. Finally let lim V (Fn (x)) = 0

n→∞

for all

x ∈ U.

Then H is an attractor with respect to (fn )n∈IN . Proof. Let x ∈ U be chosen arbitrarily. Since the sequence (Fn (x))n∈IN0 is contained in U , for every subsequence (Fni (x))i∈IN0 of (Fn (x))n∈IN0 there ´ ³ and some q ∈ U with exists a subsequence Fnij (x) j∈IN0

lim Fnij (x) = q.

j→∞

This implies

³ ´ lim V Fnij (x) = V (q) = 0,

j→∞

hence, q ∈ H, and therefore Fnij (x) → H as j → ∞. From this it follows that t Fn (x) → H which shows that H is an attractor with respect to (fn )n∈IN . u Corollary. Under the assumptions of Theorem 1.9 and Theorem 1.10 it follows that H ⊆ X is asymptotically stable. Let us demonstrate Theorem 1.9 and Theorem 1.10 by an example. We choose X = IR2 and consider a sequence (fn )n∈IN of mappings fn : IR2 → IR2 given by µ ¶ an y bn x , fn (x, y) = , (x, y) ∈ IR2 , n ∈ IN, 1 + x2 1 + y 2 where (an )n∈IN and (bn )n∈IN are sequences of real numbers with a2n ≤ 1

and b2n ≤ 1

for all

n ∈ IN.

(1.72)

Put H = {(0, 0)}. If we choose V (x, y) = x2 + y 2 ,

(x, y) ∈ IR2 ,

then V is a Lyapunov function on G = IR2 with respect to (fn )n∈IN which is positive definite with respect to H = {(0, 0)}, since ¶ ¶ µ µ a2n b2n 2 −1 y + − 1 x2 V (fn (x, y)) − V (x, y) = (1 + x2 )2 (1 + y 2 )2 ¡ ¢ ¡ ¢ ≤ a2n − 1 y 2 + b2n − 1 x2 ≤0

for all

(x, y) ∈ IR2

and n ∈ IN.


67

Now let U ⊆ IR2 be relatively compact and open with (0, 0) ∈ U be given. Then there exists some r > 0 such that © ª BU = (x, y) ∈ IR2 | x2 + y 2 < r ⊆ U. Further it follows that f1n (x, y)2 + f2n (x, y)2 ≤ x2 + y 2 < r

for all (x, y) ∈ BU ,

hence fn (BU ) ⊆ BU ⊆ U

for all

n ∈ IN.

Therefore all the assumptions of Theorem 1.9 hold true which implies that {(0, 0)} is stable with respect to (fn )n∈IN . Next we sharpen (1.72) to a2n ≤ γ < 1

and b2n ≤ γ < 1

for all

n ∈ IN.

Then it follows that V (fn (x, y)) ≤ γ V (x, y)

for all

n ∈ IN and

(x, y) ∈ IR2

which implies V (Fn (x, y)) ≤ γ n V (x, y)

for all

n ∈ IN and

(x, y) ∈ IR2

and in turn lim V (Fn (x, y)) = 0

n→∞

for all

(x, y) ∈ IR2 .

By Theorem 1.10, H = {(0, 0)} is an attractor with respect to (fn )n∈IN and by the Corollary of Theorems 1.9 and 1.10 {(0, 0)} is asymptotically stable with respect to (fn )n∈IN . 1.3.10 Linear Systems We consider a normed linear space (E, || · ||) and a sequence (fn )n∈IN of mappings fn : E → E which are given by fn (x) = An (x) + bn ,

x ∈ E, n ∈ IN,

where (An )n∈IN is a sequence of continuous linear mappings An : E → E and (bn )n∈IN is a fixed sequence in E.

68


¢ ¡ Then the pair E, (fn )n∈IN is a non-autonomous time-discrete dynamical system. The dynamics in this system is defined by the sequence (Fn )n∈IN0 of mappings Fn : E → E given by Fn (x) = fn ◦ fn−1 ◦ . . . ◦ f1 (x) = An ◦ An−1 ◦ . . . ◦ A1 (x) +

n X

An ◦ An−1 ◦ . . . ◦ Ak+1 (bk ) (1.73)

k=1

for

x∈E

and n ∈ IN.

where An ◦ An+1 = idE for all n ∈ IN and F0 (x) = x

for all x ∈ E.

(1.74)

In general there will be no common fixed point of all fn , i.e., no point x∗ ∈ E with x∗ = fn (x∗ ) for all n ∈ IN which then would also be a common fixed point of all Fn . Therefore fixed point stability is not a reasonable concept in this case. We replace it by another concept of stability which has been introduced in Section 1.3.5 already and whose definition will be repeated here. Definition. A sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, is called 1. stable, if for every ε > 0 and every N ∈ IN0 there exists some δ = δ(ε, N ) > 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN − xN || < δ it follows that ||xn − xn || < ε

for all

n ≥ N + 1,

2. attractive, if for every N ∈ IN0 there exists some δ = δ(N ) > 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN − xN || < δ it follows that lim ||xn − xn || = 0, n→∞

3. asymptotically stable, if (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, is stable and attractive. As a first consequence of this definition we have Lemma 1.4. The following statements are equivalent: (1) All sequences (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, are stable. (2) One sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, is stable. (3) The sequence (xn = An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 = (xn = ΘE )n∈IN0 is stable. The proof is the same as that of Lemma 1.3. We can also replace stable by attractive and hence by asymptotically stable. This Lemma leads to the following sufficient conditions for stability and asymptotic stability:


69

Theorem 1.11. If ||An || = sup An (x) ≤ 1

for all

n ∈ IN,

(1.75)

||x||=1

then all sequences (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, are stable. If sup ||An || < 1,

(1.76)

n∈IN

then all sequences (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, are asymptotically stable. Proof. Let us first assume that (1.75) holds true. Let ε > 0 be chosen arbitrarily. Then we put δ = ε and conclude that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN || < δ for some N ∈ IN0 it follows that ||xn − ΘE || = ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN +1 ) || ≤ ||An || · ||An−1 || · . . . · ||AN +1 || · ||xN || 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN || < δ for some N ∈ IN0 it follows that lim ||xn − ΘE || = lim ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || n→∞ ¶n−N −1 µ ≤ lim sup ||Am || ||xN ||

n→∞

n→∞

m∈IN

= 0. Thus, the sequence (xn = ΘE )n∈IN0 is attractive, hence asymptotically stable which implies that all sequences (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, are asymptotically stable. According to the above considerations it suffices to consider homogeneous systems with x ∈ E, n ∈ IN, fn (x) = An (x), such that Fn (x) = An ◦ An−1 ◦ . . . ◦ A1 (x),

x ∈ E, n ∈ IN.

Then all sequences (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, are stable or asymptotically stable, if and only if the sequence (xn = Fn (ΘE ))n∈IN0 is stable or asymptotically stable. u t

70


This leads to Theorem 1.12. Assumption: All An , n ∈ IN, are invertible. Assertion: The sequence (An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 is stable, if and only if for every N ∈ IN0 there exists a constant cN > 0 such that ||An ◦ An−1 ◦ . . . ◦ AN +1 || ≤ cN

for all

n ≥ N + 1.

(1.77)

The sequence (An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 is asymptotically stable, if and only if for every N ∈ IN0 it is lim ||An ◦ An−1 ◦ . . . ◦ AN +1 || = 0.

(1.78)

n→∞

Proof. If the sequence (An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 is stable, we choose an arbitrary ε > 0 and conclude that, for every N ∈ IN0 , there exists δ = δ(ε, N ) > 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , with x0 ∈ E, it follows that ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || < ε

for all

n ≥ N + 1.

This implies ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || < ε

for all

n ≥ N +1

and

||xN || ≤

δ , 2

hence, ||An ◦ An−1 ◦ . . . ◦ AN +1 || =

sup ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || ≤ ||xN ||≤1

for all

2ε cN δ

n≥N +1

(here the assumption that all An , n ∈ IN, are invertible is needed). Conversely, let (1.77) be true for every N ∈ IN0 . Then we choose ε > 0, N ∈ IN, define δ = ε/cN and conclude that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN || < δ it follows that ||xn − ΘE || = ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || ≤ ||An ◦ An−1 ◦ . . . ◦ AN +1 || · ||xN || 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN || < δ it follows that lim ||xn − ΘE || = 0, n→∞

i.e., for every ε > 0 there exists some n(ε) ≥ N + 1 with ||xn − ΘE || < ε for all n ≥ n(ε), hence ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || < ε

for all n ≥ n(ε) and

||xN || ≤

δ . 2

This implies ||An ◦ An−1 ◦ . . . ◦ AN +1 || =

sup ||An ◦ An−1 ◦ . . . ◦ AN +1 (xN ) || ≤ ||xN ||≤1

for all

2ε δ

n ≥ n(ε)

(here again the assumption that all An , n ∈ IN, are invertible is needed), hence lim ||An ◦ An−1 ◦ . . . ◦ AN +1 || = 0. n→∞

Conversely, let (1.78) be true for every N ∈ IN0 . Thus (1.77) is satisfied for every N ∈ IN0 and therefore the sequence (An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 is stable. Further there exists δ = δ(N ) > 0 such that for every sequence (xn = Fn (x0 ))n∈IN0 , x0 ∈ E, with ||xN || < δ it follows that lim ||xn − ΘE || ≤ lim ||An ◦ An−1 ◦ . . . ◦ AN +1 || · ||xN || = 0.

n→∞

n→∞

Thus, the sequence (An ◦ An−1 ◦ . . . ◦ A1 (ΘE ))n∈IN0 is attractive.

u t

Now we specialize to the case E = IRk equipped with any norm. Then we have fn (x) = An x + bn ,

x ∈ IRk ,

where (An )n∈IN is a sequence of real k × k−matrices and (bn )n∈IN a sequence of vectors bn ∈ IRk . This leads to Fn (x) = fn ◦ fn−1 ◦ . . . ◦ f1 (x) n X = An An−1 · · · A1 x + An An−1 · · · Aj+1 bj j=1

for x ∈ IRk and n ∈ IN where An An−1 · · · Aj+1 bj = bn for j = n.

72


Let us assume that, for some p ∈ IN, An+p = An

and bn+p = bn

for all

n ∈ IN.

The question then is whether there is a sequence (xn )n∈IN0 with xn = An xn−1 + bn

for all

n ∈ IN

(1.79)

such that xn+p = xn

for all

n ∈ IN0

(1.80)

which implies xp = x0 . Conversely, let this be the case. Then x1+p x2+p .. . xn+p

= A1+p xp + b1+p = A2+p x1+p + b2+p .. . = An+p xn+p−1 + bn+p

= A1 x0 + b1 = A2 x1 + b1 .. . = An xn−1 + bn

= x1 = x2 .. . = xn ,

hence xn+p = xn

for all

n ∈ IN0 .

Result. A sequence (xn )n∈IN0 with (1.79) satisfies (1.80), if and only if xp = x0 which is equivalent to Fp (x0 ) = x0 , i.e., x0 = Ap Ap−1 · · · A1 x0 +

p X

Ap Ap−1 · · · Aj+1 bj .

j=1

This implies that there exists exactly one sequence (xn )n∈IN0 with (1.79) and (1.80), if and only if the matrix I − Ap Ap−1 · · · A1 , I = k × k−unit matrix, is non-singular. The first vector x0 is then given by −1

x0 = (I − Ap Ap−1 · · · A1 )

p X

Ap Ap−1 · · · Aj+1 Aj .

j=1

In the homogeneous case where bn = Θk

for all

n ∈ IN

a non-zero sequence (xn )n∈IN0 with (1.79) and (1.80) can only exist, if the matrix I − Ap Ap−1 · · · A1 is singular, i.e., det (I − Ap Ap−1 · · · A1 ) = 0. This again is equivalent to the fact that λ = 1 is an eigenvalue of Ap Ap−1 · · · A1 .


73

1.3.11 Application to a Model for the Process of Hemo-Dialysis In order to describe the temporal development of the concentration of some poison like urea in the body of a person suffering from a renal disease and having to be attached to an artificial kidney a mathematical model has been proposed in [22] which can be described as follows. The human body is divided into two compartments being termed as cellular part Z and extracellular part E of volume VZ and VE , respectively. The two compartments are separated by cell membranes having the permeability CZ [ml/min]. Let t [min] denote the time and let KZ (t) [mg/min] and KE (t) [mg/min] be the concentration of the poison at the time t in Z, and E, respectively. We consider some time interval [0, T ], T > 0, and assume that the patient is attached to the artificial kidney during the subinterval [0, td ] for some td ∈ (0, T ]. We further assume that the generation rate of the poison in Z and E is L1 [mg/min] and L2 [mg/min], respectively. Then the temporal development of KZ = KZ (t) and KE = KE (t) in [0, ∞) can be described by the following system of linear differential equations VZ K˙ Z (t) = −CZ (KZ (t) − KE (t)) + L1 , VE K˙ E (t) = −CZ (KZ (t) − KE (t)) − C(t) KE (t) + L2

(1.81)

where ( C C(t) = 0

for 0 ≤ t < td , for td ≤ t < T.

C(t + T ) = C(t)

for all

(1.82)

t≥0

(1.83)

and C [ml/min] is the permeability of the membranes of the artificial kidney. By (1.83) the periodicity of the process of dialysis is expressed. It can be proved that there is exactly one pair (KZ (t), KE (t)) of positive and T −periodic solutions of (1.81) which intuitively is to be expected. For the numerical computation of these solutions Euler’s polygon method can be used. For this purpose we choose a time stepsize ∆t > 0 such that td = K · ∆t,

T = N · ∆t

for all

K, N ∈ IN with

2 0, if for every pair x0 , xT ∈ IRn there is a control u ∈ C (IR, IRm ) with u(t) ∈ Ω

for all

t ∈ [0, T ]

such that the unique solution x ∈ C 1 (IR, IRn ) of (2.2), (2.3) satisfies condition (2.4). (2) The system (2.1) is called Ω−controllable, if there exists some T > 0 such that it is Ω−controllable on the interval [0, T ]. (3) The system (2.1) is called completely Ω−controllable, if it is Ω−controllable on every interval [0, T ], T > 0. The complete Ω−controllability is obviously the strongest property. In the next section we will present sufficient conditions for linear controlled systems to be completely Ω−controllable. In the case of non-linear systems we will concentrate on the following problem: b =solution of (2.5) and a time T > 0 find a For a given x0 ∈ IRn and xT = x control function u ∈ C (IR, IRm ) such that the unique solution x ∈ C 1 (IR, IRn ) of (2.2), (2.3) satisfies condition (2.4). 2.1.2 Controllability of Linear Systems Instead of (2.1) we consider a system of the form x˙ = A x + B u

(2.6)

where x ∈ IRn , u ∈ IRm , A = real n × n−matrix and B = real n × m−matrix. If we put f (x, u) = A x + B u, x ∈ IRn , u ∈ IRm , ¡ ¢ then (2.6) is of the form (2.1) and f ∈ C 1 IRn+m , IRn . The assumption in Section 2.1.1 is satisfied. For every x0 ∈ IRn and every function u ∈ C (IR, IRm ) the unique solution x ∈ C 1 (IR, IRn ) of (2.6) with x(0) = x0 is given by ¶ µ Z t tA −sA e B u(s) ds , t ∈ IR. (2.7) x0 + x(t) = e 0

2.1 The Time-Continuous Case

We put

Y (t) = e −tA B

for all

t ∈ IR

79

(2.8)

and consider control functions of the form u(t) = Y T x,

t ∈ IR,

(2.9)

where x ∈ IRn is chosen arbitrarily. Insertion into (2.7) gives ¶ µ Z t tA T x(t) = e Y (s) Y (s) x ds , t ∈ IR. x0 + 0

Let, for some given T > 0, the n × n−matrix Z T M (T ) = Y (t) Y (t)T dt

(2.10)

0

(which is symmetric and positive semi-definite) be non-singular. Then, for every vector xT ∈ IRn , there is exactly one zT ∈ IRn such that xT = e T A (x0 + M (T ) zT )

⇐⇒

M (T ) zT = e −T A xT − x0

which implies that the system (2.6) is IRm −controllable on [0, T ]. Lemma 2.1. For some T > 0 the matrix M (T ) (2.10) is non-singular, if and only if the following implication holds true Y (t)T x = Θm

for all

t ∈ [0, T ]

=⇒

x = Θn .

(2.11)

Proof. (a) Let M (T ) be non-singular. If there were some z ∈ IRn with z 6= Θn such that Y (t)T z = Θn for all t ∈ [0, T ], then it would follow that M (T ) z = Θn which is impossible. (b) Let the implication (2.11) hold true. If M (T ) were singular, then there would exist some z ∈ IRn , z 6= Θn , with M (T ) z = Θn . This implies

Z T

z M (T ) z =

T

z T Y (t) Y (t)T z dt = 0

0

from which it follows that Y (t)T z = Θm

for all

t ∈ [0, T ].

However, this implies z = Θn contradicting z 6= Θn . Therefore, M (T ) must be non-singular. u t Theorem 2.2. The implication (2.11) holds true for every T > 0, if and only if the so called Kalman condition ¢ ¡ (2.12) rank B | AB | . . . | An−1 B = n is satisfied.

80

2 Controlled Systems

Proof. (a) Let the implication (2.11) be violated for some T > 0. Then there is some z ∈ IRn , z 6= Θn with Y (t)T z = Θm

for all

t ∈ [0, T ].

If one differentiates z T Y (t) successively by t and puts t = 0 then it follows that T T T z T B = Θm , z T AB = Θm , . . . , z T An−1 B = Θm (2.13) which contradicts condition (2.12). (b) Let condition (2.12) be violated. Then there is a vector z ∈ IRn with z 6= Θn such that (2.13) holds true. Now let Φ(−λ) = a0 + a1 (−λ) + . . . + an−1 (−λ)n be the characteristic polynomial of A. Then it follows, by the theorem of Cayley-Hamilton, that Φ(−A) = 0 and, therefore, An = b0 I + b1 A + . . . + bn−1 An−1 with suitable coefficients b0 , . . . , bn−1 ∈ IR. This implies Ak = b0 Ak−n + b1 Ak+1−n + . . . + bn−1 Ak−1 for all k ≥ n and on using (2.13) one concludes that T z T Ak B = Θm

for all

k≥0

=⇒

z T Y (t) = 0

for all

t ∈ IR,

i.e., the condition (2.11) is violated for all T > 0.

u t

As a consequence of the above considerations we obtain Theorem 2.3. If the Kalman condition (2.12) is satisfied, then the system (2.6) is completely IRm −controllable. Let us demonstrate this result by an example. We consider a moving linear pendulum as being depicted in the following picture: v(0) = v0 < 0 ¶ ¶ ¶ϕ(t)

¶

¶ m v

¶

v(t) < 0 S

S S

v(T ) = 0

ϕ(t) ϕ(0) = ϕ0 ϕ(0) ˙ = ϕ˙ 0

l

S S

ϕ(T ) = ϕ(T ˙ )=0

S vm

v m


81

The movement of the pendulum is described by the differential equation v¨(t) g , ϕ(t) ¨ = − ϕ(t) − l l

t ∈ IR,

(2.14)

with the initial conditions ϕ(0) = ϕ0 ,

ϕ(0) ˙ = ϕ˙ 0

where ϕ = ϕ(t) denotes the deviation angle at the time t and v = v(t) is the location of the suspension point at the time t. This is moving with speed v(t) ˙ and acceleration v¨(t) at the time t. The problem to be solved consists of finding a function v = v(t) by which an initial state (ϕ(0), ϕ(0), ˙ v(0), v(0)) ˙ = (ϕ0 , ϕ˙ 0 , v0 , v˙ 0 ) of the pendulum is transferred in the final (rest) state (ϕ(T ), ϕ(T ˙ ), v(T ), v(T ˙ )) = (0, 0, 0, 0) at a given time T > 0. If we define a vector function x(t) = (x1 (t), x2 (t), x3 (t), x4 (t)) by putting ˙ x3 (t) = v(t), x4 (t) = v(t), ˙ then the differential x1 (t) = ϕ(t), x2 (t) = ϕ(t), equation (2.14) can be rewritten in the form x(t) ˙ = A x(t) + B u(t),

t ∈ IR,

(2.15)

where u(t) = v¨(t), t ∈ IR, is the control and 

0  −g/l A=  0 0

1 0 0 0

0 0 0 0

 0 0 , 1 0



 0  −1/l   B=  0 . 1

In this case we obtain  0 −1/l 0 g/l2 ¢  −1/l 0 g/l2 0  ¡ , B | AB | A2 B | A3 B =   0 1 0 0  1 0 0 0 

hence

¡ ¢ rank B | AB | A2 B | A3 B = 4,

i.e., the Kalman condition (2.12) is satisfied. By Theorem 2.3 the system (2.15) is completely IR4 −controllable. This means T that every initial state (ϕ(0), ϕ(0), ˙ v(0), v(0)) ˙ ∈ IR4 of the pendulum can be controlled within every time interval [0, T ], T > 0, by a suitable control function u(t) = v¨(t), t ∈ [0, T ] with u ∈ C 1 (IR) into every final state T (ϕ(T ), ϕ(T ˙ ), v(T ), v(T ˙ )) ∈ IR4 .

82


2.1.3 Restricted Null-Controllability of Linear Systems We start with a system x(t) ˙ = A x(t),

t ∈ IR,

(2.16)

where x ∈ IRn and A =real n × n−matrix. This system has x b = Θn as rest point. In the following we make the special choice Ω = [−1, +1]m and ask for the existence of some time T > 0 and a control function u ∈ C (IR, IRm ) with u(t) ∈ Ω

for all

t ∈ [0, T ]

(2.17)

such that for every initial state x0 ∈ IRn the unique solution x ∈ C 1 (IR, IRn ) of (2.6) with (2.3) x(0) = x0 satisfies the end condition x(T ) = x b = Θn .

(2.18)

In order to give an affirmative answer to this question we replace the space C (IR, IRm ) of control functions by the space L∞ (IR, IRm ) of measurable and essentially bounded m−vector functions u on IR. Condition (2.17) is replaced by for almost all t ∈ [0, T ] (2.19) ||u(t)||∞ ≤ 1 where || · ||∞ denotes the maximum norm in IRm . For every T > 0 we define UT = {u ∈ L∞ (IR, IRm ) | u satisfies (2.19)} .

(2.20)

From the solution formula (2.7) which also holds true for the unique absolutely continuous solution x : IR → IRn of (2.6) with (2.3) in the case that u ∈ L∞ (IR, IRm ) it follows that the end condition (2.18) is equivalent to Z

T

Y (t) u(t) dt = −x0

(2.21)

0

where Y (t) is given by (2.8). This leads to the Problem of Restricted Null-Controllability: Let x0 ∈ IRn be given arbitrarily. Find some T > 0 and u ∈ UT (2.20) such that condition (2.21) is satisfied. If this problem has a solution, we call the system (2.6) restricted nullcontrollable.


83

If we define, for every T > 0, a so called reachable set ( ) Z T

E(T ) =

x=

Y (t) u(t) dt | u ∈ UT

⊆ IRn ,

(2.22)

0

then the problem of restricted null-controllability is solvable, if and only if [ E= E(T ) = IRn . (2.23) T >0

Obviously every reachable set E(T ) is convex and we have the following implication 0 ≤ T1 < T2 =⇒ E (T1 ) ⊆ E (T2 ) . This implies that E is also convex. Assumption: E 6= IRn . b 6∈ E. By a well known separation Then there exists some x b ∈ IRn with x theorem for convex sets there exists a number α ∈ IR and a vector y ∈ IRn , y 6= Θn such that b yT x ≤ α < yT x

for all

x ∈ E.

(2.24)

Because of Θn ∈ E it follows that α ≥ 0. Further the left side of (2.24) is equivalent to Z T y T Y (t) u(t) dt ≤ α for all u ∈ UT and T > 0. (2.25) 0

For the following we assume that the Kalman condition (2.12) is satisfied. Because of y 6= Θn it follows from Theorem 2.2 that for every T > 0 there exists some tT ∈ [0, T ] with T . y T Y (tT ) 6= Θm

If we put

v(t)T = y T Y (t),

t ∈ IR,

then there exists at least one component of v which does not vanish identically, say t ∈ IR, v1 (t) = y T e −tA b1 , where b1 is the first column of the matrix B. Since with u also −u belongs to UT , T > 0, it follows from (2.25) that ¯Z ∞ ¯ ¯ ¯ T ¯ for all u ∈ L∞ (IR, IRm ) with v(t) u(t) dt¯¯ ≤ α ¯ 0

||u(t)||∞ ≤ 1

for almost all t ∈ (0, ∞).

84


This implies Z

∞

||v(t)||1 dt = 0

n Z X i=1

and further

Z

∞

|vi (t)| dt ≤ α < ∞ 0

∞

|v1 (t)| dt ≤ α. 0

This implies the existence of the integrals Z ∞ v1 (s) ds for all t ∈ [0, ∞) w(t) = t

and it follows that d w(t) = −v1 (t) dt If we put D =

for all

t ∈ [0, ∞)

as well as

lim w(t) = 0.

t→∞

d , it follows that dt

¡ k ¢ D v1 (t) = y T (−A)k e −tA b1 ,

t ∈ IR,

for k = 0, 1, 2, . . .

If ψ(λ) denotes the characteristic polynomial of A, then the application of the Cayley-Hamilton-theorem implies ψ(A) = 0 and therefore (ψ(−D) v1 ) (t) = y T (ψ(A)) e −tA b1 = 0

for all

t ∈ [0, ∞).

This implies ψ(−D)(−D w)(t) = 0

for all

t ∈ [0, ∞)

and because (−D) and ψ(−D) are interchangeable we can conclude (−D ψ(−D) w)(t) = 0

for all

t ∈ [0, ∞).

The characteristic equation of this linear differential equation reads −λ ψ(−λ) = 0. From lim w(t) = 0 we infer that at least one solution −λ of the equation t→∞

ψ(−λ) = 0 must have a negative real part. Therefore the matrix A must have an eigenvalue with a positive real part. Summarizing we obtain by contraposition the Theorem 2.4. If the Kalman condition (2.12) is satisfied and if the matrix A has only eigenvalues with non-positive real parts, then the system (2.6) is restricted null-controllable. This means that for every initial state x0 ∈ IRn there is some T > 0 and a control function u ∈ UT (2.20) such that the unique absolutely continuous solution x = x(t), t ∈ IR, of (2.6), (2.3) satisfies the end condition (2.18).


85

The assumptions of this theorem are for instance satisfied for the system (2.15). It has been shown already that the Kalman condition (2.12) is satisfied. It is easy to show that the eigenvalues of the matrix A are given by r r g g , λ2 = −i , λ3 = λ4 = 0. λ1 = i l l 2.1.4 Controllability of Nonlinear Systems into Rest Points We again start with a system of the form x˙ = f (x, u) n

m

n

(2.1)

where f ∈ C (IR × IR , IR ) and f (·, u) ∈ C (IR , IR ) for every u ∈ IRm . We make the same assumptions as at the beginning of Section 2.1.1. 1

n

m

Let x b ∈ IRn be a rest point of the uncontrolled system for u ≡ Θm , i.e., a solution of the equation (2.5) f (b x, Θm ) = Θn . Given an arbitrary initial state x0 ∈ IRn we then look for some time T > 0 and a control function u ∈ C (IR, IRm ) such that the unique solution x ∈ C 1 (IR, IRn ) of (2.1) with (2.3) x(0) = x0 satisfies the end condition x(T ) = x b. If this problem has a solution for every x0 ∈ IRn we call the system (2.1) controllable to x b. Let us weaken this concept to a concept of local controllability. For this purpose we assume that fi ∈ C 1 (IRn × IRm )

for all

i = 1, 2, . . . , n

and linearize the equation (2.1) in (b x, Θm ), i.e., we replace it by x˙ = A x + B u

(2.6)

where ¢ ¡ x, Θm ) i,j=1,...,n A = fi xj (b

and

B = (fi uk (b x, Θm )) i=1,...,n . (2.26) k=1,...,m

Definition. The system (2.1) is called locally controllable to x b, if the system (2.6) with A and B by (2.26) is IRm −null-controllable, i.e., if for every x0 ∈ IRn there exists a T > 0 and a control function u ∈ C (IR, IRm ) such that the unique solution x ∈ C 1 (IR, IRn ) of (2.6), (2.3) satisfies the end condition x(T ) = Θm . An immediate consequence of Theorem 2.3 is the

(2.18)

86


Theorem 2.5. If A and B given by (2.26) satisfy the Kalman condition (2.12), then the system (2.1) is locally controllable to x b. Definition. The system (2.1) is called locally restricted controllable to x b, if the system (2.6) with A and B by (2.26) is restricted null-controllable, i.e., if for every x0 ∈ IRn there exists a T > 0 and a control function u ∈ UT (2.20) such that the unique absolutely continuous solution x = x(t), t ∈ IR, of (2.6), (2.3) satisfies the end condition (2.18). An immediate consequence of Theorem 2.4 is the Theorem 2.6. If A and B given by (2.26) satisfy the Kalman condition (2.12) and if the matrix A has only eigenvalues with non-positive real parts, then the system (2.1) is locally restricted controllable to x b. Let us demonstrate these two theorems by the example of a nonlinear pendulum with movable suspension point. We use the same notations as in the case of a linear pendulum in Section 2.1.2. The movement of the pendulum is described by the differential equation ϕ(t) ¨ =

v¨(t) g sin ϕ(t) − cos ϕ(t), l l

t ∈ IR.

(2.27)

If one defines functions x1 (t) = ϕ(t),

x2 (t) = ϕ(t), ˙

x3 (t) = v(t),

x4 (t) = v(t), ˙

then (2.27) can be rewritten in the form x˙ 1 (t) = x2 (t), g u(t) x˙ 2 (t) = − sin x1 (t) − cos x1 (t), l l x˙ 3 (t) = x4 (t), x˙ 4 (t) = u(t)

(2.28)

where u(t) = v¨(t), t ∈ IR, is the control function. If we define T

f (x, u) = (f1 (x, u), f2 (x, u), f3 (x, u), f4 (x, u)) , where f1 (x, u) = x2 (t), g u f2 (x, u) = − sin x1 − cos x1 , l l f3 (x, u) = x4 , f4 (x, u) = u,

x ∈ IR4 , u ∈ IR,


87

then the system (2.28) takes the form (2.1) and we have ¡ ¢ for i = 1, 2, 3, 4. fi ∈ C 1 IR4 , IR Obviously for u ≡ 0 the point Θ4 = (0, 0, 0, 0)T ∈ IR4 system (2.28) and we obtain  0 1  −g/l 0 ¢ ¡ A = fi xj (Θ4 , 0) i,j=1,2,3,4 =   0 0 0 0   0  −1/l   B = (fi u (Θ4 , 0))i=1,2,3,4 =   0 . 1

is a rest point of the 0 0 0 0

 0 0 , 1 0

We have already shown (see Section 2.1.2) that the Kalman condition (2.12) is satisfied for A and B and that the matrix A has only eigenvalues with real parts equal to zero (see Section 2.1.3). Theorem 2.6 then implies that the system is locally restricted controllable to x b = Θ4 . Now we return to the question of the x b−controllability of the system (2.1). We again assume a set Ω ⊆ IRn with Θm ∈ Ω to be given and define, for every T > 0, UT = {u ∈ L∞ (IR, IRm ) | u(t) ∈ Ω for almost all t ∈ [0, T ]} .

(2.29)

Further let S (b x, T ), for every T > 0, be the set of all x0 ∈ IRn such that there exists a u ∈ UT such that the unique absolutely continuous solution x = x(t), t ∈ IR, of (2.1), (2.3) satisfies the end condition x(T ) = x b. If we then define S (b x) =

[

S (b x, T ) ,

(2.30)

T >0

the set S (b x) consists of all vectors x0 ∈ IRn such that there exists a time T > 0 and a control u ∈ UT such that the unique absolutely continuous solution x = x(t), t ∈ IR, of (2.1), (2.3) satisfies the end condition x(T ) = x b. With this definition we formulate the Theorem 2.7. Let x b be an interior point of S (b x) (b x ∈ S (b x) follows from the definition of S (b x)) and let x b be attractive, i.e., there exists an open set U ⊆ IRn with x b ∈ U such that for every x0 ∈ U the unique solution x ∈ C 1 (IR, IRn ) of b. Then it x˙ = f (x, Θm ) with x(0) = x0 satisfies the statement lim x(t) = x follows that S (b x) ⊇ U.

t→∞

88


Proof. Let x0 ∈ U be chosen arbitrarily. Then for the solution x ∈ C 1 (IR, IRn ) b. Since x b is of x˙ = f (x, Θm ) with x(0) = x0 it follows that lim x(t) = x t→∞

an interior point of S (b x), there exists some t1 > 0 with x (t1 ) ∈ S (b x). This implies the existence of some time T > 0 and a function u ∈ UT such that the absolutely continuous solution x e=x e(t) of x e˙ (t) = f (e x(t), u(t)) ,

t ∈ IR with

x e(0) = x (t1 )

satisfies the end condition x e(T ) = x b. If we define

( Θm for t ≤ t1 , u (t) = u (t − t1 ) for t > t1 , ∗

then u∗ ∈ UT and it follows for the absolutely continuous solution x∗ = x∗ (t), t ∈ IR, of x˙ ∗ (t) = f (x∗ (t), u∗ (t)) , t ∈ IR, x∗ (0) = x(0) = x0 that the e(T ) = x b is satisfied which completes the proof. end condition x∗ (T + t1 ) = x u t Let us apply this result to linear systems of the form (2.6). In this case we have f (x, u) = A x + B u, x ∈ IRn , u ∈ IRm . A rest point of the uncontrolled system x˙ = f (x, Θm ) = A x,

x ∈ IRn ,

(2.16)

is given by x b = Θn . This is the only one, if the matrix A is non-singular. If all the eigenvalues of A have negative real parts (which implies that A is non-singular), then x b = Θn is globally attractive, i.e., for every x0 ∈ IRn the unique solution x ∈ C 1 (IR, IRn ) of (2.16) with x(0) = x0 satisfies the condition lim x(t) = Θn . t→∞

Theorem 2.8. If the Kalman condition (2.12) is satisfied, then x b = Θn is an interior point of S (Θn ) = E (2.23). Proof. Assume that Θn is no interior point of E. S Then Θn is also not an λ E is not equal to IRn . algebraically interior point of E and K(E) = λ≥0

Since because of E = −E the convex cone K(E) is a linear space, the set E, as subset of K(E), is contained in a hyperplane, i.e., there exists some y ∈ IRn with y 6= Θn such that yT e = 0

for all

e ∈ E.

In particular it follows for every T > 0 that Z T y T e tA B u(t) dt = 0 for all 0

u ∈ UT


and hence for all u ∈ L∞ ([0, T ], IRm ) =

S

89

λ UT .

λ≥0

This implies y T e −tA B = 0

for all

t ∈ [0, T ]

and in turn y = Θn by Theorem 2.2 which contradicts y 6= Θn .

u t

As a consequence of Theorem 2.7 and Theorem 2.8 we obtain Theorem 2.9. If the Kalman condition (2.12) is satisfied and if the matrix A has only eigenvalues with negative real parts, then for every x0 ∈ IRn there exists some T > 0 and a control function u ∈ UT (2.20) such that the unique absolutely continuous solution x = x(t), t ∈ IR, of (2.6), (2.3) (which is given by (2.7)) satisfies the end condition x(T ) = x b (2.18). This theorem is contained in Theorem 2.4. 2.1.5 Approximate Solution of the Problem of Restricted Null-Controllability We again consider the problem of restricted null-controllability as in Section 2.1.3. We assume T > 0 and x0 ∈ IRn to be given and formulate the following Approximation Problem: Find some uT ∈ UT (2.20) such that °Z ° ° °

0

T

°Z ° ° ° ° Y (t) uT (t) dt + x0 ° ≤ ° ° 2

T 0

° ° Y (t) u(t) dt + x0 ° °

for all

u ∈ UT

2

where Y (t) is given by (2.8). A solution uT ∈ UT of this approximation problem is considered as an approximation of a solution of the problem of restricted null-controllability. If one defines ( V =

Z n

y ∈ IR | y =

)

T

Y (t) u(t) dt with u ∈ UT

,

0

one obtains a convex subset of IRn . This set is weakly closed, i.e., the following implication holds true: ) ( lim ykT x = y T x for all x ∈ IRn k→∞ =⇒ y ∈ V. and yk ∈ V for all k ∈ IN

90


Proof. yk ∈ V implies for every k ∈ IN the existence of some function uk ∈ {u ∈ L∞ ([0, T ], IRm ) | ||u(t)||∞ ≤ 1 for almost all t ∈ [0, T ]} = K∞ (T ) with

Z

T

yk =

Y (t) uk (t) dt 0

which implies Z

T

uk (t)T Y (t) x dt = y T x

lim

k→∞

for all

x ∈ IRn .

0

Since the set K∞ (T ) is weak-∗ sequentially compact, there is some u ∈ K∞ (T ) and a subsequence (uki )i∈IN with Z lim

i→∞

0

Z

T

which implies Ã T

y −

T

Z uki (t)T Y (t) x dt =

T

u(t)T Y (t) x dt

0

! T

T

u(t) Y (t) dt

x=0

for all

x ∈ IRn .

0

From this it follows that Z

T

Y (t) u(t) dt

y=

⇐⇒

y∈V

0

which completes the proof.

u t

The weak closedness of the set V implies that V is also closed. As a consequence of a well known theorem in approximation theory we then obtain the existence of exactly one y ∗ ∈ V with ||y ∗ + x0 ||2 ≤ ||y + x0 ||2

for all

y∈V

which is characterized by T

T

(y ∗ + x0 ) y ∗ ≤ (y ∗ + x0 ) y

for all

y ∈ V,

if y ∗ + x0 6= Θn . Now we distinguish two cases: (a) y ∗ + x0 = Θn . Then every u∗ ∈ UT with Z T y∗ = Y (t)T u∗ (t) dt 0

is a solution of the problem of restricted null-controllability.


91

b) y ∗ + x0 6= Θn . Then for every u∗ ∈ UT as in (a) we obtain from the above characterization of y ∗ ∈ V the equation Z T Z T T T ∗ ∗ (y + x0 ) Y (t) u (t) dt = || (y ∗ + x0 ) Y (t)||1 dt. − 0

0

We try to solve this equation iteratively by starting with ¡ ¢ y 0 = Θn and u0k (t) = −sgn xT0 Y (t) k for k = 1, . . . , n, t ∈ [0, T ], ¡ ¢ (where sgn(0) = 0) and constructing a sequence y N N ∈IN0 in IRn and a ¡ N¢ sequence u N ∈IN in K∞ (T ) as follows: 0

If y N ∈ IRn is given, then we define ³¡ ´ ¢T N uN (t) = −sgn y + x Y (t) 0 k

k

and

Z y N +1 =

T

for k = 1, . . . , n, t ∈ [0, T ],

Y (t) uN (t) dt.

0

¡ ¢ If the sequence uN N ∈IN0 converges weak-∗ to u∗ ∈ K∞ (T ), then the ¡ ¢ RT sequence y N N ∈IN0 to y ∗ = 0 Y (t) u∗ (t) dt and y ∗ and u∗ satisfy the equation in (b). 2.1.6 Time-Minimal Restricted Null-Controllability of Linear Systems We replace the set UT (2.20) in Section 2.1.3 by ¾ ½ U = u ∈ L∞ (IR, IRm ) | ess sup ||u(t)||2 ≤ γ

(2.31)

t∈IR

¡ ¢1/2 where ||u(t)||2 = u(t)T u(t) , t ∈ IR. The problem of restricted null-controllability now consists of finding a time T > 0 and some u ∈ U (2.31) such that Z T Y (t) u(t) dt = −x0 (2.21) 0 n

where x0 ∈ IR is given arbitrarily. Let us assume that the problem is solvable. Then we define a so called minimum time T (γ) by T (γ) = inf{T > 0 | There is some u ∈ U which satisfies (2.21)}. At first we prove

(2.32)

92


Theorem 2.10. If restricted null-controllability is possible, then time-minimal restricted null-controllability is also possible, i.e., there exists some uγ ∈ U such that Z T (γ) Y (t) uγ (t) dt = −x0 . (2.33) 0

Proof. By the definition of T (γ) there exists a sequence (Tk )k∈IN of times Tk ≥ T (γ) with lim Tk = T (γ) and corresponding controls uk ∈ U with k→∞

Z

Tk

Y (t) uk (t) dt = −x0

for all

k ∈ IN.

0

Since the set U is sequentially weak-∗ compact, there is a subsequence (uki )i∈IN of the sequence (uk )k∈IN and some uγ ∈ U such that Z lim

i→∞

Z

T (γ)

y(t) uki (t) dt =

0

T (γ)

T

for all

y(t)T uγ (t) dt

0

y ∈ L1 ([0, T (γ)], IRm )

which implies Z lim

i→∞

Z

T (γ)

Y (t) uki (t) dt =

0

T (γ)

T

Y (t) uγ (t) dt. 0

For every i ∈ IN we then have Z

T (γ)

Y (t) uγ (t) dt 0

Z = |0

T (γ)

Z Tk Z Tk i i Y (t) (uγ (t) − uki (t)) dt + Y (t) uki (t) dt − Y (t) uki (t) dt T (γ) {z } |0 {z } = −x0

→ Θm

From this we infer because of lim Tki = T (γ) and therefore i→∞

Z lim

i→∞

Tki

T (γ)

Y (t) uki (t) dt = Θm

that (2.33) is satisfied which completes the proof.

u t

Every uγ ∈ U which satisfies (2.33) is called a time-minimal null-control. The next step is to characterize time-minimal null-controls. For this purpose we make use of the


93

Normality Condition: For every T > 0 and every y ∈ IRn the components of the vector function Y (t)T y only vanish on a subset of [0, T ] of (Lebesgue-) measure zero unless y = Θn . At first we consider the Minimum Norm Problem: For T > 0 and x0 ∈ IRn given determine u ∈ L∞ (IR, IRm ) such that (2.21) is satisfied and the functional ϕ defined by ϕ(u) = ||u||L∞ ([0,T ],IRm ) = ess sup ||u(t)||2

(2.34)

t∈[0,T ]

is minimized. In order to study the problem of minimizing ϕ(u) defined by (2.34) on the linear manifold ¯ Z ( ) ¯ T m ¯ ∞ M = u ∈ L (IR, IR ) ¯ Y (t) u(t) dt = −x0 (2.35) ¯ 0 we also consider the Dual Problem: Determine y ∈ IRn such that Z

T

¡

y T Y (t)Y (t)T y

¢1/2

dt ≤ 1

(2.36)

0

holds true and the functional ψ(y) = −xT0 y

(2.37)

N = {y ∈ IRn | (2.36) is satisfied} .

(2.38)

is minimized. Let If u ∈ M and y ∈ N , then it follows that Z ψ(y) = −xT0 y = Z

T

≤

¡

T

u(t)T Y (t)T y dt

0

u(t)T u(t)

¢1/2 ¡

0

Z

≤ ess sup ||u(t)||2 t∈[0,T ]

y T Y (t)Y (t)T y

T

¡

¢1/2

y T Y (t)Y (t)T y

dt

¢1/2

≤ ϕ(u)

0

which implies sup ψ(y) ≤ inf ϕ(u). y∈N

u∈M

(2.39)

The dual problem will now be used to solve the minimum norm problem.

94


At first we observe that the normality condition implies that for every T > 0 the function Z T ¡ T ¢1/2 y Y (t)Y (t)T y dt, y ∈ IRn , y → χ(y) = 0

is Gateaux-differentiable and its Gateaux-derivative is given by h ∈ IRn ,

Dχ(y, h) = ∇χ(y)T h, where

Z

T

∇χ(y) =

1 (y T

0

Y (t)Y

(t)T

y)

1/2

Y (t)Y (t)T y dt.

Now we can prove Theorem 2.11. Under the normality condition the minimum norm problem has exactly one solution (on [0, T ]) for every choice of T > 0 which is of the form uT (t) = ¡

−xT0 yT yTT

(t)T

T ¢1/2 Y (t) yT

Y (t)Y yT for almost all t ∈ [0, T ] and uT (t) = Θm for all t 6∈ [0, T ]

(2.40)

where yT ∈ IRn is a solution of the dual problem and satisfies Z

T 0

¡ T ¢1/2 yT Y (t)Y (t)T yT dt = 1.

(2.41)

Proof. Since the set of all y ∈ IRn which satisfy (2.36) is compact and the function ψ(y) = −xT0 y is continuous there is some yT ∈ IRn which solves the dual problem and every solution satisfies (2.41). Hence the dual problem has the same solutions, if we assume equality to hold in (2.36). Let yT ∈ IRn be a solution of the dual problem. Then by the Lagrangean multiplier rule there is a multiplier λT ∈ IR such that ∇ψ (yT ) − λT ∇χ (yT ) = Θn which is equivalent to Z

T

−x0 = λT 0

¡

1 yTT

Y (t)Y (t)T yT

T ¢1/2 Y (t)Y (t) yT dt

and implies Z −xT0

yT = λT 0

T

¡ T ¢1/2 yT Y (t)Y (t)T yT dt = λT .


95

Therefore, if we define uT = uT (t) by (2.40), then (2.21) is satisfied for u = uT . Further it follows that ¡ ¢1/2 uT (t)T uT (t) = −xT0 yT = sup ψ(y)

for almost all

t ∈ [0, T ]

y∈N

which implies ϕ (uT ) = sup ψ(y). From (2.39) it then follows that uT solves y∈N

the minimum norm problem. It remains to prove the uniqueness of uT . Let u b ∈ M be any solution of the minimum norm problem. In order to find a solution u ∈ L∞ (IR, IRm ) of (2.21) we tentatively put u(t) = Y (t)T y for t ∈ IR. Then (2.21) reads Z

T

Y (t)Y (t)T dt y = −x0 .

(2.42)

0

The normality condition implies that the row vector functions yiT = yi (t)T , t ∈ IR, i = 1, . . . , n of Y = Y (t) are linearly independent. From this it follows RT that the matrix 0 Y (t)Y (t)T dt is non-singular so that for every choice of x0 ∈ IRn the linear system (2.42) has exactly one solution y ∈ IRn . Therefore the mapping Z

T

Y (t) u(t) dt

u 7→

from

L∞ (IR, IRm )

into IRn

0

is surjective. This implies that there exists a multiplier y ∈ IRn such that ÃZ ϕ (b u) ≤ ϕ(u) − y

T

!

T

Y (t) u(t) dt + x0 0

for all u ∈ L∞ (IR, IRm ), hence Z

T

T

ϕ (b u) + y x0 ≤ ϕ(u) −

y T Y (t) u(t) dt

0

for all u ∈ L∞ (IR, IRm ) which is only possible, if Z ϕ(u) − 0

T

y T Y (t) u(t) dt ≥ 0

for all

u ∈ L∞ (IR, IRm ) .

(2.43)

96


This is equivalent to Z

T

¡ ¢1/2 u(t)T u(t)

y T Y (t) u(t) dt ≤ ess sup t∈[0,T ]

0

for all u ∈ L∞ (IR, IRm ) or Z

T

y T Y (t) u(t) dt ≤ 1 for all u ∈ L∞ (IR, IRm ) with

0

¡ ¢1/2 u(t)T u(t) = 1.

ess sup t∈[0,T ]

If we define 1

Y (t)T y 1/2 Y (t)Y (t)T y) for almost all t ∈ [0, T ] and u(t) = Θm for all t 6∈ [0, T ]

u(t) =

(y T

then it follows that ess sup

¡ ¢1/2 u(t)T u(t) =1

t∈[0,T ]

and therefore Z T

¡ T ¢1/2 y Y (t)Y (t)T y dt =

0

Z

T

y T Y (t) u(t) dt ≤ 1.

0

u) ≤ −xT0 y. Thus y ∈ N . If we choose u ≡ Θm in (2.43) it follows that ϕ (b On the other hand we have shown (in the proof of (2.39)) that −xT0 y ≤ ϕ (b u). Therefore we obtain Z T ¡ ¢1/2 b(t)T u b(t) = −xT0 y = u b(t)T Y (t)T dt y ess sup u t∈[0,T ]

0

which is only possible, if u b(t) =

−xT0 y 1/2

(y T Y (t)Y (t)T y)

Y (t)T y

for almost all t ∈ [0, T ]

This implies ¡ ¢1/2 ¡ ¢1/2 u b(t)T u b(t) = ϕ (b u) = ϕ (uT ) = uT (t)T uT (t) for almost all t ∈ [0, T ].


97

Since w(t) = 12 (b u(t) + uT (t)) , t ∈ [0, T ], is also a solution of the minimum norm problem, it follows ||b u(t)||2 = ||uT (t)||2 = ||w(t)||2

for almost all t ∈ [0, T ]

which is only possible, if u b(t) = uT (t) = w(t)

for almost all

t ∈ [0, T ].

This completes the proof.

u t

Now we make a first step towards a characterization of time-minimal nullcontrols by proving the Theorem 2.12. If restricted null-controllability holds and if the normality condition is satisfied, then the minimum time T (γ) defined by (2.32) is also given by T (γ) = inf{T > 0 | v(T ) ≤ γ} (2.44) where ( v(T ) = max Proof. Let us put

−xT0

¯ Z ) ¯ T ¡ ¢1/2 ¯ T T y Y (t)Y (t) y y¯ dt ≤ 1 ¯ 0

(2.45)

Tb(γ) = inf{T > 0 | v(T ) ≤ γ}.

Then it is clear that Tb(γ) ≤ T (γ) because of {T > 0 | (2.21) is satisfied for some u ∈ U (2.31)} ⊆ {T > 0 | v(T ) ≤ γ} which is a consequence of (2.39).

³ ´ If Tb(γ) < T (γ), then by, Theorem 2.11, for every T ∈ Tb(γ), T (γ) there is some uT ∈ U which satisfies (2.21) and for which ϕ (uT ) = v(T ) ≤ γ holds true. This, however, contradicts the definition (2.32) of T (γ) which completes the proof. u t Theorem 2.11 can be strengthened to Theorem 2.13. In addition to the assumption of Theorem 2.12 let the function T → v(t) (2.45) from (0, ∞) into (0, ∞) be continuous. Then it follows that v(T (γ)) = γ (2.46) and for each uγ ∈ U with (2.33) it follows that ϕ (uγ ) = v(T (γ)), i.e., every time-minimal null-control is a minimum norm control of [0, T (γ)] (we tacitly assume that x0 6= Θn ).

98


Proof. x0 6= Θn implies T (γ) > 0 and lim v(T ) = ∞. T →0+

By virtue of the continuity of T → v(T ) it follows from (2.44) that v(T (γ)) ≤ γ. Let us assume that v(T (γ)) < γ. By the intermediate value theorem for continuous functions then it follows that there is some T ∗ ∈ (T, T (γ)) (where v(T ) > γ) with v (T ∗ ) = γ. This implies the existence of uT ∗ ∈ U which satisfies (2.21) and ϕ (uT ∗ ) = v (T ∗ ) = γ. This is a contradiction to the definition (2.32) of T (γ). Hence (2.46) must be true from which the last assertions follow immediately. u t In view of Theorem 2.11 we have the following Corollary 2.14. Under the assumptions of Theorem 2.13 there is exactly one time-minimal control on [0, T (γ)] which is the minimum norm control uT (γ) (t) ∈ U on [0, T (γ)] and is given by γ T uT (γ) (t) = ³ ´1/2 Y (t) yT (γ) T yT (γ) Y (t)Y (t)T yT (γ) for almost all

(2.47)

t ∈ [0, T (γ)]

where yT (γ) ∈ IRn is a solution of the dual problem to the minimum norm problem for T = T (γ). The continuity assumption for the function T → v(T ) in Theorem 2.13 can be dispensed with, since it is a consequence of the normality condition. In order to show this we at first observe that in the case x0 6= Θn , for every T > 0, v(T ) defined by (2.45) is also given by v(T ) = 1/λ(T ) where ¯ ) (Z ¯ T ¡ ¢1/2 ¯ T T T y Y (t)Y (t) y dt ¯ − x0 y = 1 (2.48) λ(T ) = min ¯ 0 Theorem 2.15. If the normality condition holds, then the function T → λ(T ) (2.48), T > 0, is strictly increasing and continuous. Proof. Let T2 > T1 be given. Let Z λ (Tk ) = 0

Tk

¡

yTTk Y (t)Y (t)T yTk

¢1/2

dt

for k = 1, 2

with −xT0 yTk = 1. For brevity we put, for every y ∈ IRn , ¡ ¢1/2 , ||Y (t)T y||2 = y T Y (t)Y (t)T y

t ∈ IR.


99

At first we get Z

T1

λ (T1 ) = Z

0

||Y (t)T yT1 ||2 dt Z

T1

||Y (t) yT2 ||2 dt ≤

≤ 0

T2

T

||Y (t)T yT2 ||2 dt = λ (T2 ) .

0

It even follows that Z

T2

λ (T2 ) = Z

0

||Y (t)T yT2 ||2 dt Z

T1

= 0

||Y (t) yT2 ||2 dt + Z

T2

≥ λ (T1 ) + T1

where

T2

T

Z

T2 T1

T1

||Y (t)T yT2 ||2 dt

||Y (t)T yT2 ||2 dt

||Y (t)T yT2 ||2 dt > 0,

since otherwise Y (t) yT2 = Θm for all t ∈ [T1 , T2 ] which, by the normality condition, implies yT2 = Θn and contradicts −xT0 yT2 = 1. As a result we obtain λ (T2 ) > λ (T1 ) which shows that T → λ(T ) is strictly increasing. On the other hand we also have Z

T2

λ (T2 ) = 0

Z =

0

T1

Z ||Y (t)T yT2 ||2 dt ≤ ||Y (t)T yT1 ||2 dt + Z

T2

= λ (T1 ) + T1

T2

0 Z T2 T1

||Y (t)T yT1 ||2 dt ||Y (t)T yT1 ||2 dt

||Y (t)T yT1 ||2 dt,

hence

Z

T2

0 ≤ λ (T2 ) − λ (T1 ) ≤ T1

||Y (t)T yT1 ||2 dt

which implies lim

T2 →T1 +0

λ (T2 ) = λ (T1 )

and shows the right continuity of the function T → λ(T ). The proof of the left continuity requires some preparation.

100


We choose T0 ∈ (0, T1 ) arbitrarily and consider the function Z T0 ||Y (t)T y||2 dt y→ 0 n

from IR into (0, ∞) for which (Z

¯ ) ¯ ¡ T ¢1/2 ¯ ||Y (t) y||2 dt¯ ||y||2 = y y >0 ¯

T0

mT0 = inf

T

0

holds true. This implies Z mT0 ||y||2 ≤

T0

||Y (t)T y||2 dt

y ∈ IRn .

for all

0

In particular we obtain Z mT0 ||yT1 ||2 ≤ Z

T0

0 T1

≤ 0

||Y (t)T yT1 ||2 dt ||Y (t)T yT1 ||2 dt = λ (T1 ) ≤ λ (T2 )

and further Z

T2

0 ≤ λ (T2 ) − λ (T1 ) ≤ Z

T1 T2

≤ T1

||Y (t)T yT1 ||2 dt

λ (T2 ) ||Y (t) ||2 dt ||yT1 ||2 ≤ mT0

Z

T

with

 ||Y (t)T ||2 = 

n X

T2

||Y (t)T ||2 dt

T1

1/2 yjk (t)2 

.

j,k=1

From this we infer lim

T1 →T2 −0

λ (T1 ) = λ (T2 )

which shows the left-continuity of the function and completes the proof.

u t

Under the assumptions of Theorem 2.15 it follows that the function T → v(T ) = 1/λ(T ), T > 0, is strictly decreasing and continuous. Thus we can strengthen Theorem 2.13 to Theorem 2.16. If restricted null-controllability holds and if the normality condition is satisfied, then the equation (2.46) for the minimum time T (γ) holds true and there is exactly one time-minimal control on [0, T (γ)] which is the unique minimum norm control uT (γ) ∈ L∞ (IR, IRm ) on [0, T (γ)] and is given by (2.47).

2.2 The Time-Discrete Autonomous Case

101

2.2 The Time-Discrete Autonomous Case 2.2.1 The Problem of Fixed Point Controllability We begin with a system of difference equations of the form x(t + 1) = g(x(t), u(t)),

t ∈ IN0

(2.49)

where g : IRn × IRm → IRn is a continuous mapping. The functions x : IN0 → IRn and u : IN0 → IRm are considered as state and control functions, respectively. For every control function u : IN0 → IRm and every vector x0 ∈ IRn there exists exactly one state function x : IN0 → IRn which satisfies (2.49) and (2.50) x(0) = x0 . If we fix the control function u : IN0 → IRm and define ft+1 (x) = g(x, u(t)),

x ∈ IRn , t ∈ IN0 ,

(2.51)

n n for every ¢ t ∈ IN0 , ft : IR → IR is a continuous mapping and ¡then, n IR , (ft )t∈IN is a non-autonomous time-discrete dynamical system which is controlled by the function u : IN0 → IRm . If

u(t) = Θm = zero vector of IRm for all t ∈ IN0 , then the system (2.49) is called uncontrolled. Let us assume that the uncontrolled system (2.49) admits fixed points x b ∈ IRn which then solve the equation (2.52) x b = g (b x, Θm ) . Now let Ω ⊆ IRm be a subset with Θm ∈ Ω. Then we define the set of admissible control functions by U = {u : IN0 → IRm | u(t) ∈ Ω

for all

t ∈ IN0 } .

(2.53)

After these preparations we are in the position to formulate the b ∈ IRn of the Problem of Fixed Point Controllability: Given a fixed point x system t ∈ IN0 , (2.54) x(t + 1) = g (x(t), Θm ) , i.e., a solution x b of equation (2.52) and an initial state x0 ∈ IRn find some N ∈ IN0 and a control function u ∈ U with u(t) = Θm

for all

t≥N

(2.55)

such that the solution x : IN0 → IRn of (2.49), (2.50) satisfies the end condition x(N ) = x b.

(2.56)

102


(This implies x(t) = x b for all t ≥ N .) From (2.49) and (2.50) it follows that ¢ ¢ ¢ ¢ ¡ ¡ ¡ ¡ x(N ) = g g · · · g x0 , u(0) , u(1) , · · · , u(N − 1) | {z } N −times

= GN (x0 , u(0), . . . , u(N − 1)) .

(2.57)

Let N ∈ IN be given. If u(0), . . . , u(N − 1) ∈ Ω are solutions of the system b GN (x0 , u(0), . . . , u(N − 1)) = x

(2.58)

and one defines u(t) = Θm

for all

t ≥ N,

then one obtains a control function u : IN0 → IRm which solves the problem of fixed point controllability. From the definition (2.57) it follows that ¡ ¢ GN (x0 , u(0), . . . , u(N − 1)) = G1 GN −1 (x0 , u(0), . . . , u(N − 2)) , u(N − 1) . ¡ ¢ Conversely now let us assume that we are given a sequence GN N ∈IN of vector functions GN : IRn × IRm·N → IRn with this property. Then we define, for every t ∈ IN, x(t) = Gt (x0 , u(0), . . . , u(t − 1)) for x0 ∈ IRn and u(s) ∈ IRm for s = 0, . . . , t − 1 and conclude ¡ ¢ x(t) = G1 Gt−1 (x0 , u(0), . . . , u(t − 2)) , u(t − 1) = G1 (x(t − 1), u(t − 1)) for all t ∈ IN, x) ⊆ IRn be the set of all vectors x0 ∈ IRn if we define G0 (x0 ) = x0 . Now let S (b ¢T ¡ such that there exists a time N ∈ IN and a solution u(0)T , . . . , u(N − 1)T ∈ b ∈ S (b x) (choose N = 1 and u(0) = Θn ). Ω N of the system (2.58). Obviously, x A first simple sufficient condition for the solvability of the problem of fixed point controllability is then given by Proposition 2.17. Let x b ∈ S (b x) be an interior point of S (b x) and let x b be an attractor of the uncontrolled system (2.54), i.e., there exists an open set b ∈ U such that lim x(t) = x b where x : IN0 → IRn is a solution U ⊆ IRn with x t→∞

of (2.54) with (2.50) for any x0 ∈ U . Then it follows that S (b x) ⊇ U which implies that for every choice of x0 ∈ U the problem of fixed point controllability has a solution.


103

Proof. Since x b ∈ S (b x) is an interior point of S (b x), there is an open neighborb with W (b x) ⊆ S (b x). hood W (b x) ⊆ IRn of x Now let x0 ∈ U be chosen arbitrarily. Then there is some N1 ∈ IN with x) where x : IN0 → IRm is a solution of (2.54), (2.50). x (N1 ) ∈ W (b ¡ This implies the existence of some N2 ∈ IN and a solution u(0)T , . . . , ´T T u (N2 − 1) ∈ Ω N2 with GN2 (x (N1 ) , u(0), . . . , u (N2 − 1)) = x b. If we define

( ∗

u (t) =

Θm for t = 0, . . . , N1 − 1, u (t − N1 ) for t = N1 , . . . , N1 + N2 − 1,

then it follows that GN (x0 , u∗ (0), . . . , u∗ (N − 1)) = x b where N = N1 + N2 , i.e., x0 ∈ S (b x) which completes the proof.

u t

The essential assumption in Proposition 2.17 is that the fixed point x b of the uncontrolled system (2.54) is an interior point of the controllable set S (b x). In order to find sufficient conditions for that¡ we assume that Ω is open and ¢ g ∈ C 1 (IRn × IRm ) which implies GN ∈ C 1 IRn × IRm·N for every N ∈ IN and GN x (x, u(0), . . . , u(N − 1)) ¡ ¢ = gx GN −1 (x0 , u(0), . . . , u(N − 2)) , u(N − 1) ¡ ¢ × gx GN −2 (x0 , u(0), . . . , u(N − 3)) , u(N − 2) .. × . × gx (x, u(0)), and GN u(k) (x, u(0), . . . , u(N − 1)) ¡ ¢ = gx GN −1 (x0 , u(0), . . . , u(N − 2)) , u(N − 1) ¡ ¢ × gx GN −2 (x0 , u(0), . . . , u(N − 3)) , u(N − 2) .. × . ¡ k+1 ¢ (x, u(0), . . . , u(k)) , u(k + 1) × gx G ¡ ¢ × gu(k) Gk (x, u(0), . . . , u(k − 1)) , u(k) for k = 0, . . . , N − 1.

104


Let us assume that gx (b x, Θm ) is non-singular. Then it follows, for all N ∈ IN, x, Θm , . . . , Θm ) is also non-singular, since that GN x (b N

x, Θm , . . . , Θm ) = gx (b x, Θm ) . GN x (b Since

x b = GN (b x, Θm , . . . , Θm ) ,

there exists, by the implicit function theorem, an open set V ⊆ Ω N with N ∈ V and a function h : V → IRn with h ∈ C 1 (V ) such that Θm ¡ N¢ h Θm =x b and GN (h(u(0), . . . , u(N − 1)), u(0), . . . , u(N − 1)) = x b for all (u(0), . . . , u(N − 1)) ∈ V . Moreover, ¡ N¢ −1 hu(k) Θm = −GN x, Θm , . . . , Θm ) Gu(k) (b x, Θm , . . . , Θm ) x (b −N

x, Θm ) = −gx (b

−k

x, Θm ) = −gx (b

N −k

gx (b x, Θm )

gu(k) (b x, Θm )

gu(k) (b x, Θm )

for k = 0, . . . , N − 1. Result. If gx (b x, Θm ) is non-singular, then, for every N ∈ IN, there is an N ∈ VN and a function hN : VN → IRn with open set VN ⊆ Ω N with Θm 1 hN ∈ C (VN ) such that hN (u(0), . . . , u(N − 1)) ∈ S (b x)

for all

(u(0), . . . , u(N − 1)) ∈ VN .

We next assume that, for some N ∈ IN, ¡ N¢ ¡ N ¢¢ ¡ | . . . | hu(N −1) Θm = n. rank hu(0) Θm Then there are n columns in the n × m · N −matrix ¡ ¡ N¢ ¡ N ¢¢ hu(0) Θm | . . . | hu(N −1) Θm which are linearly independent. Now let E be the n−dimensional subspace of IRm·N consisting of all vectors whose components vanish that do not correspond to the above linearly independent columns. If we put U = E ∩ VN , then U is an open subset of E and the restriction N at Θm consists of the of h to U is a C 1 −function on U whose¡ Jacobi ¡ matrix ¢ ¡ N ¢¢ N and above linearly independent columns of hu(0) Θm | . . . | hu(N −1) Θm is therefore invertible. By the inverse function theorem there exist open sets N e and x e ⊆ E ∩ VN and Ve ⊆ IRn with Θm ∈U b ∈ Ve such that h is homeomorU ³ ´ e and h U e = Ve . phic on U


105

This implies that x b is an interior point of S (b x). Let us demonstrate this result by the predator prey model that was investigated with respect to asymptotical stability in Section 1.3.7. We consider this model as a controlled system of the form x1 (t + 1) = x1 (t) + a x1 (t) − e x1 (t)2 − b x1 (t)x2 (t) − x1 (t) u1 (t), x2 (t + 1) = x2 (t) − c x2 (t) + d x1 (t)x2 (t) − x2 (t) u2 (t), t ∈ IN0 . The mapping g : IR2 × IR2 → IR2 in (2.49) is therefore given by ! Ã (1 + a) x1 − e x21 − b x1 x2 − x1 u1 , x, u ∈ IR2 . (1 − c) x2 + d x1 x2 − x2 u2 We have seen in Section 1.3.7 that µ ¶T c 1³ ce ´ x b= , a− d b d is the only fixed point of the uncontrolled system (with u1 = u2 = 0) in ◦

◦

IR2+ × IR2+ , if a > ce/d. One calculates

 bc c  1−ed − d  x, Θ2 ) =  d ³ gx (b  ce ´ a− 1 b d which implies that gx (b x, Θ2 ) is non-singular, if and only if ³ ce ´ c 6= 0. 1−e +c a− d d 

Further we obtain

(∗)

 c  0 − gu (b x, Θ 2 ) =  d 1 ³ ce ´ 0 − a− b d

which implies ¢ ¡ ¢ ¡ x, Θ2 ) = 2. rank hu(0) (Θ2 ) = rank −gu(0) (b Hence x b is an interior point of S (b x), if (∗) is satisfied. This example is a special case of the following situation: Let g(x, u) = f (x) + F (x) u = f (x) +

m X

fi (x) ui ,

i=1

x ∈ IRn , u1 , . . . , um ∈ IR, where f, fi ∈ C 1 (IRn ) , i = 1, . . . , m.

106


Let x b ∈ IRn be a fixed point of f , i.e., x b = f (b x) = g (b x, Θm ) . Then, x, Θm ) = fx (b x) gx (b

and

gu (b x, Θm ) = F (b x)

and hence ¡ ¡ N¢ ¡ N ¢¢ hu(0) Θm | . . . | hu(N −1) Θm ´ ³ −1 −N +1 x) F (b x) | . . . | fx (b x) F (b x) . = − F (b x) | fx (b In the example we have m = n = 2 and the 2 × 2−matrix F (b x) is nonsingular. Next we come back to the solution of (2.58) which we replace by an optimization problem. For this purpose we define a cost functional ϕ : IRm·N → IR by putting ° °2 ϕ(u(0), . . . , u(N − 1)) = °GN (x0 , u(0), . . . , u(N − 1)) − x b°2 for (u(0), . . . , u(N − 1)) ∈ IRm·N

(|| · ||2 =Euclidian norm in IRn )

and try to find (u(0), . . . , u(N − 1)) ∈ Ω N such that ϕ(u(0), . . . , u(N − 1)) ≤ ϕ (e u(0), . . . , u e(N − 1)) for all (e u(0), . . . , u e(N − 1)) ∈ Ω N . If ϕ(u(0), . . . , u(N − 1)) = 0, then ϕ(u(0), . . . , u(N − 1)) ∈ Ω N solves the equation (2.58). Otherwise no such solution exists. We again assume that g ∈ C 1 (IRn × IRm ). Let Ω ⊆ IRn be open. Then a necessary condition for u(0), . . . , u(N − 1)) ∈ Ω N to minimize ϕ on Ω N is given by T

ϕu(k) (u(0), . . . , u(N − 1)) = 2 GN u(k) (x0 , u(0), . . . , u(N − 1)) ¡ N ¢ × G (x0 , u(0), . . . , u(N − 1)) − x b (OC) = Θm for all k = 0, . . . , N − 1. For the determination of (u(0), . . . , u(N − 1)) ∈ Ω N with (OC) one can apply Marquardt’s algorithm: Let (u(0), . . . , u(N − 1)) ∈ Ω N be chosen. If (OC) is satisfied, then (u(0), . . . , u(N − 1)) is taken as a solution of the optimization problem. Otherwise, for every k ∈ {0, . . . , N − 1}, a vector hλ (k) ∈ IRm is determined as a solution of the linear system ³ ´T N 2 GN u(k) (x0 , u(0), . . . , u(N − 1)) Gu(k) (x0 , u(0), . . . , u(N − 1)) + λ Im hλ (k) ³ ´ T GN b = 2 GN u(k) (x0 , u(0), . . . , u(N − 1)) u(k) (x0 , u(0), . . . , u(N − 1)) − x where λ > 0 and Im is the m × m−unit matrix.


107

Then one can show (see, for instance [22]) that for sufficiently large λ > 0 it follows that (u(0) + hλ (0), . . . , u(N − 1) + hλ (N − 1)) ∈ Ω and ϕ (u(0) + hλ (0), . . . , u(N − 1) + hλ (N − 1)) < ϕ (u(0), . . . , u(N − 1)) . The algorithm is then continued with (u(0) + hλ (0), . . . , u(N − 1) + hλ (N − 1)) instead of (u(0), . . . , u(N − 1)). Special Case: Modelling of Conflicts Now let us consider a special case which is motivated by a situation which occurs in the modelling of conflicts. We begin with an uncontrolled system of the form ¡ ¢ x1 (t + 1) = g1 x1 (t), x2 (t) , ¡ ¢ x2 (t + 1) = g2 x1 (t), x2 (t) ,

t ∈ IN0 ,

where gi : IRn1 × IRn2 → IRni , i = 1, 2, are given continuous mappings and xi : IN0 → IRni , i = 1, 2, are considered as state functions. For t = 0 we assume initial conditions x1 (0) = x10 ,

x2 (0) = x20

(2.59)

where x10 ∈ IRn1 and x20 ∈ IRn2 are given vectors with Θn2 ≤ x20 ≤ x2∗ for some x2∗ ≥ Θn2 which is also given. We further assume that the above ¡ 1 T 2 T ¢T b ∈ IRn1 × IRn2 with system admits fixed points x b ,x b2 ≤ x2∗ Θn2 ≤ x which are then solutions of the system ¡ 1 2¢ x b1 = g1 x b ,x b , Now we consider the following

¡ 1 2¢ x b2 = g2 x b ,x b .

108


Problem: Find vector functions x1 : N0 → IRn1 and x2 : IN0 → IRn2 with Θn2 ≤ x2 (t) ≤ x2∗

for all

t ∈ IN0

which satisfy the above system equations and initial conditions and b1 , x1 (t) = x

x2 (t) = x b2

for all

t≥N

where N ∈ IN0 is a suitably chosen integer. In general this problem will not have a solution. Therefore we replace the uncontrolled system by the following controlled system ¡ ¢ x1 (t + 1) = g1 x1 (t), x2 (t) + u(t) , ¡ ¢ x2 (t + 1) = g2 x1 (t), x2 (t) + u(t) , t ∈ IN0 , (2.60) where u : IN0 → IRn2 is a control function. Then we consider the problem of finding a control function u : IN0 → IRn2 such that the solutions x1 : IN0 → IRn1 and x2 : IN0 → IRn2 of (2.60) and (2.59) satisfy the conditions Θn2 ≤ x2 (t) + u(t) ≤ x2∗

for all

t ∈ IN0

and b1 , x1 (t) = x

x2 (t) + u(t) = x b2

for all

t≥N

where N ∈ IN0 is a suitably chosen integer. Let us assume that we can find a vector function v : IN0 → IRn with Θn2 ≤ v(t) ≤ x2∗

for t = 0, . . . , N − 1

v(t) = x b2

and

for t ≥ N

such that the solution x1 : IN0 → IRn1 of x1 (0) = x10 , ¡ ¢ x1 (t + 1) = g1 x1 (t), v(t) ,

t ∈ IN0

satisfies b1 x1 (t) = x

for all

t≥N

where N ∈ IN is a suitably chosen integer. Then we put x2 (0) = x20 , ¡ ¢ x2 (t + 1) = g2 x1 (t), v(t) ,

t ∈ IN0

and define u(t) = v(t) − x2 (t)

for

t ≥ N0 .

With these definitions we obtain a solution of the above control problem.


109

Thus, in order to find such a solution, we have to find a vector function v : IN0 → IRn2 with Θn2 ≤ v(t) ≤ x2∗ v(t) = x b2

for t = 0, . . . , N − 1, for t ≥ N

such that the solution x1 : IN0 → IRn1 of x1 (0) = x10 , ¡ ¢ x1 (t + 1) = g1 x1 (t), v(t) ,

t ∈ IN0

satisfies b1 x1 (t) = x

for all

t≥N

where N ∈ IN is a suitably chosen integer. Let us demonstrate all this by an emission reduction model (1.65) to which we add the conditions 0 ≤ Mi (t) ≤ Mi∗

for all

t ∈ IN0

and i = 1, . . . , r

and the initial conditions Ei (0) = E0i

and Mi (0) = M0i

for i = 1, . . . , r

where E0i ∈ IR and M0i ∈ IR with 0 ≤ M0i ≤ Mi∗ for i = 1, . . . , r are given. The corresponding controlled system (2.60) reads in this case

Ei (t + 1) = Ei (t) +

r X

emij (Mj (t) + uj (t)) ,

j=1

Mi (t + 1) = Mi (t) + ui (t) − λi (Mi (t) + ui (t)) (Mi∗ − Mi (t) − ui (t)) Ei (t) for i = 1, . . . , r and t ∈ IN0 . The control functions ui : IN0 → IR, i = 1, . . . , r, must satisfy the conditions 0 ≤ Mi (t) + ui (t) ≤ Mi∗

for i = 1, . . . , r

and t ∈ IN0 .

´T ³ bT , Θ bT b ∈ IRr Fixed points of the system (1.65) are of the form E with E r arbitrary.

110


We have to find a vector function v : IN0 → IRr with Θr ≤ v(t) ≤ M ∗ v(t) = Θr

for t = 0, . . . , N − 1, for t ≥ N

such that the solution E : IN0 → IRr of E(0) = E0 , E(t + 1) = E(t) + C v(t), satisfies

b E(t) = E

t ∈ IN0 ,

´ ³ C = (emij )i,j=1,...,r

for all

t≥N

where N ∈ IN is a suitably chosen integer. First of all we observe that for every N ∈ IN ÃN −1 ! X v(t) . E(N ) = E0 + C t=0

Let us assume that C is invertible and C −1 is positive. Further we assume b ≥ E0 . that E b if and only if Then E(N ) = E, N −1 X

³ ´ b − E0 ≥ Θr . v(t) = C −1 E

t=0

If we define then

v(t) = Θr

for all

t ≥ N,

b E(t) = E

for all

t ≥ N.

Let us put vN =

N −1 X

³ ´ b − E0 . v(t) = C −1 E

t=0

If we define v(t) =

1 vN N

for t = 0, . . . , N − 1

then N −1 X

³ ´ b − E0 v(t) = C −1 E

and

Θr ≤ v(t) ≤ M ∗

t=0

for sufficiently large N , if Mi∗ > 0 for all i = 1, . . . , r.

for

t = 0, . . . , N − 1


111

b = (10, 10, 10)T , We finish with a numerical example: r = 3, E0 = (0, 0, 0)T , E ∗ T M = (1, 1, 1) , and   1 −0.8 0 1 −0.8 . C= 0 −0.1 −0.5 1 Then we have to solve the linear system vN1 − 0.8 vN2 = 10, vN2 − 0.8 vN3 = 10, vN3 = 10. −0.1 vN1 − 0.5 vN2 + The solution reads vN1 = 38.059701, vN2 = 35.074627, vN3 = 31.343284. We choose N = 39. Then we have to put   0.9758898 1 v(t) = vN = 0.8993494 N 0.803674

for

t = 0, . . . , 38.

2.2.2 Null-Controllability of Linear Systems Instead of (2.49) we consider a linear system of the form x(t + 1) = A x(t) + B u(t),

t ∈ IN0 ,

(2.61)

where A is a real n × n−matrix and B a real n × m−matrix and where u : IN0 → IRm is a given control function. The corresponding uncontrolled system reads (2.62) x(t + 1) = A x(t), t ∈ IN0 , and admits x b = Θn as a fixed point. The problem of fixed point controllability is then equivalent to the Problem of Null-Controllability: Given x0 ∈ IRn find some N ∈ IN0 and a control function u ∈ U (2.53) with (2.55) such that the solution x : IN0 → IRn of (2.61), (2.50) satisfies the end condition x(N ) = Θn (which implies x(t) = Θn for all t ≥ N ).

(2.63)

112


From (2.61) and (2.50) it follows that x(N ) = AN x0 +

N X

AN −t B u(t − 1)

(2.64)

t=1

so that (2.63) turns out to be equivalent to N X

AN −t B u(t − 1) = −AN x0 .

(2.65)

t=1

Now let A be non-singular. Then the set S (Θn ) of all vectors x0 ∈ IRn such ¢T ¡ ∈ ΩN that there exists a time N ∈ IN and a solution u(0)T , . . . , u(N − 1)T of the system (2.61) is given by [ S (Θn ) = E(N ) N ∈IN

where, for every N ∈ IN, ( E(N ) =

x=

N X

A

t=1

N −t

¯ ) ¯ ¯ B u(t − 1)¯ u ∈ U (2.53) . ¯

Next we assume that Ω ⊆ IRm is convex, has Θm as interior point and satisfies u∈Ω

=⇒

−u ∈ Ω.

(2.66)

Then, for every N ∈ IN, the set E(N ) is convex and E(N ) = −E(N ). This implies because of E(N ) ⊆ E(N + 1)

for all

N ∈ IN

that S (Θn ) is also convex and S (Θn ) = −S (Θn ). Further we assume Kalman’s condition, i.e., there exists some N0 ∈ IN such that ¢ ¡ (2.67) rank B | AB | . . . | AN0 −1 B = n. Then we can prove Theorem 2.18. If A is non-singular, Ω ⊆ IRm is convex, has Θm as interior point and satisfies (2.66) and if Kalman condition (2.67) is satisfied for some N0 ∈ IN, then Θn is an interior point of S (Θn ). Proof. Let us assume that Θn is not an interior point of S (Θn ). Then S (Θn ) must be contained in a hyperplane through Θn , i.e., there must exist some y ∈ IRn , y 6= Θn , with yT x = 0

for all

x ∈ S (Θn ) .


113

This implies Ã y

T

N X

! A

N −t

t=1

for all

B u(t − 1)

=0

¢T ¡ u(0)T , . . . , u(N − 1)T ∈ IRm·N

and all

N ∈ IN,

hence T y T AN −t B = Θm

for all

N ∈ IN.

In particular for N = N0 this implies y = Θn due to the Kalman condition (2.67) which contradicts y 6= Θn . Therefore the assumption that Θn is not an u t interior point of S (Θn ) is false. In addition to the assumption of Theorem 2.18 we assume that all the eigenvalues of A are less than 1 in absolute value. Then according to the Corollary following Theorem 1.6 Θn is a global attractor of the uncontrolled system (2.62), i.e., lim x(t) = Θn where x : IN0 → IRn is a solution (2.62) with t→∞

(2.50) for any x0 ∈ IRn . By Proposition 2.17 therefore the problem of nullcontrollability has a solution for every choice of x0 ∈ IRn . If the set Ω of control vector values has the form Ω = {u ∈ IRm | ||u|| ≤ γ}

(2.68)

for some γ > 0 where || · || is any norm in IRm , then this result can be strengthened to Theorem 2.19. Let the Kalman condition (2.67) be satisfied for some N0 ∈ IN. Further let Ω be of the form (2.68). Finally let all the eigenvalues of AT be less than or equal to one in absolute value and the corresponding eigenvectors be linearly independent. Then the problem of null-controllability has a solution for every choice of x0 ∈ IRn , if A is non-singular. Proof. We have to show that for every choice of x0 ∈ IRn there is some N ∈ IN0 and a control function u ∈ U (2.53) such that (2.65) is satisfied. Since A is non-singular, (2.65) is equivalent to N X t=1

A−t B u(t − 1) = −x0 .

114


For every N ∈ IN we define the convex set ¯ ) ( N ¯ X ¯ −t R(N ) = x = A B u(t − 1)¯ u ∈ U ¯ t=1

and put

[

R∞ =

R(N ).

N ∈IN

Because of R(N ) ⊆ R(N + 1)

for all

N ∈ IN0

the set R∞ is also convex. We have to show that R∞ = IRn . Let us assume that R∞ 6= IRn . Then there exists some x e ∈ IRn with x e 6∈ R∞ which can be separated from R∞ by a hyperplane, i.e., there exists a number α ∈ IR and a vector y ∈ IRn , y 6= Θn such that yT x ≤ α ≤ yT x e

for all x ∈ R∞ .

Since Θn ∈ R∞ , it follows that α ≥ 0. Further it follows from the implication u ∈ Ω ⇒ − u ∈ Ω that ¯ ¯N ¯ ¯X ¯ ¯ T −t for all N ∈ IN and all u ∈ U. y A B u(t − 1)¯ ≤ α ¯ ¯ ¯ t=1

This implies N ° ° X °¡ T −t ¢T ° ° y A B ° ≤α d

t=1

for all

N ∈ IN

where || · ||d is the norm in IRm which is dual to || · ||. This in turn implies T lim y T A−t B = Θm .

t→∞

(2.69)

From the Kalman condition (2.67) it follows that there exist n linearly independent vectors in IRn of the form ci = Ati bji

for i = 1, . . . , n

where ti ∈ {0, . . . , N0 − 1} and ji ∈ {1, . . . , m} and bji denotes the ji −th column vector of B. From (2.69) it follows that lim y T A−t ci = 0

t→∞

This implies

for i = 1, . . . , n.

lim y T A−t = ΘnT

t→∞


or, equivalently,

¡ ¢−t lim AT y = Θn .

115

(2.70)

t→∞

Now let λ1 , . . . , λn ∈ C be eigenvalues of AT and y1 , . . . , yn ∈ Cn corresponding linearly independent eigenvectors. Then there is a unique representation n X

y=

αj yj

where not all

αj ∈ C

are zero

j=1

and

¡

hence

¡

AT

¢−t

¢ T −t

A

µ yj =

y=

n X

1 λj

¶t yj µ

αj

j=1

1 λj

for j = 1, . . . , n,

¶t yj

for all

t ∈ IN.

From (2.70) we therefore infer that |λj | > 1

for all

j ∈ {1, . . . , n}

with

αj 6= 0.

This is a contradiction to |λj | ≤ 1

for all

j = 1, . . . , n.

Hence the assumption R∞ 6= IRn is false.

u t

Remark. If we define 

0

λ1

Y = (y1 | y2 | . . . | yn )

and

 ∧=

.. 0

then it follows that

.

  ,

λn

AT Y = Y ∧

which implies Y

−1

AT Y = ∧

and in turn ³ T ´−1 T Y A Y =∧ from which

follows.

³ ¢T ¡ ¢−1 ´ ¡ since A−1 = AT

³ T ´−1 ³ T ´−1 = Y ∧ A Y

116


This implies that A has the same eigenvalues as AT (which holds for arbi³ T ´−1 . trary matrices) and the eigenvectors of A are the column vectors of Y Therefore AT in Theorem 2.19 could be replaced by A. For the following let us assume that Ω = IRm . For every N ∈ IN let us define ¡ ¢ Y (N ) = B | AB | . . . | AN −1 B . Since U (2.53) consists of all functions u : IN0 → IRm , it follows, for every N ∈ IN, that ¯ ) ( N ¯ X ¯ m AN −t B u(t − 1)¯ u : IN0 → IR . E(N ) = x = ¯ t=1

Further we can prove Proposition 2.20. The following statements are equivalent: (i) rank Y (N ) = rank Y (N + 1), (ii) E(N ¢ E(N + 1), ¡ )= (iii) AN B IRm ⊆ E(N ), (iv) rank Y (N ) = rank Y (N + j)

for all

j ≥ 1.

Proof. (i) ⇒ (ii): This is a consequence of the fact¡ that E(N ) ⊆¢E(N + 1). (ii) ⇒ (iii): This follows from Y (N + 1) = Y (N ) | AN B . ¡ ¢ (iii) ⇒ (i): Y (N + 1) = Y (N ) | AN B shows that (iii) ⇒ (ii) and obviously we have (ii) ⇒ (i). (i) ⇒ (iv): Since (i) implies (iii), it follows that ¡ N +1 ¢ m B IR ⊆ A E(N ) ⊆ E(N + 1) A which implies E(N + 1) = E(N + 2) and hence ¡ ¢ ¡ ¢ rank Y (N + 1) = rank Y (N + 2) . (iv) ⇒ (i): is obvious. This completes the proof.

u t r−1

are linearly indeNow let r be the smallest integer such that I, A, . . . , A pendent in IRn·n and hence there are numbers αr−1 , αr−2 , . . . , α0 ∈ IR such that Ar + αr−1 Ar−1 + . . . + α0 I = 0. Defining

Φ0 (λ) = λr + αr−1 λr−1 + . . . + α0 ,

we have Φ0 (A) = 0. This monic polynomial (leading coefficient 1) is the monic polynomial of least degree for which Φ0 (A) = 0 and is called the minimal polynomial of A.


117

The polynomial Φ(λ) = det(λ I − A)

with degree

n

is called characteristic polynomial of A, and the Cayley-Hamilton Theorem states that Φ(A) = 0 which implies r ≤ n. This leads to Proposition 2.21. Let s be the degree of the minimal polynomial of A (s ≤ n). Then there is an integer k ≤ s such that rank Y (1) < rank Y (2) < . . . < rank Y (k) = rank Y (k+j)

for all

j ∈ IN.

Proof. Proposition 2.20 implies the existence of such an integer k, since rank Y (N ) ≤ n for all N ∈ IN. We have to show that k ≤ s. Let ψ(λ) = λs + αs−1 λs−1 + . . . + α0 be the minimal polynomial of A. Then ψ(A) B = 0 and As B IRm ⊆ E(s) which implies (by Proposition 2.20) that u t rank Y (s) = rank Y (s + j) for all j ∈ IN, hence k ≤ s. As a consequence of Proposition 2.21 we obtain Proposition 2.22. If the Kalman condition (2.67) is satisfied for some N0 ∈ IN, then necessarily N0 ≤ n and ¡ ¢ rank B | AB | . . . | An−1 B = n, hence E(n) = IRn . Conversely, if E(n) = IRn , then Kalman’s condition (2.67) is satisfied for all N ≥ n and E(N ) = IRn

for all

N ≥ n.

Proposition 2.21 also implies that, if rank Y (n) < n, then rank Y (N ) < n for all N ≥ n and E(N ) = E(n) 6= IRn

for all

N ≥ n.

If we define, for every N ∈ IN, the n × n−matrix W (N ) = Y (N )Y (N )T =

N −1 X

¡ ¢T Aj BB T Aj ,

j=0

then it follows that W (N ) IRn ⊆ E(N )

for every N ∈ IN.

Now let us assume that rank Y (N ) = n which is equivalent to E(N ) = IRn .

118


Then W (N ) is non-singular and W (N ) IRn = IRn = E(N ). Let y ∗ ∈ IRn be the unique solution of W (N ) y ∗ = −AN x0 . Further let u ∈ IRm·N be any solution of Y (N ) u = −AN x0 If we put u∗ = Y (N )T y ∗

(see (2.65)). ¡

¢ ∈ IRm·N ,

then Y (N )T u∗ = −AN x0 and ||u∗ ||22 = u∗ T u∗ = y ∗ T Y (N ) u∗ = y ∗ T W (N )y ∗ = −y ∗ T AN x0 = y ∗ Y (N ) u = u∗ T u ≤ ||u∗ ||2 ||u||2 which implies ||u∗ ||2 ≤ ||u||2 . 2.2.3 A Method for Solving the Problem of Null-Controllability Let us equip IRm with the Euclidian norm || · ||2 and consider the following Problem (P) For a given N ∈ IN find u : {0, . . . , N − 1} → IRm such that N X

AN −t B u(t − 1) = −AN x0

(2.71)

t=1

and ϕN (u) = max ||u(t − 1)||2 t=1,...,N

is as small as possible. If the Kalman condition (2.67) is satisfied for some N0 ∈ IN and if N ≥ N0 , then Problem (P) has a solution uN ∈ IRm·N . If ϕN (uN ) ≤ γ (see (2.68)), then we obtain solution uN : IN0 → IRm of the problem of null-controllability if we define uN (t) = Θm for all t ≥ N. If ϕN (uN ) > γ, then N must be increased.


119

If the matrix A is non-singular, then (2.65) is equivalent to N X

A−t B u(t − 1) = −x0

t=1

which implies ϕN +1 (uN +1 ) ≤ ϕN (uN )

for all

N ≥ N0 .

Under the assumptions of Theorem 2.19 there exists, for every ε > 0, some N (ε) ∈ IN such that ¡ ¢ ϕN (ε) uN (ε) ≤ ε which implies lim ϕN (uN ) = 0.

N →∞

So we can be sure, for every choice of γ > 0, to find a solution UN (γ) ∈ ¡ ¢ IRm·N (γ) of Problem (P) with ϕN (γ) uN (γ) ≤ γ which leads to a solution uN (γ) : IN0 → IRm of the problem of null-controllability, if we define uN (γ) (t) = Θm

for all

t ≥ N (γ).

In order to solve Problem (P) we replace it by Problem (D) Minimize χ(y) =

N ° ° X ° T ¡ N −k ¢T ° y° , °B A k=1

subject to

y ∈ IRn ,

2

¡ ¢N cT y = −xT0 AT y = 1.

(2.72)

Let u : {0, . . . , N − 1} → IRm be a solution of (2.65) and let y ∈ IRn satisfy (2.72). Then it follows that N X

y T AN −k B u(k − 1) = y T c = 1

k=1

which implies max ||u(k − 1)||2 ≥

k=1,...,N n

1 . χ(y)

Now let yb ∈ IR be a solution of Problem (D). Then there is a multiplier λ ∈ IR such that ∇χ (b y ) = λ c. (2.73)

120


This yields ∇χ (b y) =

X

1 T ||B T (AN −k )

k∈I(b y)

with

yb||2

¡ ¢T AN −k BB T AN −k yb

o n ¯° ¡ ¢T ° ° ¯° I (b y ) = k ¯ °B T AN −k yb° > 0 . 2

This implies λ = ybT ∇χ (b y ) = x (b y) . If we define  ¡ ¢T  B T AN −k yb  1 uN (k − 1) = x (b y ) ||B T (AN −k )T yb||2   Θm then it follows that

N X

if k ∈ I (b y) , else,

(2.74)

k = 1, . . . , N,

AN −k B uN (k − 1) = c

k=1

and ||uN (k − 1)||2 =

1 χ (b y)

for all

which implies max ||uN (k − 1)||2 =

k=1,...,N

k ∈ I (b y)

1 . χ (b y)

Hence u : {0, . . . , N −1} → IRm solves Problem (P). This result is summarized as Theorem 2.23. If yb ∈ IRn solves Problem (D), then u : {0, . . . , N −1} → IRm defined by (2.74) solves Problem (P). In order to solve Problem (D) we apply the well known gradient projection method which is based on the following iteration step: Let y ∗ ∈ IRn with cT y ∗ = 1 be given. (At the beginning we take y ∗ = c/||c||22 .) Then we calculate µ ¶ 1 T ∗ h= c ∇χ (y ) c − ∇χ (y ∗ ) ||c||22 and see that ch = 0 and T

∇χ (y ∗ ) h =

¢2 1 ¡ T c ∇χ (y ∗ ) − ||∇χ (y ∗ ) ||22 ≤ 0. ||c||22


121

T

If ∇χ (y ∗ ) h = 0, then there exists some λ ≥ 0 such that (2.73) holds true T which is equivalent to y ∗ being optimal. If ∇χ (y ∗ ) h < 0, then h is a feasible b > 0 such that direction of descent. If we determine λ ³ ´ b h = min χ (y ∗ + λ h) , χ y∗ + λ

(2.75)

λ>0

then

³ ´ b h < χ (y ∗ ) χ y∗ + λ

³ ´ b h = 1. The next step is then performed with y ∗ + λ b h instead and cT y ∗ + λ b > 0 to satisfy (2.75) is of y ∗ . A necessary and sufficient condition for λ ³ ´T d ³ ∗ b ´ bh χ y + λ h = ∇χ y ∗ + λ h=0 dλ which is equivalent to ´ ¡ ¢T ³ ∗ bh y +λ hT AN −k BB T AN −k ° ³ ´° =0 ° T N −k T ∗+λ bh ° y (A ) °B ° ∗ b k∈I (y +λ h) X

2

and in turn to the fixed point equation ³ ´ b=ψ λ b λ with P ψ(λ) =

k∈I(y ∗ +λ h)

P

k∈I(y ∗ +λ h)

T

hT AN −k BB T (AN −k ) y ∗

k

B T (AN −k )T

hT

(y ∗ +λ h)

k2

AN −k BB T (AN −k )T

h kB T (AN −k )T (y∗ +λ h)k

.

2

In order to solve this equation we apply the iteration procedure λk+1 = ψ (λk ) ,

k ∈ IN0

starting with λ0 = 0. Let us return to the problem of fixed point controllability in Section 2.2.1 and let us assume that g : IRn × IRm → IRn is continuously Fréchet differentiable.

122


Then it follows that GN (x0 , u(0), . . . , u(N − 1)) − x b x, Θm , . . . , Θm ) = GN (x0 , u(0), . . . , u(N − 1)) − GN (b x ≈ JG x, Θm , . . . , Θm ) (x0 − x b) + N (b

N X

u(k−1)

JGN

(b x, Θm , . . . , Θm ) u(k − 1)

k=1

=

Jgx

N

(b x, Θm )

(x0 − x b) +

N X

Jgx (b x, Θm )

N −k

Jgu (b x, Θm ) u(k − 1)

k=1

where

¡ ¢ x, Θm ) = gixj (b x, Θm ) i,j=1,...,n Jgx (b

and

x, Θm ) = (giuk (b x, Θm )) i=1,...,n . Jgu (b k=1,...,m

Therefore we replace equation (2.58) by N X

N −k

Jgx (b x, Θm )

N

Jgu (b x, Θm ) u(k − 1) = −Jgx (b x, Θm )

(x0 − x b)

(2.76)

k=1

and solve the problem of finding u : {0, . . . , N − 1} → IRm which solves (2.76) and minimizes ϕN (u) = max ||u(k − 1)||2 . k=1,...,N

Such a u : {0, . . . , N − 1} → IR (2.58).

m

is then taken as an approximate solution of

The above problem has the form of Problem (P) at the beginning of this section and can be solved by the method described above. Finally we consider a special case in which the problem of fixed point controllability is reduced to a sequence of such problems which can be solved more easily. For this purpose we consider the system x(t + 1) = g0 (x(t)) +

m X

gj (x(t)) uj (t),

t ∈ IN0 ,

(2.77)

j=1

where gj : IRn → IRn , j = 1, . . . , m, are continuous vector functions. For every control function u : IN0 → IRm there is exactly one function x : IN0 → IRm which satisfies (2.77) and the initial condition x(0) = x0 ,

x0 ∈ IRn given.

(2.78)


123

We denote it by x = x(u). We assume that the uncontrolled system x(t + 1) = g0 (x(t)),

t ∈ IN,

n

has a fixed point x b ∈ IR which then solves the system x) . x b = g0 (b We again assume that the set U of admissible control functions is given by (2.53) where Ω ⊆ IRm is a subset with Θm ∈ Ω. Let us define ge0 (x) = g0 (x) − x,

x ∈ IRn .

Then (2.77) can be rewritten in the form x(t + 1) = x(t) + ge0 (x(t)) +

m X

gj (x(t)) uj (t),

t ∈ IN0 .

(2.79)

j=1

In order to find some N ∈ IN0 and a control function u ∈ U with (2.55) such that the solution x : IN0 → IRn of (2.77) satisfies the end condition (2.56) we apply an iterative method: Starting with some N0 ∈ IN0 and some u0 ∈ U (for instance u0 (t) =¡ Θm ¢ for all t ∈ IN0 ) we construct a sequence (Nk )k∈IN in k IN0 and a sequence u k∈IN in U as follows: ¢ ¡ If Nk−1 ∈ IN0 and uk−1 ∈ U are determined, we calculate x uk−1 : IN0 → IRn as the solution of (2.78) and (2.79) for u = uk−1 . Then we determine Nk ∈ IN0 and uk ∈ U such that for all t ≥ Nk uk (t) = Θm ¡ k¢ n and the solution x u : IN0 → IR of (2.78) and ¡ ¢ ¡ ¡ ¢ ¢ ¡ ¢ x uk (t + 1) = x uk (t) + ge0 x uk−1 (t) m X ¡ ¡ ¢ ¢ gj x uk−1 (t) uk+1 (t), + j

(2.55)k

t ∈ IN0 ,

(2.79)k

j=1

satisfies the end condition

¡ ¢ b. x uk (Nk ) = x

(2.56)k

If we put x k = x0 +

Nk X

¡ ¡ ¢ ¢ ge0 x uk−1 (t − 1)

t=1

and

¢ ¢ ¡ ¡ ¢ ¢¢ ¡ ¡ ¡ B k (t − 1) = g1 x uk−1 (t − 1) | . . . | gm x uk−1 (t − 1) ,

then the end condition (2.56)k is equivalent to N X t=1

B k (t − 1) uk−1 (t − 1) = x b − xk .

124


2.2.4 Stabilization of Controlled Systems Let g : IRn × IRm → IRn be a continuous mapping and let H be a family of continuous mappings h : IRn → IRm . If we define, for every h ∈ H, the mapping fh : IRn → IRn by fh (x) = g(x, h(x)),

x ∈ IRn ,

then fh is continuous and (IRn , fh ) is a time-discrete autonomous dynamical system. Let x b ∈ IRn be a fixed point of f (x) = g (x, Θm ) ,

x ∈ IRn .

Further we assume that h (b x) = Θm

for all

h∈H

which implies that x b is a fixed point of all fh , h ∈ H. After these preparations we can formulate the Problem of Stabilization Find h ∈ H such that {b x} is asymptotically stable with respect to fh . We assume that g : IRn × IRm → IRn and every mapping h ∈ H are continuously Fréchet differentiable. Then every mapping fh : IRn → IRn , h ∈ H, is also continuously Fréchet differentiable and, for every x ∈ IRn , its Jacobi matrix is given by Jfh (x) = Jgx (x, h(x)) + Jgu (x, h(x)) Jhx (x) where

¡ ¢ Jgx (x, h(x)) = gixj (x, h(x)) i=1,...,n

j=1,...,n

and Jgu (x, h(x)) = (giuk (x, h(x))) i=1,...,n , k=1,...,m

¡ ¢ Jhx (x) = hixj (x) i=1,...,m . j=1,...,n

From the Corollary of Theorem 1.5 we then obtain the Theorem 2.24. (a) Let the spectral radius % (Jfh (b x)) < 1. Then x b is asymptotically stable with respect to fh . x)) be invertible and let all the eigenvalues of % (Jfh (b x)) be larger (b) Let (Jfh (b than 1 in absolute value. Then x b is unstable with respect to fh .


125

Special cases: (a) Let

x ∈ IRn , u ∈ IRm ,

g(x, u) = A x + B u,

where A is a real n × n−matrix and B a real n × m−matrix, respectively. Further let H be the family of all linear mappings h : IRn → IRm which are given by h(x) = C x, x ∈ IRn , where C is an arbitrary real m × n−matrix, respectively. If we choose x b = Θn , then f (Θn ) = g (Θn , Θm ) = Θn and h (Θn ) = Θm

for all

h ∈ H.

Finally we have Jh (x) = C, Jgx (x, h(x)) = A

and

Jgu (x, h(x)) = B

for all x ∈ IRn and h ∈ H which implies Jfh (x) = A + BC

for all

x ∈ IRn

and h ∈ H.

Thus x b = Θn is asymptotically stable with respect to fh , if %(A + BC) < 1, and unstable with respect to fh , if all the eigenvalues of A + BC are larger than one in absolute value. (b) Let g(x, u) = F (x) + B(x) u, x ∈ X, u ∈ IRm , where F : X → X, X ⊆ IRn open, is continuously Fréchet differentiable and B(x) = (b1 (x), . . . , bm (x)) , x ∈ X, where bj : X → IRn , j = 1, . . . , n, are also continuously Fréchet differentiable. Let again H be the family of all linear mappings h : IRn → IRm which are given by h(x) = C x,

x ∈ IRn .

Finally, we assume that Θn ∈ X and F (Θn ) = Θn . If we choose x b = Θn , then f (Θn ) = g (Θn , Θm ) = Θn and h (Θn ) = Θm

for all

h ∈ H.

126


Further we obtain Jhx (x) = C, m X

Jgx (x, h(x)) = JF (x) +

Jbj (x) hj (x)

and

Jgu (x, h(x)) = B(x)

j=1

for all x ∈ X and h ∈ H which implies Jfh (x) = JF (x) +

m X

Jbj (x) hj (x) + B(x)C

for

x ∈ X, h ∈ H,

j=1

hence Jfh (Θn ) = JF (Θn ) + B (Θn ) C

for all

h ∈ H.

Thus x b = Θn is asymptotically stable with respect to fh , if % (JF (Θn ) + B (Θn ) C) < 1 and unstable with respect to fh , if all the eigenvalues of JF (Θn )+B (Θn ) C are larger than one in absolute value. 2.2.5 Applications a) An Emission Reduction Model We pick up the emission reduction model that was treated as uncontrolled system in Section 1.3.6 and as controlled system in Section 2.2.1. Here we concentrate on the controlled system which we linearize at a fixed point ´T ³ b ∈ IRr , of the uncontrolled system which leads to a linear bT , ΘT , E E r

control system of the form t ∈ IN0 ,

x(t + 1) = A x(t) + B u(t), with

! Ir C , 0r D

Ã ! C B= , D

Ã A=

where Ir and 0r is the r × r−unit and zero-matrix, respectively, and  em1 1 · · · em1 r  ..  , .. C =  ... . .  emr 1 · · · emr r 

  D=

b1 1 − λ1 M1∗ E 0

0 ..

. br 1 − λr Mr∗ E

  .


127

This implies Ã k

A B=

¢! ¡ C Ir + D + . . . + D k Dk+1

for all

k ∈ IN0 .

We consider the problem of null-controllability as being discussed in Section 2.2.2. Let us assume that C and D are non-singular. Then it follows that the matrices A and ! Ã C C (Ir + D) D D2 are non-singular which implies that the Kalman condition (2.67) is satisfied for N0 = 2. Let d1 , . . . , dr be the diagonal elements of D. Thus the non-singularity of D is equivalent to di 6= 0

for all

i = 1, . . . , r.

If all di 6= 1 for i = 1, . . . , r, then it follows (see Section 1.3.6) that the eigenvectors corresponding to the eigenvalues µi = 1

for i = 1, . . . , r

and

µi+r = di

for i = 1, . . . , r

of A are linearly independent which also holds true for AT (which has the same eigenvalues) (see Section 2.1.2). If |di | ≤ 1 and

for all

i = 1, . . . , r

Ω = {u ∈ IRr | ||u|| ≤ γ}

for some γ > 0 where || · || is any norm in IRr , then by Theorem 2.19 the problem of null-controllability has a solution for every choice of x0 = ¡ 1 T 2 T ¢T x0 , x0 ∈ IR2r . This problem can be solved with the aid of Problem (P) in Section 2.2.3 which reads as follows in this case: For a given N ∈ IN find u : {0, . . . , N − 1} → IRr such that Ã ¡ ¢! N X C Ir + D + . . . + D k u(k − 1) Dk+1 k=1 Ã ¢! Ã 1 ! ¡ x0 Ir C Ir + D + . . . + DN −1 =− N x20 0r D and ϕN (u) =

max ||u(k − 1)||2

k=1,...,N

is minimized (where || · ||2 denotes the Euclidean norm in IRr ).

128


Finally we illustrate the method by two numerical examples. Let m = 3. In both cases we choose x10 T = x20 T = (1, 1, 1). At first we choose  0.8 C =  0.2 0.4

 0.5 −0.5 0.2 0.3  0.3 0.2

(2.80) 

 0.1 0 0 D =  0 −0.2 0  0 0 0.1

and

and obtain Fig. 2.1: 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15

20

25

30

35

Fig. 2.1. Ordinate, ϕN (uN ) = Abscissa, N

Next we choose 

 0.8 0.5 0.5 C =  0.2 0.2 0.3  0.4 0.1 0.2

40

45

50

1 χ(ˆ yN )



and

 0.2 0 0 D =  0 0.8 0  0 0 0.6

and get Fig. 2.2. b) A Controlled Predator-Prey Model We pick up the predator prey model that has been discussed in Section 1.3.7 a) whose controlled version we assume to be of the form x1 (t + 1) = x1 (t) + a x1 (t) − b x1 (t)x2 (t) − x1 (t) u1 (t), x2 (t + 1) = x2 (t) − c x2 (t) + d x1 (t)x2 (t) − x2 (t) u2 (t)

t ∈ IN0 ,


129

2.5

2

1.5

1

0.5

0

0

5

10

15

20

25

30

Fig. 2.2. Ordinate, ϕN (uN ) = Abscissa, N

35

40

45

50

1 χ(ˆ yN )

where a > 0, 0 < c < 1, b > 0, d > 0, x1 (t) and x2 (t) denote the density of the prey and predator population at time t, respectively, and u1 , u2 : IN0 → IR are control functions. If we define

Ã

!

! Ã −x1 (t) g1 (x1 (t), x2 (t)) = , 0

Ã

a x1 (t) − b x1 (t)x2 (t) −c x1 (t) + d x1 (t)x2 (t)

ge0 (x1 (t), x2 (t)) = and

g2 (x1 (t), x2 (t)) =

! 0 , −x2 (t)

then the system can be rewritten in the form x(t + 1) = x(t) + ge0 (x(t)) +

2 X

gj (x(t)) uj (t),

t ∈ IN0 ,

(2.81)

j=1 T

with x(t) = (x1 (t), x2 (t)) . This is exactly the system (2.79) for m = 2. In addition we assume an initial condition Ã ! x01 x(0) = = x0 . x02

(2.82)

130


The uncontrolled system x(t + 1) = x(t) + ge0 (x(t)),

t ∈ IN0 ,

³ c a ´T , as fixed point. d b We assume the set U of admissible control functions to be given by

has x b=

© ª U = u : IN0 → IR2 | ||u(t)||2 ≤ γ for all t ∈ IN0 where γ > 0 is a given constant and || · ||2 is the Euclidean norm. For every u ∈ U we denote the unique solution x : IN0 → IR2 of (2.81) and (2.82) by x(u). Our aim now consists of finding some N ∈ IN0 and a control function u ∈ U with for all t ≥ N (2.83) u(t) = Θ2 such that the solution x : IN0 → IR2 of (2.81), (2.82) satisfies the end condition x(N ) = x b.

(2.84)

In order to find a solution of this problem we apply the iteration method described in Section 2.2.3. In the k−th step of this procedure we have, for a given uk−1 ∈ U , to find some Nk ∈ IN0 and a uk ∈ U with uk (t) = Θ2

for t ≥ Nk

such that !Ã ! Ã ¡ ¢ Nk X uk1 (t − 1) 0 x1 uk−1 (t − 1) ¡ k−1 ¢ = xk − x b, k (t − 1) u u (t − 1) 0 x 2 2 t=1 where xk = x 0 +

Nk X

¡ ¡ ¢ ¢ ge0 x uk−1 (t − 1) ,

t=1

and ||uk (t − 1)||2 ≤ γ

for

t = 1, . . . , Nk .

For this we can apply the method developed in Section 2.2.3. c) Control of a Planar Pendulum with Moving Suspension Point We consider a non-linear planar pendulum of length l(> 0) whose movement is controlled by moving its suspension point with acceleration u = u(t) along a horizontal straight line.


131

If we denote the deviation angle from the orthogonal position of the pendulum by ϕ = ϕ(t), then the movement of the pendulum is governed by the differential equation ϕ(t) ¨ =−

u(t) g sin ϕ(t) − cos ϕ(t), l l

t ∈ IR,

(2.85)

where g denotes the gravity constant. For t = 0 initial conditions are given by ϕ(0) = ϕ0

and

ϕ(0) ˙ = ϕ˙ 0 .

Now we discretize the differential equation by introducing a time step length 1 h > 0 and replacing the second derivative ϕ(t) ¨ by (ϕ(t+2h)−2 ϕ(t+h)+ϕ(t)) h thus obtaining the difference equation ϕ(t + 2h) = 2 ϕ(t + h) − ϕ(t) −

gh2 u(t)h2 sin ϕ(t) − cos ϕ(t), l l

t ∈ IR.

If we define y1 (t) = ϕ(t)

and

y2 (t) = ϕ(t + h)

then we obtain y1 (t + h) = y2 (t), y2 (t + h) = 2 y2 (t) − y1 (t) −

gh2 u(t)h2 sin y1 (t) − cos y1 (t), l l

t ∈ IR.

Finally we define functions x1 : IN0 → IR and x2 : IN0 → IR by putting x1 (n) = y1 (n · h) and

x2 (n) = y2 (n · h),

n ∈ IN0 ,

and obtain the system x1 (t + 1) = x2 (t), x2 (t + 1) = 2 x2 (t) − x1 (t) −

gh2 u(t)h2 sin x1 (t) − cos x1 (t), l l

t ∈ IN0 which is of the form (2.79) for m = 1 with   x2 (t) − x1 (t) , ge0 (x1 (t), x2 (t)) =  gh2 sin x1 (t) x2 (t) − x1 (t) − l   0 , t ∈ IN0 . g1 (x1 (t), x2 (t)) =  h2 cos x1 (t) − l

(2.49’)

132


In addition we assume initial conditions x1 (0) = ϕ0

and

x2 (0) = ϕ˙ 0 .

(2.50’)

The uncontrolled system x1 (t + 1) = x2 (t), x2 (t + 1) = 2 x2 (t) − x1 (t) −

gh2 sin x1 (t) l

has x b = (0, 0) as fixed point. We assume the set U of admissible control functions to be given by U = {u : IN0 → IR | |u(t)| ≤ γ for all t ∈ IN0 } where γ > 0 is a given constant. For every u ∈ U we denote the unique solution x : IN0 → IR2 of (2.49’) and (2.50’) by x(u). Our aim consists of finding some N ∈ IN0 and a control function u ∈ U with u(t) = 0

for all t ≥ N

(2.55’)

such that the solution x : IN0 → IR2 of (2.49’), (2.50’) satisfies the end condition x(N ) = Θ2 . This condition is equivalent to x2 (N ) = x2 (N − 1) = 0. The k−th step of the iteration method described in Section 2.2.3 for the solution of this problem then reads as follows: Let uk−1 ∈ U be given. Then we determine Nk ∈ IN0 and uk ∈ U such that uk (t) = 0

for all t ≥ Nk ¢ and the solution x uk : IN0 → IRn of (2.50’) and ¡

¡ ¡ ¢ ¡ ¢ ¡ ¢ ¢ x1 u1 (t + 1) = x1 uk (t) + x2 uk−1 (t) − x1 uk−1 (t), ¡ ¡ ¢ ¡ ¢ ¡ ¢ ¢ x2 u1 (t + 1) = x2 uk (t) + x2 uk−1 (t) − x1 uk−1 (t) ¡ ¢ ¡ ¢ h2 gh2 sin x1 uk−1 (t) − cos x1 uk−1 (t) uk (t), − l l t ∈ IN0 , satisfies the end conditions ¡ ¢ ¡ ¢ x2 uk (Nk ) = x2 uk (Nk − 1) = 0.


133

These are equivalent to −

Nk −1 ¡ ¢ 1 X cos x1 uk−1 (t − 1) uk (t − 1) l t=1 Nk −1 ¡ ¢ ¡ ¢ g X = −ϕ˙ 0 − x2 uk−1 (Nk − 2) + ϕ0 − sin x1 uk−1 (t − 1) l t=1 ¡ ¢ 1 − cos x1 uk−1 (Nk − 1) uk (Nk − 1) l ¡ ¢ ¡ ¢ = −x2 uk−1 (Nk − 1) + x2 uk−1 (Nk − 2) ¡ ¢ g − sin x1 uk−1 (Nk − 1) . l

Let us consider the special case Nk = 2. Then it follows that 1 g − cos ϕ0 uk (0) = ϕ0 − sin ϕ0 , l l 1 gh2 g h2 k − cos ϕ0 u (1) = −ϕ˙ 0 + ϕ0 + sin ϕ0 − sin ϕ˙ 0 − cos ϕ˙ 0 uk−1 (0). l l l l Let us assume that cos ϕ0 6= 0

and

cos ϕ˙ 0 6= 0.

Then it follows for all k ≥ 1 that lϕ0 + g tan ϕ0 , cos ϕ0 l (ϕ0 − ϕ˙ 0 ) sin ϕ0 − gh2 + g tan ϕ˙ 0 + h2 uk−1 (0). uk (1) = cos ϕ˙ 0 cos ϕ˙ 0

uk (0) = −

134


2.3 The Time-Discrete Non-Autonomous Case 2.3.1 The Problem of Fixed Point Controllability We consider a system of difference equations of the form x(t + 1) = gt (x(t), u(t)),

t ∈ IN0 ,

(2.86)

where (gt )t∈IN0 is a sequence of continuous vector functions gt : IRn × IRm → IRn , u : IN0 → IRm is a given vector function which is called a control function. The vector function x : IN0 → IRn which is called a state function is uniquely defined by (2.86), if we require an initial condition (2.87)

x(0) = x0 for some given vector x0 ∈ IRn . If we fix the control function u : IN0 → IRm and define x ∈ IRn , t ∈ IN0 ,

ft (x) = gt (x, u(t)),

(2.88)

n n for every ¢ t ∈ IN0 , ft : IR → IR is a continuous mapping and ¡then, n IR , (ft )t∈IN0 is a non-autonomous time-discrete dynamical system which is controlled by the function u : IN0 → IRm . If

u(t) = Θm

for all

t ∈ IN0 ,

then the system (2.88) is called uncontrolled. Let us assume that the uncontrolled system (2.88) admits a fixed point x b ∈ IRn which then solves the equations x, Θm ) , for all t ∈ IN. (2.89) x b = gt (b Now let Ω ⊆ IRm be a subset with Θm ∈ Ω. Then we define the set of admissible control functions by U = {u : IN0 → IRm | u(t) ∈ Ω for all t ∈ IN0 } .

(2.90)

After these preparations we can formulate the Problem of Local Fixed Point Controllability Given a fixed point x b ∈ IRn of the system x(t + 1) = gt (x(t), Θm ) ,

t ∈ IN0 ,

(2.91)

i.e., a solution x b of the equations (2.89) and an initial state x0 ∈ IRn , find some N ∈ IN0 and a control function u ∈ U with u(t) = Θm ,

for all

t≥N

(2.92)

2.3 The Time-Discrete Non-Autonomous Case

135

such that the solution x : IN0 → IRn of (2.86), (2.87) satisfies the end condition x(N ) = x b

(2.93)

(which implies x(t) = x b for all t ≥ N ). Let us assume that, for every t ∈ IN0 , the Jacobi matrices At =

∂gt (b x, Θm ) ∂x

and

Bt =

∂gt (b x, Θm ) ∂x

exist. Then it follows that, for every t ∈ IN0 , x(t + 1) − x b = gt (x(t), u(t)) − gt (b x, Θ m ) ≈ At (x(t) − x b) + Bt u(t). Therefore we replace the system (2.86) by h(t + 1) = At h(t) + Bt u(t),

t ∈ IN0 ,

(2.94)

and the initial condition (2.87) by h(0) = x0 − x b.

(2.95)

The end condition (2.93) is replaced by h(N ) = Θn = zero vector of IRn .

(2.96)

Now we consider the Problem of Local Controllability Find N ∈ IN0 and u ∈ U with u(t) = Θm

for all

t≥N

such that the corresponding solution h : IN0 → IRn of (2.94), (2.95) satisfies the end condition (2.96) which implies h(t) = Θn

for all

t ≥ N.

From (2.94) and (2.95) we conclude that, for every N ∈ IN, h(N ) = AN −1 · · · A0 (x0 − x b) +

N X

AN −1 · · · Ak Bk−1 u(k − 1),

k=1

if, for k = N , we put AN −1 · · · Ak = I = n × n−unit matrix.

136


Therefore the end condition (2.96) is equivalent to N X

AN −1 AN −2 · · · Ak Bk−1 u(k − 1) = −AN −1 AN −2 · · · A0 (x0 − x b) . (2.97)

k=1

Let us assume that, for some N0 ∈ IN, rank (BN0 −1 | AN0 −1 BN0 −2 | . . . | AN0 −1 · · · A1 B0 ) = n.

(2.98)

¡

Then, for N = N0 , the system (2.97) has a solution u(0)T , u(1)T , . . . , ´T T u (N0 − 1) ∈ IRm·N0 where x0 ∈ IRN can be chosen arbitrarily. If we define for all t ≥ N0 , (2.99) u(t) = Θm then we obtain a control function u : IN0 → IRm such that the corresponding solution h : IN0 → IRn of (2.94) and (2.95) satisfies the end condition (2.96) for N = N0 . If Ak is non-singular for all k ∈ IN0 , then the assumption (2.97) implies that (2.98) holds true for all N ≥ N0 instead of N0 . For instance, if we replace N0 by N0 + 1, then (BN0 | AN0 BN0 −1 | AN0 AN0 −1 BN0 −2 | . . . | AN0 · · · A1 B0 )   = BN0 | |

AN |{z}0



 (BN0 −1 | AN0 −1 BN0 −2 | . . . | AN0 −1 · · · A1 B0 ) . {z } |

non-singular

{z

rank = n

}

=⇒ rank = n

´T ³ T So the system (2.97) has a solution u(0)T , u(1)T , . . . , u (N0 − 1) for all N ≥ N0 . Next we assume that Ω ⊆ IRm is convex, has Θm as interior point and satisfies u ∈ Ω ⇒ − u ∈ Ω. Let us define, for every N ∈ IN, the set ¯ ) ( N ¯ X ¯ AN −1 AN −2 · · · Ak Bk−1 u(k − 1) ¯ u ∈ U . R(N ) = x = ¯ k=1

Then we can prove (see Theorem 2.18) Theorem 2.25. If, for some N0 ∈ IN, condition (2.98) is satisfied and if Ak is non-singular for all k ∈ IN0 , then Θn is an interior point of R(N ) for all N ≥ N0 .


137

Proof. Let us assume that Θn (∈ R(N )) is not an interior point of R(N ) for some N ≥ N0 . Thus R(N ) must be contained in a hyperplane through Θn , i.e., there must exist some y ∈ IRn , y 6= Θn , with yT x = 0

for all

x ∈ R(N ).

This implies N X

y T AN −1 AN −2 · · · Ak Bk−1 uk = 0

for all

¡

u1 T , . . . , uN T

¢T

N

∈ (IRm ) ,

k=1

hence T y T AN −1 AN −2 · · · Ak Bk−1 = Θm

for all

k = 1, . . . , N.

Since (2.98) also holds true for all N ≥ N0 instead of N0 , it follows that y = Θn which contradicts y 6= Θn . Hence the assumption is false and the proof is complete. u t As a consequence of Theorem 2.25 we obtain Theorem 2.26. In addition to the assumption of Theorem 2.25 let sup ||Ak || < 1

where || · || denotes the spectral norm.

(2.100)

k∈IN0

Then there is some N ∈ IN and some u ∈ U such that (2.98) holds true. Proof. Assumption (2.100) implies b) = Θn . lim AN −1 AN −2 · · · A0 (x0 − x

N →∞

Hence Theorem 2.25 implies that there is some N ∈ IN with N ≥ N0 such that b) ∈ R(N ) −AN −1 AN −2 · · · A0 (x0 − x which completes the proof.

u t

138


2.3.2 The General Problem of Controllability We consider the same situation as at the beginning of Section 2.3.1. However, we do not assume the existence of a fixed point x b ∈ IRn of the uncontrolled system (2.91), i.e., a solution of (2.89). Instead we assume a vector x1 ∈ IRn to be given and consider the general Problem of Controllability Find some N ∈ IN0 and a control function u ∈ U (2.90) such that the solution x : IN0 → IRn of (2.86), (2.87) satisfies the end condition x(N ) = x1 .

(2.101)

From (2.86) and (2.87) we infer x(N ) = gN −1 (gN −2 (· · · (g0 (x0 , u(0)) , u(1)) , . . .) , u(N − 1)) = GN (x0 , u(0), . . . , u(N − 1)) . (2.102) Hence the end condition (2.101) is equivalent to GN (x0 , u(0), . . . , u(N − 1)) = x1 .

(2.103)

So we have to find vectors u(0), . . . , u(N −1) ∈ Ω such that (2.103) is satisfied. For every N ∈ IN we define the controllable set SN (x1 ) = {x ∈ IRn | there exists some u ∈ U ª such that GN (x, u(0), . . . , u(N − 1)) = x1 and put S (x1 ) =

[

SN (x1 ) .

IN∈IN

Now let x0 ∈ S (x1 ). Then we ask the question under which conditions is x0 an interior point of S (x1 )? In order to find an answer to this question we assume that Ω is open

and

Then it follows that GN

gN ∈ C 1 (IRn × IRm ) for all N ∈ IN0 . ¡ ¢ ∈ C 1 IRn × IRm·N for every N ∈ IN and

GN x (x, u(0), . . . , u(N − 1)) ¡ ¢ = (gN −1 )x GN −1 (x, u(0), . . . , u(N − 2)) , u(N − 1) ¡ ¢ × (gN −2 )x GN −2 (x, u(0), . . . , u(N − 3)) , u(N − 2) .. × . × (g0 )x (x, u(0)),


139

and GN u(k) (x, u(0), . . . , u(N − 1)) ¡ ¢ = (gN −1 )x GN −1 (x, u(0), . . . , u(N − 2)) , u(N − 1) ¡ ¢ × (gN −2 )x GN −2 (x, u(0), . . . , u(N − 3)) , u(N − 2) .. × . ¡ k+1 ¢ (x, u(0), . . . , u(k)) , u(k + 1) × (gk+1 )x G ¡ ¢ × (gk )u(k) Gk (x, u(0), . . . , u(k − 1)) , u(k) for k = 0, . . . , N − 1, x ∈ IRn and u ∈ U . Let us assume that x0 ∈ SN0 (x1 ) for some N0 ∈ IN, i.e., GN0 (x0 , u0 (0), . . . , u0 (N0 − 1)) = x1 for some u0 ∈ U . Further let (gN )x (x, u0 ) be non-singular for all N ∈ IN0 , for all x ∈ IRn and all u ∈ Ω. 0 Then GN x (x0 , u0 (0), . . . , u0 (N0 − 1)) is also non-singular and, by the implicit function theorem, there exists an open set V ⊆ Ω N0 with (u0 (0), . . . , u0 (N0 − 1)) ∈ V and a function h : V → IRn with h ∈ C 1 (V ) such that

h (u0 (0), . . . , u0 (N0 − 1)) = x0 and GN0 (h (u0 (0), . . . , u0 (N0 − 1)) , u(0), . . . , u (N0 − 1)) = x1 for all

(u(0), . . . , u (N0 − 1)) ∈ V

which means h (u(0), . . . , u (N0 − 1)) ∈ SN0 (x1 )

for all

(u(0), . . . , u (N0 − 1)) ∈ V.

Moreover, −1

0 hu(k) (u0 (0), . . . , u0 (N0 − 1)) = −GN x (x0 , u0 (0), . . . , u0 (N0 − 1))

0 ×GN u(k) (x0 , u0 (0), . . . , u0 (N0 − 1)) .

Next we assume that ¡ rank hu(0) (u0 (0), . . . , u0 (N0 − 1)) | . . .

¢ | hu(N0 −1) (u0 (0), . . . , u0 (N0 − 1)) = n.

140


Then it follows with the aid of the inverse function theorem that there exists an n−dimensional relatively open set Ve ⊆ V with (u0 (0), . . . , u0 (N0 − 1)) ∈ Ve e such ³ ´that the restriction of h to V is a homeomorphism which implies that h Ve ⊆ SN0 (x1 ) is open. ³ ´ Therefore x0 ∈ h Ve is an interior point of S (x1 ). Now we consider the special case where there exists some x b ∈ IRn with

which implies

x, Θm ) = x b gN (b

for all

N ∈ IN

¡ ¢ N b, Θm =x b GN x

for all

N ∈ IN.

x), hence x b ∈ S (b x). Then, for every N ∈ IN, it follows that x b ∈ SN (b Let us assume that x, Θm ) is non-singular (gN )x (b

for all

N ∈ IN0 .

Then ¡ ¢ N b, Θm = (gN −1 )x (b x, Θm ) · (gN −2 )x (b x, Θm ) · · · (g0 )x (b x, Θm ) GN x x is also non-singular for all N ∈ IN. By the implicit function theorem we therefore conclude, for every N ∈ IN, N ∈ VN and a function hN : that there exists an open set VN ⊆ Ω N with Θm n 1 VN → IR with hN ∈ C (VN ) such that ¡ N¢ =x b and GN (hN (u(0), . . . , u (N − 1)) , u(0), . . . , u(N − 1)) = x b hN Θm for all (u(0), . . . , u(N − 1)) ∈ VN which means x) hN (u(0), . . . , u(N − 1)) ∈ SN (b

for all

(u(0), . . . , u(N − 1)) ∈ VN .

Moreover, ¡ N¢ ¡ ¢ ¡ ¢ N −1 N = −GN b, Θm b, Θm . · GN (hN )u(k) Θm x x u(k) x Next we assume that, for some N0 ∈ IN, ³ ¡ N0 ¢ ¡ N0 ¢´ | . . . | (hN0 )u(N0 −1) Θm = n. rank (hN0 )u(0) Θm Then it follows with the aid of the inverse function theorem that there exists an N0 ∈ VeN0 such³that ´the n−dimensional relatively open set VeN0 ⊆ VN0 with Θm restriction of hN0 to VeN0 is a homeomorphism which implies that hN0 VeN0 ⊆ ³ ´ x). This SN0 (b x) is open. Therefore x b ∈ hN0 VeN0 is an interior point of S (b result is a generalization of Theorem 2.25, if Ω in addition is open.


141

2.3.3 Stabilization of Controlled Systems Let (gt )t∈IN be a sequence of continuous mappings gt : IRn × IRm → IRn and let H be a family of continuous mappings h : IRn → IRm . If we define, for every h ∈ H and t ∈ IN, the mapping fth : IRn → IRn by fth (x) = gt (x, h(x)),

x ∈ IR,

³ then we obtain a non-autonomous time-discrete dynamical system IRn , ´ ¡ h¢ ft t∈IN . The dynamics in this system is defined by the sequence F h = ¡ h¢ Ft t∈IN of mappings Fth : IRn → IRn given by h ◦ . . . ◦ f1h (x) Fth (x) = fth ◦ ft−1

and

for all

x ∈ IRn

and t ∈ IN

for all x ∈ IRn . ´ ³ ¡ ¢ We also obtain the dynamical system IRn , fth t∈IN , if we replace the control function u : IN0 → IRm in the system (2.86) by the feedback controls h(x) : IN0 → IRm , x ∈ IRn . F0h (x) = x

The problem of stabilization of the controlled system (2.86) by the feedback controls h(x), x ∈ IRn , then reads as follows: Given x0 ∈ IRn such that the limit set LF h (x0 ) defined by (1.70) (see Section 1.3.8) is non-empty and compact for all h ∈ H. Find a mapping h ∈ H such that ¡ L¢F h (x0 ) is stable, an attractor or asymptotically stable with respect to fth t∈IN . Let us consider the special case gt (x, u) = At (x) x + Bt (x) u

for x ∈ IRn , u ∈ IRm ,

(2.104)

where (At (x))t∈IN and (Bt (x))t∈IN is a sequence of real, continuous n × n− and n × m−matrix functions on IRn , respectively. Let H be the family of all linear mappings h : IRn → IRm (which are automatically continuous). Every h ∈ H is then representable in the form h(x) = C h x,

x ∈ IRn ,

where C h is a real m × n−matrix. For every t ∈ IN and h ∈ H we therefore obtain ¢ ¡ x ∈ IRn . (2.105) fth (x) = At (x) + Bt (x) C h x, Let us put Dth (x) = At (x) + Bt (x) C h

for all

x ∈ IRn .

142


If we choose x0 ∈ Θn = zero vector of IRn , then we conclude Fth (x0 ) = x0

for all

t ∈ IN0 , h ∈ H,

and therefore LF h (x0 ) = {x0 }. The problem of stabilization of the controlled system (2.86) with gt , t ∈ IN, given by (2.104) in this situation consists of finding an m × n−matrix C h such that ¡ {x¢0 = Θn } is stable, an attractor or asymptotically stable with respect to fth t∈IN with fth given by (2.105). Now let us assume that ||Dth (x)|| ≤ 1

for all

x ∈ IRn

and t ∈ IN

(2.106)

where || · || denotes the spectral norm. Let U ⊆ IRn be a relatively compact open set with x0 = Θn ∈ U . Then there is some r > 0 such that BU = {x ∈ IRn | ||x||2 < r} ⊆ U. Hence BU is open, x0 ∈ BU , and assumption (2.106) implies fth (BU ) ⊆ BU

for all t ∈ IN.

If we define V (x) = ||x||22 = xT x for

x ∈ IRn ,

then V (x) ≥ 0

for all

x ∈ IRn

and

(V (x) = 0 ⇔ x = x0 = Θn )

and ¢ ¡ V fth (x) − V (x) = xT Dth (x)T Dth (x) x − xT x ¡ ¢ = ||Dth (x) x||22 − ||x||22 ≤ ||Dth (x) x|| − 1 ||x||22 ≤ 0 for all x ∈ IRn and t ∈ IN.

¡ ¢ This shows that V is a Lyapunov function with respect to fth t∈IN on G = IRn which is positive definite with respect to {x0 = Θn }. By ¡Theorem 1.9 we ¢ therefore conclude that {x0 = Θn } is stable with respect to fth t∈IN with fth given by (2.105). Next we assume that sup ||Dth (x)|| < 1

t∈IN

for all

x ∈ IRn .

(2.107)


143

Then it follows from ¡ ¡ h ¡ h ¢ ¢T ¢ V Fth (x) = xT D1h (x)T · · · Dth Ft−1 (x) Dth Ft−1 (x) · · · D1h (x) x ¡ h ¢ (x) · · · D1h (x) x||22 = ||Dth Ft−1 ¡ h ¢ ≤ ||Dth Ft−1 (x) x||2 · · · ||D1h (x) x||2 ||x||2 for all x ∈ IRn and t ∈ IN that ¡ ¢ lim V Fth (x) = 0

t→∞

for all

x ∈ IRn .

This implies lim Fth (x) = Θn

t→∞

for all

x ∈ IRn

¡ ¢ and shows that {x0 = Θn } is an attractor with respect to fth t∈IN with fth given by (2.105). Result. Under the assumption (2.107) the set {x0 = Θn } is asymptotically ¡ ¢ stable with respect to fth t∈IN with fth given by (2.105). 2.3.4 The Problem of Reachability We again consider the situation at the beginning of Section 2.3.1 without necessarily assuming the existence of a fixed point x b ∈ IRn of the uncontrolled m system (2.91). Let Ω ⊆ IR be a non-empty subset. For a given x0 ∈ IRn we then define the set of states that are reachable from x0 in N ∈ IN steps by n RN (x0 ) = x = GN (x0 , u(0), . . . , u(N − 1)) ¯ o ¯ ¯ u(k) ∈ Ω, k = 0, . . . , N − 1

(2.108)

where the map GN : IRm·N → IRn is defined by (2.102). Further we define the set of states reachable from x0 in a suitable number of steps by [ RN (x0 ) . (2.109) R (x0 ) = N ∈IN

The question we are interested in now is: Under which conditions does R (x0 ) have a non-empty interior? A simple answer to this question gives

144


Theorem 2.27. Let Ω be open. If there is some N ∈ IN and there exist u(0), . . . , u(N − 1) ∈ Ω such that ³ rank GN u(0) (x0 , u(0), . . . , u(N − 1)) | . . . ´ (2.110) | GN u(N −1) (x0 , u(0), . . . , u(N − 1)) = n, then RN (x0 ) has a non-empty interior and therefore also R (x0 ). Proof. Condition (2.110) implies that the n × N · m−matrix ´ ³ N (x , u(0), . . . , u(N − 1)) | . . . | G (x , u(0), . . . , u(N − 1)) GN 0 0 u(0) u(N −1) has n linearly independent column vectors. Let E be the n−dimensional subset of Ω N consisting of all vectors whose components which do not correspond to these linearly independent column vectors are equal to the ones of ¡ ¢T u(0)T , . . . , u(N − 1)T . If we restrict the mapping GN to E, then the Jacobi matrix of this restriction consists of these linearly independent column vectors and is therefore non-singular. By the inverse function theorem therefore there exists an open set (with re¢T ¡ ∈ U which is mapped spect to E) U ⊆ Ω N with u(0)T , . . . , u(N − 1)T homeomorphically by GN on an open set V ⊆ RN (x0 ) with GN (x0 , u(0), . . . , u(N − 1)) ∈ V . This completes the proof. u t Next we consider the linear case where x ∈ IRn , u ∈ IRm ,

gt (x, u) = At x + Bt u,

with n × n−matrices At and Bt , respectively, for every t ∈ IN0 . Then, for every N ∈ IN and every x0 ∈ IRn , we obtain GN (x0 , u(0), . . . , u(N − 1)) = AN −1 · · · A0 x0 +

N X

AN −1 · · · Ak Bk−1 u(k − 1)

k=1

where for k = N we put AN −1 · · · Ak = I = n × n−unit matrix. Further we have, for every N ∈ IN and every x0 ∈ IRn , ½ RN (x0 ) =

x = AN −1 · · · A0 x0 +

N X k=1

AN −1 · · · Ak Bk−1 u(k − 1)

¯ ¾ ¯ ¯ u(k) ∈ Ω, k = 0, . . . , N − 1 . ¯


145

Because of GN u(k−1) (x0 , u(0), . . . , u(N − 1)) = AN −1 · · · Ak Bk−1

for

k = 1, . . . , N

it follows that the condition (2.110) for N = N0 coincides with the condition (2.98). If this is satisfied, then by Theorem 2.27 the set R (x0 ) (2.109) of states reachable from x0 has a non-empty interior, if Ω is open. If Ω = IRm , it follows in addition that R (x0 ) = IRn for all x0 ∈ IRn . Proof. Let x, x0 ∈ IRn be given arbitrarily. Then condition (2.98) implies the existence of u(k) ∈ IRm for k = 0, . . . , N0 − 1 such that x − AN0 −1 · · · A0 x0 =

N0 X

AN0 −1 · · · Ak Bk−1 u(k − 1)

k=1

holds true which shows that x ∈ RN0 (x0 ) ⊆ R (x0 ). For every k = 1, . . . , N let us define an n × m−matrix C k by C k = AN −1 · · · Ak Bk−1

for k = 1, . . . , N − 1

and C N = BN −1 . The condition (2.98) implies the existence of n column vectors  ck1jl k  . l  .   .  cknjl k 

for

l = 1, . . . , n

which are linearly independent.

l

Let us define the n × n−matrix C and a vector u ∈ IRn by   C= 

ck1j1k

1

cknj1 k 1

 · · · ck1jnkn  ..  .  · · · cknjnkn



and

 uj kl (k1 − 1)   .. u=  , respectively, . uj kn (kn − 1)

and put uj (k − 1) = 0

for

k 6= kl , j 6= jkl , l = 1, . . . , n.

u t

146


Then we obtain GN (x0 , u(0), . . . , u(N − 1)) = AN −1 · · · A0 x0 + C u which implies ³ ´ u = C −1 GN (x0 , u(0), . . . , u(N − 1)) −AN −1 · · · A0 x0 . {z } | =x

Now let n E = u = (u(0), . . . , u(N − 1)) ∈ IRm·N ¯ o ¯ ¯ uj (k − 1) = 0 for k 6= kl and j 6= jkl , l = 1, . . . , n . Then GN (x0 , ·) is a linear isomorphism from E on IRn . Therefore GN (x0 , u) = x

for some

u∈E

implies −1

u = GN (x0 , ·)

(x), −1

N

G (x0 , u) = AN −1 · · · A0 x0 + C · GN (x0 , ·)

(x).

If all Ak , k ∈ IN0 are invertible it follows that −1 −1 −1 N x0 = A−1 0 · AN −1 x − A0 · AN −1 C · G (x0 , ·)

−1

(x).

In the nonlinear case we have the following situation: If the condition (2.110) is satisfied, there exists an n−dimensional subset E of Ω N and a set U ⊆ E which is open with respect to E and contains ¡ ¢T u(0)T , . . . , u(N − 1)T and which is mapped homeomorphically on an open V ⊆ RN (x0 ) by the restriction of GN (x0 , ·) to E. If x = GN (x0 , u(0), . . . , u(N − 1)) , then

¢T ¡ −1 u(0)T , . . . , u(N − 1)T = GN (x0 , ·) (x).

If in addition GN x (x0 , u(0), . . . , u(N − 1)) is non-singular, then by the implicit function theorem there exists an open set W ⊆ Ω N which contains ¡ ¢T u(0)T , . . . , u(N − 1)T and a function h : W → IRn with h ∈ C 1 (W ) such that h(u(0), . . . , u(N − 1)) = x0 and


147

GN (h (e u(0), . . . , u e(N − 1)) , u e(0), . . . , u e(N − 1)) = x ¢ ¡ T T T e(N − 1) ∈ W. for all u e(0) , . . . , u This implies

³ ´ −1 x0 = h GN (x0 , ·) (x) .

Since hu(k) (u(0), . . . , u(N − 1)) = −Gx (x0 , u(0), . . . , u(N − 1))

−1

0 × GN u(k) (x0 , u(0), . . . , u(N − 1))

for k = 0, . . . , N − 1, it follows from (2.110) that ¡ ¢ rank hu(0) (u(0), . . . , u(N − 1)) | . . . | hu(N −1) (u(0), . . . , u(N − 1)) = n which implies that h maps U ∩ W homeomorphically onto an open set −1 Ve ⊆ RN (x0 ) which contains x. Therefore h ◦ GN (x0 , ·) maps V ∩ Ve homee omorphically on an open set Ve which contains x0 and is contained in x ∈ IRn | there exists some u e ∈ U (2.53) SN (x) = {e ª with GN (e x, u e(0), . . . , u e(N − 1)) = x . Finally let us assume (as in Theorem 2.27) that Ω is open and (for given x0 ∈ IRn ) there exists some N ∈ IN such that the condition (2.110) is satisfied ¢T ¡ ∈ Ω N . Then it follows from the proof of for all u(0)T , . . . , u(N − 1)T Theorem 2.27 that every x ∈ RN (x0 ) is an interior point of RN (x0 ), i.e., RN (x0 ) is open. This implies in the linear case with Ω being an open subset of IRm that RN (x0 ) is open for every x0 ∈ IRn , if the condition (2.98) is satisfied.

3 Chaotic Behavior of Autonomous Time-Discrete Systems

3.1 Chaos in the Sense of Devaney Let f : X → X be a continuous mapping of a metric space X into itself. Then by the definition π(x, n) = f n (x)

for all

x∈X

and n ∈ IN0

(3.1)

with f 0 (x) = x

and

f n+1 (x) = f (f n (x)) ,

n ∈ IN0 ,

(3.2)

we obtain an autonomous time-discrete dynamical system (see Section 1.3.1). A point x ∈ X is called period point of f : X → X, if there exists an n ∈ IN with f n (x) = x. The smallest n ∈ IN with this property is called f −period of x ∈ X. Definition. A continuous mapping f : X → X is called topologically transitive, if for every pair of non-empty open subsets U, V ⊆ X there is an n ∈ IN with f n (U ) ∩ V 6= ∅ where f n (U ) = {f n (x) | x ∈ U } . Graphically this means that a topologically transitive mapping f : X → X causes a complete mixing of the metric space X, if it is applied sufficiently often. On using the two concepts “period points” and “topological transitivity” we now define chaos as follows: Definition. A continuous mapping f of a metric space X into itself is called chaotic, if the following two conditions are satisfied: (1) The set of period points of f is dense in X. (2) The mapping f : X → X is topologically transitive. S. Pickl and W. Krabs, Dynamical Systems: Stability, Controllability and Chaotic Behavior, 149 DOI 10.1007/978-3-642-13722-8_3, © Springer-Verlag Berlin Heidelberg 2010

150


Every period point x ∈ X of f can be assigned an orbit © ª O(x) = f k (x) | k = 0, . . . , n for some n ∈ IN0 whose points are also period points of f . Each such “periodic” orbit can be considered as an ordered movement in the time-discrete dynamical system that is defined by f . Every period point therefore defines a certain order; the topological transitivity, however, is the expression of complete disorder. The above chaos definition therefore states that order and complete disorder are immediately neighboured, if the mapping f : X → X is chaotic. This definition goes back to Devaney (see [5]) who adds a third property which is defined as follows: Definition. A continuous mapping f : X → X is called sensitively depending on initial values, if there exists some δ > 0 such that for every x ∈ X and every neighborhood V (x) of x there exists some y ∈ V (x) and some n ∈ IN0 with d (f n (x), f n (y)) ≥ δ where d denotes the metric of X. This property, however, is superfluous, if X consists of infinitely many elements, because of the Theorem 3.1. Let the metric space X consist of infinitely many elements and let f : X → X be chaotic in the sense of the above definition. Then f is also sensitively depending on initial values (see [2]). Proof. Since X consists of infinitely many points, there are two period points x1 , x2 ∈ X of f with O (x1 ) 6= O (x2 ) where O(x) = {f n (x) | n ∈ IN0 }

for every x ∈ X.

Let δ0 = d (O (x1 ) , O (x2 )) = min {d (y1 , y2 ) | yi ∈ O (xi ) , i = 1, 2} . Let x ∈ X be chosen arbitrarily. Then either δ0 (a) d (x, O (x1 )) = min {d(x, y) | y ∈ O (x1 )} ≥ or 2 δ0 (b) d (x, O (x1 )) < . 2 Let d (x, yb1 ) = d (x, O (x1 )). Then it follows in case (b), for every y2 ∈ O (x2 ), the chain of inequalities y1 , y2 ) − d (x, yb1 ) > δ0 − d (x, y2 ) ≥ d (b

δ0 δ0 δ0 = , hence d (x, O (x2 )) > . 2 2 2

3.1 Chaos in the Sense of Devaney

151

Result. There is a number δ0 > 0 such that for every x ∈ X there exists a δ0 period point x b ∈ X with d (x, O (b x)) ≥ . 2 δ0 . Then we choose x ∈ X arbitrarily and an arbitrary 8 neighbourhood V (x). Further we put

Let us put δ =

U = V (x) ∩ Bδ (x)

where

Bδ (x) = {y ∈ X | d(y, x) < δ}.

By property (1) there is a period point p ∈ U of f . Let f n (p) = p. By the δ0 = 4δ. above result there is a period point q ∈ X of f with d(x, O(q)) ≥ 2 We put n \ V = f −i (Bδ (f (q))) i=0

where, for every non-empty subset W of X, we have © ª f −i (W ) = y ∈ X | f i (y) ∈ W , i ∈ IN0 . Then V is open and non-empty, since q ∈ V . Since f is topologically transitive, there is some y ∈ U and some k ∈ IN with f k (y) ∈ V . · ¸ k Now let j = + 1 ⇒ 1 ≤ n · j − k ≤ n. By construction we therefore have n ¡ ¡ ¢ ¢ f n·j (y) = f n·j−k f k (y) ∈ f n·j−k (V ) ⊆ Bδ f n·j−k (q) . Now we have f n·j (p) = p and hence ¡ ¢ ¡ ¢ d f n·j (p), f n·j (y) = d p, f n·j (y) ´ ³ ¡ ¢ ≥ d x, f n·j−k (q) − d f n·j−k (q), f n·j (y) − d(p, x) | {z } ∈ O(q)

≥ 4δ − δ − δ = 2δ because of p ∈ Bδ (x). Therefore we further have either ¢ ¡ (α) d f n·j (x), f n·j (y) ≥ δ or ¢ ¡ (β) d f n·j (x), f n·j (y) < δ where in the latter case we conclude that ¡ ¢ ¡ ¢ ¡ ¢ d f n·j (x), f n·j (p) ≥ d f n·j (p), f n·j (y) − d f n·j (y), f n·j (x) > 2δ − δ = δ. Because of y ∈ U ¢and p ∈ U we have found in each case a z ∈ V (x) with ¡ d f n·j (x), f n·j (z) ≥ δ which completes the proof. u t

152


A paradigmatic example for a chaotic mapping is the so called shift-mapping in the space of 0 − 1−sequences, i.e., in the space Σ = {s = (s0 , s1 , s2 , . . .) | sj ∈ {0, 1} for j ∈ IN0 } . If one defines a function d : Σ × Σ → IR+ by d(s, t) =

∞ X |si − ti |

2i

i=0

,

s, t ∈ Σ,

then d turns out to be a metric. In the following we need the Lemma 3.2. Let s, t ∈ Σ be given. Then it follows that (a) si = ti for i = 0, . . . , n =⇒ d(s, t) ≤ 2−n . Conversely it follows that (b) d(s, t) < 2−n =⇒ si = ti for i = 0, . . . , n. Proof. (a) Let si = ti for i = 0, . . . , n. Then d(s, t) =

∞ ∞ ∞ X X 1 X 1 1 |si − ti | 1 ≤ = = n. i i n+1 i 2 2 2 2 2 i=n+1 i=n+1 i=0

1 for some n ∈ IN0 . If si 6= ti for some i ∈ {0, . . . , n}, then 2n it follows that d(s, t) ≥ 2−i ≥ 2−n .

(b) Let d(s, t)
0 be given. Then we choose n ∈ IN0 1 1 such that n < ε. Let δ = n+1 . If we then, for some t = (t0 , t1 , t2 , . . .) ∈ Σ, 2 2 have that d(s, t) < δ, it follows from Lemma 3.2 that si = ti

for i = 0, . . . , n + 1,

hence σ(s)i = σ(t)i for i = 1, . . . , n which again by Lemma 3.2 implies that 1 u t d(σ(s), σ(t)) ≤ n < ε. This completes the proof. 2

3.1 Chaos in the Sense of Devaney

153

For every n ∈ IN0 there are 2n different points s ∈ Σ of the form s = (s0 , . . . , sn−1 , s0 , . . . , sn−1 , . . .) for which it follows that σ n (s) = s which shows that they are period points of σ. Next we prove Theorem 3.4. The set Per(σ) of all period points of σ is dense in Σ. Proof. Let s = (s0 , s1 , s2 , . . .) ∈ Σ be given arbitrarily. Then we define for every n ∈ IN0 the sequence τn = (s0 , s1 , . . . , sn , s0 , s1 , . . . , sn , . . .) ,

n ∈ IN0 .

Therefore we have τn ∈ Per(σ)

for all

n ∈ IN0

and τni = si

for i = 0, . . . , n. 1 The Lemma 3.2 implies d (τn , s) ≤ n . 2 Let ε > 0 be given. Then we choose n ∈ IN0 such that that d (τn , s) ≤ ε which completes the proof.

1 ≤ ε and conclude 2n u t

s))n∈IN0 is dense Theorem 3.5. There is some sb ∈ Σ such that the orbit (σ n (b in Σ. Proof. Let us define sb = (01 00011011 {z. . . 111} . . .). | | {z } 000001 double blocs

triple blocs

n

s) coincides, for sufficiently large n, with every t ∈ Σ in the Obviously σ (b 1 s) , t) ≤ n and shows first n bits. This implies by the Lemma 3.2 that d (σ n (b 2 the assertion. u t As a consequence we obtain the Theorem 3.6. The shift-mapping (3.3) is topologically transitive and therefore chaotic. Proof. Let U, V ⊆ Σ be open subsets of Σ. Take any sb ∈ U . Then there is some δ1 > 0 with Bs (δ1 ) = {t ∈ Σ | d (t, sb) < δ1 } ⊆ U . Theorem 3.5 implies s) ∈ Bs (δ1 ). Now take any t ∈ V . the existence of some n ∈ IN0 such that σ n (b Then there is some δ2 > 0 with Bt (δ2 ) ⊆ V and there exists some m ∈ IN0 such that σ n+m (b s) ∈ Bt (δ2 ). Since σ n+m (b s) ∈ σ m (Bs (δ1 )), it follows that σ m (Bs (δ1 )) ∩ Bt (δ2 ) 6= ∅ which completes the proof.

=⇒

σ m (U ) ∩ V 6= ∅ u t

154


3.2 Topological Conjugacy Let X and Y be two metric spaces and let f : X → X and g : Y → Y be two continuous mappings. Definition. f and g are called topologically conjugated, if there exists a homeomorphism h : X → Y such that h ◦ f = g ◦ h,

(3.4)

i.e., if the following diagram is commutative X

f

h

² Y

g

/X ² /Y

h

Theorem 3.7. Let f : X → X and g : Y → Y be continuous mappings of metric spaces X and Y , respectively, and let f be chaotic. If then there exists a continuous mapping h : X → Y which is surjective and for which (3.4) holds true, g is also chaotic. Proof. Let y ∈ Y be an arbitrary point and let W (y) be an arbitrary neighbourhood of y. Then there is some x ∈ X with h(x) = y and h−1 (W (y)) = V (x) is a neighborhood of x. Since f is chaotic, there is a period point p ∈ V (x) of f with f n (p) = p for some n ∈ IN. If we put q = h(p), then q ∈ W (y) and g n (q) = g n (h(p)) = g n−1 (g(h(p))) = g n−1 (h(f (p))) = h (f n (p)) = h(p) = q, i.e., q is a period point of g. The set of period points is therefore also dense in Y . Now let V and W be two non-empty open subsets of Y . Then h−1 (V ) and open h−1 (W ) are two non-empty ¡ ¢ subsets of X. Since f is chaotic, there exists some n ∈ IN with f n h−1 (V ) ∩ h−1 (W ) 6= ∅. Now we have ¡ ¢ ¡ ¢ ¡ ¢ g n (V ) = g n ◦ h h−1 (V ) = g n−1 ◦ g ◦ h h−1 (V ) = g n−1 ◦ h ◦ f h−1 (V ) ¡ ¢ = . . . = h ◦ f n h−1 (V ) which implies g n (V ) ∩ W 6= ∅. Hence g is also topologically transitive and therefore chaotic which completes the proof. u t Corollary 3.8. Let the continuous mappings f : X → X and g : Y → Y be topologically conjugated. Then f is chaotic, if and only if g is chaotic.

3.2 Topological Conjugacy

155

Proof. If f is chaotic, then by Theorem 3.7 also g is chaotic. If conversely g is chaotic, then, since h−1 : Y → X is continuous and surjective and h−1 ◦ f = u t g ◦ h−1 , again by Theorem 3.7 f is also chaotic. Let us demonstrate this Corollary by an example. For that purpose we consider at first the mapping f : IR → IR which is defined by f (x) = µ x(1 − x),

x ∈ IR.

(3.5)

µ ¶ 1 µ of f is larger = 2 4 1 than 1 and there is an open interval A0 ⊆ [0, 1] with as midpoint such that 2

We assume that µ > 4. Then the maximum value f

f (x) > 1,

f 2 (x) < 0

lim f n (x) = −∞

and

n→∞

for all x ∈ A0 . If one defines A1 = {x ∈ [0, 1] | f (x) ∈ A0 } , then it follows that f 2 (x) > 1,

f 3 (x) < 0

lim f n (x) = −∞

and

n→∞

for all x ∈ A1 . Graphically we have the following situation: 6 I

1 N

N

¡

¡

¡ ¡ ¡

¡

¡ ¡

¡

I

¡ ¡

¡ ¡ ¡

H

H

¡

N ¡

¡ ¡ ¡¡ ¢ A1

³

´ A0

¡ ¢ A1 1

-

156


Next we define © ª A2 = x ∈ [0, 1] | f i (x) ∈ Ai−1 for i = 1, 2 . Then it follows that f 3 (x) > 1 for all x ∈ A2 . If one defines, for an arbitrary n ∈ IN, © ª An = x ∈ [0, 1] | f i (x) ∈ Ai−1 for i = 1, . . . , n , then it follows that f n+1 (x) > 1 for all x ∈ An . Now we put

Ã A = [0, 1] \

∞ [

! An

=

n=0

∞ \

([0, 1] \ An ) .

(3.6)

n=0

and [0, 1]\(A0 ∩ A1 ) of 4 = Obviously [0, 1]\A0 consists of two closed µ n intervals ¶ S 2 2 closed intervals. In general [0, 1]\ Ak consists of 2n+1 closed intervals k=0

1 which are located symmetrically with respect to . Every set [0, 1] \ An is 2 closed. Therefore A is non-empty and closed. From the construction it follows that f (A) ⊆ A. In addition we have the following Theorem 3.9. The set A defined by (3.6) is a Cantor set, i.e., it is closed, totally unconnected (does not contain √ intervals) and perfect (every point of A is an accumulation point), if µ > 2 + 5. Proof. The closedness of A has been shown already. f (x) > 1 for all x ∈ A0 implies that r r µ ¶ 1 1 4 1 1 4 − A0 = 1− , + 1− . 2 2 µ 2 2 µ | {z } | {z } x1

Further we obtain

x2

r

f 0 (x) ≥ f 0 (x1 ) = µ (1 − 2x1 ) = µ and

1−

r 0

0

f (x) ≤ f (x2 ) = µ (1 − 2x2 ) = −µ From µ > 2 +

1−

4 µ

for all

x ∈ [0, x1 ]

4 µ

for all

x ∈ [0, x2 ] .

√ 5 it follows that µ2 − 4µ − 1 > 0 and further r µ ¶ 4 4 2 µ 1− > 1 =⇒ µ 1 − > 1. µ µ


157

This implies |f 0 (x)| > 1

for all x ∈ [0, 1] \ A0

and further the existence of some λ > 1 such that |f 0 (x)| > λ

for all

x ∈ A ⊆ [0, 1] \ A0 .

The chain rule implies for every n ∈ IN 0

| (f n ) (x)| > λn

for all

x ∈ A.

Now let x, y ∈ A with x 6= y and [x, y] ⊆ A be given. Then it follows for all n ∈ IN that 0 for all α ∈ [x, y]. | (f n ) (α)| > λn Choose n ∈ IN such that λn (y − x) > 1. Then the mean value theorem implies |f n (y) − f n (x)| ≥ λn (y − x) > 1 and hence f n (x) 6∈ [0, 1] or f n (y) 6∈ [0, 1] which contradicts x, y ∈ A. Therefore A does not contain an interval. In order to show the perfectness of A we at first observe that every end point of an interval in Ak , k ∈ IN0 , is mapped into 0 after finitely many steps, i.e., belongs to A. Now let p ∈ A be an isolated point of A. Then there is in every neighbourhood of p a point x 6= p with f k+1 (x) > 1 for some k ∈ IN0 , hence x ∈ Ak . Now we distinguish two cases: Either there is a sequence of such points x which are all end points of intervals in some Ak and hence belong to A. This would be a contradiction to the isolation of p. Or every sequence of such points x is mapped elementwise after finitely many steps into the exterior of [0, 1]. We can even assume that for every element xk of the sequence there exists some n ∈ IN with f n (xk ) < 0 so that f n attains a maximum ¡ ¢ in p which implies 0 (f n ) (p) = 0. By the chain rule it follows that f 0 f j (p) = 0 for some j < n 1 and hence f j (p) = which implies f j+1 (p) > 1 and f n (p) → ∞ for n → ∞. 2 This is a contradiction to p ∈ A. Hence the set A does not contain isolated points which completes the proof. u t √ Remark. Theorem 3.9 also holds true for µ ∈ (4, 2 + 5]. However, the proof is more complicated. For every x 6∈ A we have lim f n (x) = −∞ and for all x ∈ A it follows that n→∞

f n (x) ∈ A

for all n ∈ IN0 . √ Question: Is f : A → A chaotic, if µ > 2 + 5?

158


In order to answer this question in the affirmation sense we show that the mapping f : A → A and the shift-mapping σ : Σ → Σ defined by (3.3) are topologically conjugated so that the Corollary 3.8 implies that f is chaotic. At first we define a mapping S : A → Σ by S(x) = (s0 , s1 , s2 , . . .) where ( 0, if f j (x) ∈ I0 (3.7) sj = 1, if f j (x) ∈ I1 , and I0 , I1 are the two intervals with I0 ∪I1 = [0, 1]\(A0 = {x ∈ [0, 1] | f (x) > 1}). For this mapping we prove the √ Theorem 3.10. If µ > 2 + 5, then S : A → Σ is a homeomorphism. Proof. At first we prove that S is injective. Let x, y ∈ A with x 6= y be given. Assumption: S(x) = S(y). Then, for every n ∈ IN0 , it follows that (f n (x) ∈ I0

and f n (y) ∈ I0 )

(f n (x) ∈ I1

or

and f n (y) ∈ I1 ) .

This implies [x, y] ⊆ A or [y, x] ⊆ A which contradicts the fact that by Theorem 3.9 A is totally disconnected. In order to show that S is surjective we define, for every closed interval J ⊆ [0, 1], the n−th counterimage by f −n (J) = {x ∈ [0, 1] | f n (x) ∈ J} . The counterimage f −1 (J) consists of two intervals, one in I0 and another one in I1 as being shown in the following picture: 6 1

J

£

f −1 (J)

£

I0

¤ ¤

f −1 (J)

£ £

I1

¤

¤ 1

-


159

Now let s = (s0 , s1 , s2 , . . .) ∈ Σ be given. Then we define for every n ∈ IN0 Is0 s1 ...sn = {x ∈ [0, 1] | x ∈ Is0 , f (x) ∈ Is1 , . . . , f n (x) ∈ Isn } = Is0 ∩ f −1 (Is1 ) ∩ . . . ∩ f −n (Isn ) = Is0 ∩ f −1 (Is1 s2 ...sn ) .

(3.8)

According to the above remark f −1 (Is1 s2 ...sn ) consists of two closed intervals, one in I0 and another one in I1 . Therefore Is0 ∩ f −1 (Is1 s2 ...sn ) is a closed interval in Is0 . Further we have Is0 ...sn = Is0 ...sn−1 ∩ f −n (Isn ) ⊆ Is0 ...sn−1 . T T Is0 s1 ...sn is non-empty. Let x ∈ Is0 s1 ...sn . Then it This implies that follows that

n∈IN0

n∈IN0

x ∈ Is0 , f (x) ∈ Is1 , f 2 (x) ∈ Is2 , . . . ,

hence S(x) = s, which shows the surjectivity of S. Further we have \ Is0 s1 ...sn = {x}, since S is injective. n∈IN0

In order to show the continuity of S we choose an arbitrary x ∈ A and put (s0 , s1 , . . .) = S(x). Let ε > 0 be given. Then we choose n ∈ IN0 such 1 that n ≤ ε. Now consider the intervals It0 t1 ...tn defined by (3.8) for all 2n+1 2 combinations of t0 , . . . , tn . These are all different from each other and Is0 s1 ...sn is one of these. Now choose δ > 0 such that |x − y| < δ

and y ∈ A

=⇒

y ∈ Is0 s1 ...sn .

Then it follows that S(x)i = S(y)i

for i = 0, . . . , n

and hence by Lemma 3.2 d(S(x), S(y)) ≤

1 ≤ε 2n+1

which proves the continuity of S. In order to show the continuity of the inverse mapping S −1 : Σ → A we show that S maps closed subsets of A into closed subsets of Σ. Let B ⊆ A be closed and non-empty. Then B is compact, since A is compact as a closed subset of [0, 1] which is compact. Since S is continuous S(B) is compact in Σ and therefore closed. This completes the proof of Theorem 3.10. After these preparations we come to

u t

160


Theorem 3.11. The mappings f : A → A and σ : Σ → Σ are topologically √ conjugated, if µ > 2 + 5. In particular we have S◦f =σ◦S where S : A → Σ is the homeomorphism defined by (3.7). Proof. From the proof of Theorem 3.10 we obtain that every x ∈ A can be represented uniquely in the form \ Is0 s1 ...sn where (s0 , s1 , . . .) = S(x). {x} = n∈IN0

From (3.8) it follows because of f (Is0 ) = [0, 1] that f (Is0 s1 ...sn ) = Is1 ∩ f −1 (Is2 ) ∩ . . . ∩ f −n+1 (Is1 s2 ...sn ) . Therefore it follows that ! Ã ! Ã \ \ Is0 s1 ...sn = S Is1 s2 ...sn = (s1 , s2 , . . .) = σ(S(x)) S(f (x)) = S◦f n∈IN0

n∈IN

which completes the proof.

u t

Corollary 3.8, Theorem 3.6 and Theorem 3.11 now imply Theorem 3.12. The mapping √ f : A → A defined by (3.5) with A defined by (3.6) is chaotic, if µ > 2 + 5. Next we consider the mapping f (3.5) for µ = 4, i.e, f (x) = 4x(1 − x),

x ∈ IR.

Obviously f maps the interval [0, 1] into itself. Since f is continuous, it defines a time-discrete dynamical system on X = [0, 1]. In order to show that f is chaotic we at first consider the space Σ2 of twofoldinfinite 0 − 1−sequences, i.e., n o Σ2 = s = (sj )j∈Z | sj = 0 or 1 . If we equip Σ2 with the metric d(s, t) =

∞ X

1 |s − tj |, |j| j 2 j=−∞

s, t ∈ Σ2 ,

then Σ2 becomes a compact metric space which is homeomorphic to the interval [0, 1] where the homeomorphism is given by the mapping s = (sj )j∈Z →

X 1 sj , 2|j| j∈Z

s ∈ Σ2 .


161

Lemma 3.13. Given s, t ∈ Σ2 . Then the following two statements are true: 1 (a) si = ti for i = −n, . . . , n =⇒ d(s, t) ≤ n−1 . 2 1 (b) d(s, t) < n =⇒ si = ti for i = −n, . . . , n. 2 The proof of this lemma is the same as that for sequences in Σ. We again define a shift-mapping σ : Σ2 → Σ2 by σ(s)j = sj+1

for all

j ∈ Z and all s ∈ Σ2 .

This mapping is even a homeomorphism, i.e., bi-unique, surjective and continuous together with its inverse mapping σ −1 : Σ2 → Σ2 which is defined by for all j ∈ Z and all t ∈ Σ2 . σ −1 (t) = tj−1 For every n ∈ IN0 there are 22n+1 different points s ∈ Σ2 of the form (. . . , sn , s−n+1 , . . . , s0 , . . . , sn−1 , sn , . . .) for which it follows that σ 2n+1 (s) = s, i.e., which are period points. Similar to Theorem 3.4 and Theorem 3.5 one can prove Assertion 1: The set Per(σ) of period points is dense in Σ2 . Assertion 2: There is some s ∈ Σ2 such that the orbit (σ n (s))n∈IN0 is dense in Σ2 . As a consequence we obtain the statement (see Theorem 3.6): The shift-mapping σ : Σ2 → Σ2 is topologically transitive and hence chaotic. Since Assertion 1 and 2 are also true for the inverse mapping σ −1 : Σ2 → Σ2 , it follows that σ −1 is also chaotic. Next the mapping g : S 1 → S 1 of the unit circle S 1 = ª © iα we consider e | α ∈ IR into itself which is defined by ¡ ¢ g e iα = e 2iα ,

α ∈ IR.

Now let h : Σ2 → S 1 be defined by h(s) = e iα

where α =

X sj . 2j

j∈Z

Then h is continuous and surjective. Further we have h ◦ σ −1 = g ◦ h. Theorem 3.7 implies then that g is also chaotic.

162


If we define a mapping h1 : S 1 → [−1, 1] by ¡ ¢ h1 e iα = cos α,

α ∈ IR

and a mapping h2 [−1, 1] → [0, 1] by h2 (x) =

1 (1 − x), 2

x ∈ [−1, 1],

then h2 ◦ h1 : S 1 → [0, 1] is continuous and surjective. Further we have ¡ ¢ 1 h2 ◦ h1 ◦ g e iα = (1 − cos 2α) = sin2 α 2 and ¶ µ ¡ iα ¢ 1 (1 − cos α) = f ◦ h2 (cos α) = f f ◦ h2 ◦ h1 e 2 µ ¶ 1 = 2 (1 − cos α) 1 − (1 − cos α) 2 = 1 − cos2 α = sin2 α hence h2 ◦ h1 ◦ g = f ◦ h2 ◦ h1 . Again Theorem 3.7 implies that f is chaotic.

for

α ∈ IR,

3.3 The Topological Entropy as a Measure for Chaos

163

3.3 The Topological Entropy as a Measure for Chaos 3.3.1 Definition and Invariance The following representation is based on the paper [28] by A. Mielke. Let X be a compact metric space whose metric is denoted by d and let f : X → X be a continuous mapping. Definition. Choose n ∈ IN and ε > 0 arbitrarily. A non-empty subset M ⊆ X is called (n, ε)−separated with respect to f , if for every pair x 6= y in M there exists a number m ∈ {1, 2, . . . , n} with d (f m (x), f m (y)) > ε. Assertion: If M ⊆ X is (n, ε)− separated with respect to f , then M consists at most of finitely many elements. Proof. If M consisted of infinitely many elements, then there would exist an accumulation point x b ∈ X of M and a sequence (xi )i∈IN of different points in b for all i ∈ IN and lim d (xi , x b) = 0 which in particular implies M with xi 6= x i→∞

lim d (xi , xi+1 ) = 0. Because of the continuity of f (which implies uniform

i→∞

continuity) there exists, for all m ∈ {1, 2, . . . , n}, some i(ε) ∈ IN with d (f m (xi ) , f m (xi+1 )) ≤ ε

for all

m ∈ {1, . . . , n} and all i ≥ i(ε).

This contradicts the (n, ε)−separatedness of M with respect to f which completes the proof. u t For every n ∈ IN and ε > 0 we put s(f, n, ε) = sup{#M | M ⊆ X is (n, ε) − separated with respect to f } where # denotes the number of elements of M and define h(f, ε) = lim sup n→∞

1 ln(s(f, n, ε)). n

Let ε2 > ε1 > 0. Then every (n, ε2 ) −separated subset of X with respect to f is also (n, ε1 ) −separated which implies s (f, n, ε1 ) ≥ s (f, n, ε2 )

=⇒

h (f, ε1 ) ≥ h (f, ε2 ) .

The topological entropy of f is then defined by h(f ) = lim h(f, ε) = sup h(f, ε) + ε→0

0 0 therefore exists a δ(ε) > 0 with lim δ(ε) = 0

ε→0+

and

d2 (g(x), g(y)) ≤ δ(ε)

=⇒

d1 (x, y) ≤ ε.

The latter implication is equivalent to d1 (x, y) > ε

=⇒

d2 (g(x), g(y)) > δ(ε).

Now let M1 ⊆ X1 be (n, ε)−separated with respect to f1 . Then for x 6= y in M1 there is some m ∈ {1, . . . , n} with d1 (f1m (x), f1m (y)) > ε. This implies d2 (f2m (g(x)), f2m (g(y))) = d2 (g (f1m (x)) , g (f1m (y))) > δ(ε). Because of g(x) 6= g(y) ⇔ x 6= y it follows that g (M1 ) ⊆ g (X1 ) is (n, δ(ε))−separated with respect to f2 . This implies s (f1 , n, ε) ≤ s (f2 , n, δ(ε)) and further h (f1 , ε) ≤ h (f2 , δ(ε)) which finally implies h (f1 ) ≤ h (f2 ). This completes the proof.

u t

Theorem 3.15. We assume that under the assumption of Theorem 3.14 there exists a surjective continuous mapping g : X1 → X2 with f2 ◦ g = g ◦ f1 . Then it follows that h (f2 ) ≤ h (f1 ). Proof. Since g : X1 → X2 is uniformly continuous, there exists for every ε > 0 some δ(ε) such that lim δ(ε) = 0

ε→0+

and

d1 (x, y) ≤ δ(ε)

=⇒

d2 (g(x), g(y)) ≤ ε.

The latter implication is equivalent to d2 (g(x), g(y)) > ε

=⇒

d1 (x, y) > δ(ε).

Now let M2 ⊆ X2 be (n, ε)−separated with respect to f2 . Then for u 6= v in M2 there exists some m ∈ {1, . . . , n} with d2 (f2m (u), f2m (v)) > ε. Let x, y ∈ X1 with u = g(x) and v = g(y) be given. Then it follows that x 6= y and d2 (g (f1m (x)) , g (f1m (y))) = d2 (f2m (u), f2m (v)) > ε.


165

This implies d1 (f1m (x), f1m (y)) > δ(ε) for some m ∈ {1, . . . , n}. Therefore g −1 (M2 ) ⊆ X2 is (n, δ(ε))−separated with respect to f . Because of #M2 ≤ #g −1 (M2 ) we obtain s (f2 , n, ε) ≤ s (f1 , n, δ(ε)) and further h (f2 , ε) ≤ h (f1 , δ(ε)) which finally implies h (f2 ) ≤ h (f1 ).

u t

As a consequence of Theorem 3.14 and Theorem 3.15 we obtain the Theorem 3.16. Let the continuous mappings f1 : X1 → X1 and f2 : X2 → X2 of two compact metric spaces X1 and X2 be topologically conjugated. Then it follows that h (f1 ) = h (f2 ). This property of the topological entropy is called topological invariance. Definition. A continuous mapping f : X → X of a compact metric space X into itself is called disorder-chaotic, if the topological entropy h(f ) of f is positive (h(f ) ≥ 0 follows from the definition). 3.3.2 The Topological Entropy of the Shift-Mapping In Section 3.1 we have considered the space Σ of the 0 − 1−sequences s = (s0 , s1 , s2 , . . .) with sj = 1 or sj = 0 for every j ∈ IN0 endowed with the metric d(s, t) =

X |si − ti | 2i

for s, t ∈ Σ.

i∈IN0

This space is compact, since by virtue of the mapping X si , s = (s0 , s1 , s2 , . . .) 7−→ d(s) = 2i i∈IN0

it turns out to be homeomorphic to the compact interval [0, 2]. The shiftmapping σ : Σ → Σ is defined by σ (s0 , s1 , s2 , . . .) = (s1 , s2 , s3 , . . .) and is continuous according to Theorem 3.3. We even have d(σ(s), σ(t)) ≤ 2d(s, t)

for

s, t ∈ Σ

which shows that σ is Lipschitz-continuous. In the following we look for the topological entropy h(σ) of σ. For that purpose we consider for every n ∈ IN the subset Σn = {s ∈ Σ | sk+n = sk

for all

k ∈ IN0 } of Σ.

166


Let s, t ∈ Σn be given with s 6= t. Then it follows for every m ∈ {1, . . . , n} that X 1 |sk+m − tk+m |, d (σ m (s), σ m (t)) = 2k k∈IN0

hence,

d (σ m (s), σ m (t)) ≥ 1,

if sm 6= tm .

Therefore Σn is (n, ε)−separated with respect to σ for every ε ∈ (0, 1). This implies for all ε ∈ (0, 1) 2n = #Σn ≤ s(σ, n, ε) and further ln 2 ≤

1 lim sup s(σ, n, ε) = h(σ, ε) n n→∞

for all

ε ∈ (0, 1)

which implies ln 2 ≤ h(σ) and shows that σ is disorder-chaotic. Next we will determine an upper bound for h(σ). For that purpose we start with the implication d (σ m (s), σ m (t)) > 21−N for some m ∈ {1, . . . , n} =⇒ d(s, t) > 21−(N +m) for every pair

n, N ∈ IN.

If we choose ε = 21−N and assume that M ⊆ Σ is (n, ε)−separated with respect to σ, then, for every pair s, t ∈ M , there is some m ∈ {1, . . . , n} with d (σ m (s), σ m (t)) > ε which implies d(s, t) > 21−(N +m) . Therefore Lemma 3.2 implies sk 6= tk

for some

From this it follows that µ s σ, n, and it is

1 2N −1

k ∈ {0, . . . , N + n − 1}.

¶ ≤ #ΣN +n = 2N +n

µ µ ¶¶ µ ¶ 1 N 1 ln s σ, n, N −1 ≤ 1+ ln 2. n 2 n

Finally this implies µ µ µ ¶ ¶¶ 1 1 1 h σ, N −1 = lim sup ln s σ, n, N −1 ≤ ln 2 2 2 n→∞ n

for all N ∈ IN

and therefore h(σ) ≤ ln 2 which, with the above result, implies h(σ) = ln 2.


167

3.3.3 Disorder-Chaos for One-Dimensional Mappings In order to show disorder-chaos for one-dimensional mappings it is useful to determine the topological entropy of suitable subspaces of Σ. For that purpose we introduce 2 × 2−matrices ¶ µ a00 a01 A= a10 a11 with aij = 0 or 1 for i, j = 0, 1. Then we define, for a given such matrix, a subspace Σ A of Σ by ª © Σ A = (si )i∈IN0 ∈ {0, 1}IN | asi si+1 = 1 for all i ∈ IN0 . Assertion: Σ A is closed and therefore compact. ¡ ¢ Proof. Let sk k∈IN be a sequence in Σ A with sk → t for some t ∈ Σ. Assumption: t 6∈ Σ A . Then there is a smallest i0 ∈ IN0 with ati0 ti0 +1 = 0. Because of sk → t there is some k0 ∈ IN with ¢ ¡ 1 for all k ≥ k0 . d sk , t < i0 +1 2 By Lemma 3.2 it follows that ski = ti

for i = 0, . . . , i0 + 1.

Because of sk ∈ Σ A for all k ∈ IN this implies atio tio +1 = 1 which leads to a contradiction and concludes the proof. u t ¡ A¢ Obviously we have σ Σ ⊆ Σ A so that σ also defines a dynamical system. We denote the restriction of σ to Σ A by σA and call this mapping a subshift. Question. How does one calculate the topological entropy of σA ? From the above considerations we at first deduce |ΣnA | ≤ s (σA , n, ε) and

µ s σA , n,

1 2N −1

for all ε ∈ (0, 1)

and all

n ∈ IN

¶ A ≤ |ΣN +n |

for all

n, N ∈ IN,

where for every n ∈ IN ΣnA = {(s0 , s1 , . . . , sn−1 , s0 , s1 , . . . , sn−1 , . . .) ∈ Σ | as0 s1 = 1, . . . , ª asn−2 sn−1 = 1 . Let e = (1, 1)T and for every n ∈ IN let µ ¶ a0 (n) a(n) = = An−1 e. a1 (n) Assertion: For every n ∈ IN it is true that |ΣnA | = kAn−1 ek1 = a0 (n) + a1 (n).

168


Proof. Let n = 1. Then we have Σ1A = {(s0 , s1 , . . .) ∈ Σ | s0 = 0 or s0 = 1} and

|Σ1A | = 2 = a0 (1) + a1 (1).

Let n = 2. Then |Σ2A | = a00 + a01 + a10 + a11 = kAek1 = a0 (2) + a1 (2). Let n = 3. Then |Σ3A | = a200 + a01 a10 + a10 a00 + a11 a10 + a00 a01 + a01 a11 + a10 a01 + a211 . Now we have

µ 2

A = which implies

a200 + a01 a10 a00 a01 + a01 a11 a10 a00 + a11 a10 a10 a01 + a211

¶

|Σ3A | = kA2 ek1 = a0 (3) + a1 (3).

In general for n ≥ 2 we have |ΣnA | = sum of all products as0 s1 as1 s2 · · · asn−2 sn−1 with si = 0 or 1 for i = 0, . . . , n − 1 = kAn−1 ek1 = a0 (n) + a1 (n). u t The first of the above two statements therefore implies, for all ε ∈ (0, 1) and all n ∈ IN, 1 1 ln (a0 (n) + a1 (n)) ≤ ln (s (σA , n, ε)) n n and from the second we infer for all n, N ∈ IN that µ µ ¶¶ 1 1 1 ln s σA , n, N −1 ≤ ln (a0 (N + n) + a1 (N + n)) . n 2 n Now we have ¶ µ ¶ Ã (N ) (N ) ! µ ¶ µ b01 b a0 (n) a0 (N + n) N a0 (n) =A = 00 , (N ) (N ) a1 (N + n) a1 (n) a1 (n) b10 b11 hence

´ ´ ³ ³ (N ) (N ) (N ) (N ) a0 (N + n) + a1 (N + n) = b00 + b10 a0 (n) + b01 + b11 a1 (n) ´ ³ (N ) (N ) (N ) (N ) ≤ max b00 + b10 , b01 + b11 (a0 (n) + a1 (n)) = γN (a0 (n) + a1 (n)) .


This implies 1 1 1 ln (a0 (N + n) + a1 (N + n)) ≤ ln (a0 (n) + a1 (n)) + ln γN n n n and further lim sup n→∞

µ ¶ 1 1 1 s σA , n, N −1 ≤ lim sup ln (a0 (n) + a1 (n)) n 2 n→∞ n for all N ∈ IN.

On the other hand we have lim sup n→∞

1 1 ln (a0 (n) + a1 (n)) ≤ lim sup ln (s (σA , n, ε)) n n→∞ n

for all ε ∈ (0, 1). Therefore we obtain for all N ≥ 2 µ ¶ 1 1 h σA , N −1 = lim sup ln (a0 (n) + a1 (n)) 2 n n→∞ and therefore h (σA ) = lim sup n→∞

1 ln (a0 (n) + a1 (n)) . n

Without proof we use the statement 1 ¡ n−1 ¢ ln kA ek1 = ln(λ(A)) n→∞ n lim

where λ(A) is the largest eigenvalue of A. As a result we obtain the Theorem 3.17. The topological entropy of σA is given by h (σA ) = ln(λ(A)) where λ(A) is the largest eigenvalue of A. µ ¶ 01 Example. A = . 11 √ 1 In this case we have λ(A) = (1 + 5) and therefore 2 µ ¶ √ 1 (1 + 5) > 0. h (σA ) = ln 2

169

170


We will now apply this result in order to show that a continuous mapping f : IR → IR which has a period point of f −period 3 is disorder-chaotic on a suitable subset J ⊆ IR. Let there be given three points a, b, c ∈ IR with a < b < c and f (a) = b, f (b) = c and f (c) = a. We put I0 = [a, b] and I1 = [b, c]. Then it follows that I1 ⊆ f (I0 )

and

I0 ∪ I1 ⊆ f (I1 ) ,

however in general

I0 6⊆ f (I0 ) .

We even assume that I0 ∩ f (I0 ) = {b}. If we define for i, j = 0, 1 ( 0, if Ij ⊆ 6 f (Ii ) , aij = 1, if Ii ⊆ f (Ij ) , then we obtain the matrix A=

µ ¶ 01 . 11

If we define for every n ∈ IN0 the set where J0 = [a, c] = I0 ∪ I1 , Jn = {x ∈ J0 | f n (x) ∈ J0 } T Jn is a non-empty compact subset of IR with f (J) ⊆ J, then J = n∈IN0

since every Jn is a counterimage of a continuous mapping of J0 which T is a Jn compact interval. If J were empty, then already a finite intersection n∈IN

were empty which is not the case. Now let x ∈ J be given. Then it follows that f n (f (x)) ∈ J0 , i.e., f (x) ∈ J and therefore f (J) ⊆ J. Finally we define the mapping S : J → Σ by S(x) = (s0 , s1 , s2 , . . .) where ( 0, if f j (x) ∈ [a, b), sj = 1, if f j (x) ∈ [b, c). Obviously it follows that sj = 0

=⇒

sj+1 = 1 and sj = 1

=⇒

sj+1 = 0 or sj+1 = 1.

This implies f (J) ⊆ Σ A . Assertion 1: The mapping S : J → Σ A is surjective. Proof. Let (s0 , s1 , s2 , . . .) ∈ Σ A be given. Then it follows that ¢ ¡ for every n ∈ IN. Is1 ⊆ f (Is0 ) , Is2 ⊆ f (Is1 ) , . . . , Isn ⊆ f Isn−1 If we put Ai = Isi for i ∈ IN0 , then it follows that Ai+1 ⊆ f (Ai )

for all

i ∈ IN0 .


171

Then, for every i ∈ IN0 , there exists a closed set Bi ⊆ A0 = Is0

with Bi+1 ⊆ Bi

and f i (Bi ) = Ai = Isi .

In order to see that we put B0 = A0 and obtain f 0 (B0 ) = A0 . Now assume Bi for some i ∈ IN0 already to be defined such that Bi ⊆ A0 and f i (Bi ) = Ai . From Ai+1 ⊆ f (Ai ) it follows that Ai+1 ⊆ f i+1 (Bi ). This implies the existence of a closed set Bi+1 ⊆ Bi with f i+1 (Bi+1 ) = Ai+1 (Simply put Bi+1 = f −i−1 (Ai+1 )). T If we put B = Bn then B is non-empty and compact (by the same n∈IN0

arguments as for J) and it follows that f n (x) ∈ Isn

for all

x ∈ B and all n ∈ IN0 .

This shows that B ⊆ J and (s0 , s1 , s2 , . . .) = S(x) for all x ∈ B. Assertion 2: The mapping S : J → Σ

A

u t

is continuous.

Proof. Let x ∈ J and ε > 0 be chosen arbitrarily. Then we choose n ∈ IN0 1 such that n < ε. Let S(x) = (s0 , s1 , s2 , . . .). For this sequence in Σ A we 2 assume a sequence Bi ⊆ Is0 , i ∈ IN0 , to be T constructed as in the proof of Bi and Assertion 1 so that in particular x ∈ B = i∈IN0 i

f (x) ∈ Isi

for all i ∈ IN0 .

Now let y ∈ Bn be chosen arbitrarily. Then it follows that f i (y) ∈ Isi

for all

i = 0, . . . , n

and therefore S(y)j = sj = S(x)j

for j = 0, . . . , n.

By Lemma 3.2 this implies d(S(x), S(y)) ≤

1 0

(3.9)

lim inf |f n (p) − f n (q)| = 0.

(3.10)

n→∞

and n→∞

(b) For every p ∈ S and every period point q ∈ J it follows that lim sup |f n (p) − f n (q)| > 0. n→∞

In a paper with the title “Period Three Implies Chaos” (see [27]) upon which the following is essentially based they have proved the following theorem: Theorem 3.18. If there are three points a, b, c ∈ J with a < b < c such that f (a) = b, f (b) = c and f (c) = a, then f is chaotic in the above sense. In connection with this theorem the following theorem is proved: Theorem 3.19. Under the assumptions of Theorem 3.18 there is, for every k ∈ IN, a point xk ∈ J with f k (xk ) = xk and f j (xk ) 6= xk for all j = 1, . . . , k − 1. For the proof of this theorem we need three lemmata. Lemma 3.20. Let I, J ⊆ IR be two closed genuine intervals and f : J → IR a continuous function with f (I) ⊇ J. Then there is a closed subinterval I1 ⊆ I with f (I1 ) = J. Proof. Let J = [f (p), f (q)] for p, q ∈ I. If p < q, then there is some rb ∈ [p, q] with rb = max{r ∈ [p, q] | f (r) = f (p)} and some sb ∈ [b r, q] with sb = min {s ∈ [b r, q] | f (s) = f (q)} . r, sb], then I1 ⊆ I and f (I1 ) = J. Similar arguments apply If one puts I1 = [b in the case p > q. u t

3.4 Chaos in the Sense of Li and Yorke

173

Lemma 3.21. Let J ⊆ IR be a closed genuine interval and let f : J → J be continuous. Further let (In )n∈IN0 be a sequence of closed genuine intervals In ⊆ J with In+1 ⊆ f (In ) for all n ∈ IN0 . Then there is a sequence (Qn )n∈IN0 of closed intervals Qn ⊆ J with Qn ⊆ Qn−1 ⊆ I0

and

f n (Qn ) = In

for all

n ∈ IN0

which implies that f n (x) ∈ In

for all

n ∈ IN0

and

\

x∈Q=

Qn .

n∈IN0

Proof. If we put Q0 = I0 , then it follows that f 0 (Q0 ) = I0 . Now let Qn−1 for some n ∈ IN be constructed such that Qn−1 ⊆ I0

and

f n−1 (Qn−1 ) = In−1 .

By assumption we have In ⊆ f n (Qn−1 ). Lemma 3.20 with I = Qn−1 , J = In and f n instead of f then implies the existence of a closed interval Qn ⊆ Qn−1 ⊆ I0 with f n (Qn ) = In . The existence of the sequence (Qn )n∈IN0 with the above properties follows by the principle of induction. u t Lemma 3.22. Let I ⊆ J ⊆ IR be closed genuine intervals. Further let g : J → IR be continuous and let I ⊆ g(I). Then there is a point x b ∈ I with g (b x) = x b. Proof. Let I = [β0 , β1 ]. Then choose α0 , α1 ∈ I such that β0 = g (α0 ) and β1 = g (α1 ). Then it follows that g (α0 ) − α0 ≤ 0

and

g (α1 ) − α1 ≥ 0.

Since g is continuous there exists some x b ∈ I with g (b x) − x b = 0 which completes the proof. u t After these preparations we come to the Proof of Theorem 3.19. The assumptions of the theorem imply [b, c] ⊆ f ([a, b])

and

[a, c] ⊆ f ([b, c])

(∗)

by virtue of the intermediate value theorem for continuous functions. Since [b, c] ⊆ [a, c], it follows that [b, c] ⊆ f ([b, c]). By Lemma 3.22 we obtain the existence of some x1 ∈ [b, c] with f (x1 ) = x1 . Therefore the assertion is true for k = 1. Now let k > 1 be given.

174


Then we define a sequence (In )n∈IN0 of intervals In ⊆ J by In = [b, c] for

n = 0, . . . , k − 2,

Ik−1 = [a, b]

and In+k = In

for n = 0, 1, 2, . . .

From (∗) it then follows that In+1 ⊆ f (In ) for all n ∈ IN0 . By Lemma 3.21 there exists a closed interval Qk ⊆ I0 with f k (Qk ) = Ik = I0 . By Lemma 3.22 with g = f k , I = Qk , J = I0 there exists some xk ∈ Qk with f k (xk ) = xk . Without loss of generality we can assume that k 6= 3. Assumption: f j (xk ) = xk for some j ∈ {1, . . . , k − 1}. Because of f j (Qj ) = Ij = I0 for j = 0, . . . , k − 2 it follows that f k−1 (xk ) = f k−j−1 ◦ f j (xk ) = f k−j−1 (xk ) ∈ I0 = [b, c]. On the other hand we have f k−1 (xk ) ∈ f k−1 (Qk−1 ) = Ik−1 = [a, b]. This implies f k−1 (xk ) = b and therefore xk = f k (xk ) = c. This is only possible for k = 3. Therefore the assumption is false for k 6= 3 and Theorem 3.19 is proved. u t This theorem is contained in the Theorem of Sarkovski: Let f : IR → IR be continuous and have a period point of period k. Then f also has, for every l ∈ IN with k B l, a period point with period l where “B” denotes an ordering of the natural numbers by virtue of 2 2 2 {z7 B . .}. B 2 · 3 B 2 · 5 B 2 · 7 B . . . B 2 · 3 B 2 · 5 B 2 · 7 B . . . |3 B 5 B

odd numbers

. . . B 23 · 3 B 23 · 5 B 23 · 7 B . . . B 23 B 22 B 2 B 1. We will not give the proof of this theorem and refer to the book “Introduction to Chaotic Dynamics” by Robert L. Devaney (see [5]). An example for Theorem 3.19 is the quadratic function f (x) = µx(1 − x), x ∈ J = [0, 1]

for µ ≈ 3.839.

One confirms by calculation that f has a period three which is given approximately by a = 0.149888, b = 0.489172, c = 0.959299.


175

Proof of Theorem 3.18. Let M be the set of all sequences M = (Mn )n∈IN of closed intervals with Mn = [a, b]

or

Mn ⊆ [b, c]

and

f (Mn ) ⊇ Mn+1 .

(A.1)

If Mn = [a, b], then n is the square of a natural number and Mn+1 , Mn+2 ⊆ [b, c].

(A.2)

If n is the square of a natural number, then this is not the case for n + 1 and n + 2 so that the requirement Mn+1 , Mn+2 ⊆ [b, c] in (A.2) is redundant. For every M ∈ M let P (M, ¶ the number of those i ∈ {1, . . . , n} with µ n) be 3 Mi = [a, b]. For every r ∈ , 1 let M r = (Mnr )n∈IN be a sequence in M 4 with ¢ ¡ P Mnr , n2 = r. (A.3) lim n→∞ n ¯ µ ¶¾ ½ ¯ 3 Let M0 = M r ¯¯ r ∈ ,1 . Then M0 is uncountable, since M r1 6= M r2 4 for r1 6= r2 . By Lemma 3.21 there exists for every M r ∈ M0 a point xr ∈ [a, c] with for all n ∈ IN. f n (xr ) ∈ Mnr Put

¯ µ ¶¾ ¯ 3 ¯ ,1 . xr ∈ [a, c] ¯ r ∈ 4

½ S=

Then S is uncountable. For every x ∈ S let P (x, n) be the number of those i ∈ {1, . . . , n} for which f i (x) ∈ [a, b]. Certainly we never have f k (xr ) = b for xr ∈ S, since otherwise f k would have a period 3 which because of (A.2) is impossible. This implies

P (xr , n) = P (M r , n)

for all

n ∈ IN

for all

r∈

and therefore ¢ ¡ P xr , n2 % (xr ) = lim =r n→∞ n

µ

¶ 3 ,1 . 4

Assertion: For p, q ∈ S with p 6= q there are infinitely many n ∈ IN such that f n (p) ∈ [a, b]

and

f n (q) ∈ [b, c]

or vice versa.

We can assume that %(p) > %(q). Then it follows that lim (P (p, n) − P (q, n)) = ∞.

n→∞

This implies that there are infinitely many n ∈ IN with (A.4).

(A.4)

176


Since f 2 (b) = a and f 2 is continuous, there exists some δ > 0 such that f 2 (x)
δ. By Assertion (A.4) we conclude that for every pair p, q ∈ S with p 6= q it follows that lim sup |f n (p) − f n (q)| > δ. n→∞

This shows that (3.9) is true. In order to prove (b) we at first show that S does not contain period points. Let p ∈ J be a period point of the period k ∈ IN. Then we distinguish the following two cases: ª © (α) f (p), . . . , f k (p) ∩ [a, b]¡ = ∅. ¢ Then it follows that P p, n2 = 0 for all n ∈ IN and hence %(p) = 0. This shows ¶ p cannot belong to S, since for every xr ∈ S we have µ that 3 ,1 . % (xr ) ∈ 4 ª © (β) f (p), . . . , f k (p) ∩ [a, b] 6= ∅. Let b k be the number of those j ∈ {1, . . . , k} with f j (p) ∈ [a, b]. If we choose in particular n = nl = l · k for l ∈ IN, then it follows that ¡ ¡ ¢ ¢ P p, n2l P p, n2 l2 k b = k = lb k→∞ = n nl lk for l → ∞. Therefore it is impossible that ¢ µ ¡ ¶ P p, n2 3 lim ∈ ,1 n→∞ n 4 which shows that p 6∈ S. Result: S does not contain period points. Now let p ∈ J be a period point and let q ∈ S be given. We at first assume the case (α), i.e., f j (p) 6∈ [a, b] for all j ∈ IN.


177

Now let f j (p) < a Then we put

for some

j ∈ {1, . . . , k}.

ª © δ = min |a − f j (p)| | f j (p) < a .

Since f n (q) ∈ [a, c] for all n ∈ IN, it follows that |f n (p) − f n (q)| > δ

for infinitely many n.

This implies lim sup |f n (p) − f n (q)| > δ. n→∞

Let f j (p) > b

for all

j ∈ {1, . . . , k}.

Then it follows that f n (p) > b for all n ∈ IN. Let us put

ª © δ = min f j (p) − b | j = 1, . . . , k .

Since f n (q) ∈ [a, b] for infinitely many n, it follows that |f n (p) − f n (q)| > δ

for infinitely many n.

This implies lim sup |f n (p) − f n (q)| > δ. n→∞

In case (β) we infer from %(p) = ∞ that lim (P (p, n) − P (q, n)) = ∞.

n→∞

This implies that there are infinitely many n ∈ IN with (A.4). The proof can now be continued as the proof of (3.9). For the proof of (3.10) we at first observe that because of f (b) = c and f (c) = a we can construct intervals [bn , cn ] , n = 0, 1, 2, . . . with the following properties (see Lemma 3.20): ¤ £ ¤ £ (α) [b, c] = b0 , c0 ⊇ b1 , c1 ⊇ . . . ⊇ [bn , cn ] ⊇ . . ., ¢ ¡ (β) f (x) ∈ (bn , cn ) for all x ∈ bn+1 , cn+1 , ¢ ¢ ¡ ¡ (γ) f bn+1 = cn , f cn+1 = bn . ∞ T Let A = [bn , cn ] , b∗ = inf A and c∗ = sup A. Then it follows that n=0

f (b∗ ) = c∗ and f (c∗ ) = b∗ as a consequence of (γ). In order to prove (3.10) we have to make a careful choice of the sequences M r ∈ M. In addition to the above requirements we assume the following:

178


If Mk = [a, b] for some k = n2 and k = (n + 1)2 , then we choose i h Mn2 +(2j−1) = b2n−(2j−1) , b∗

¤ £ Mn2 +2j = c∗ , c2n−2j

and

for j = 1, . . . , n. For the remaining k ∈ IN which are no squares we choose Mk = [b, c]. One can check that these additional requirements are consistent with (A.1) and (A.2) and that for every r ∈ (3/4, 1) one can find a sequence M r ∈ M which satisfies (A.3). µ ¶ ¶ µ 3 3 ∗ From %(r) = r for all r ∈ , 1 it follows for r, r ∈ , 1 the existence of 4 4 ∗ infinitely many n ∈ IN with Mkr = Mkr = [a, b] for k = n2 and k = (n + 1)2 . Now let xr and xr∗ ∈ S be given. Because of bn → b∗ and cn → c∗ for n → ∞, for every ε > 0, there is some N ∈ IN with |bn − b∗ |
N.

= [a, b] for k = n2 and

¤ £ ∗ (xr∗ ) ∈ Mkr = b2n−1 , b∗

¯ ¯ 2 ¯ 2 ¯ for k = n2 + 1 which implies ¯f n +1 (xr ) − f n +1 (xr∗ )¯ < ε. From this it follows that lim inf |f n (xr ) − f n (xr∗ ) | = 0 n→∞

which concludes the proof of Theorem 3.18.

u t

With the aid of the Sarkovski Theorem it is possible to derive from Theorem 3.18 the following Theorem 3.23. Let J ⊆ IR be a finite closed interval and let f : J → J be a continuous function with a period point of period k with k 6= 2i , i ∈ IN. Then f is chaotic in the sense of Li and Yorke. Proof. Because of k 6= 2i , i ∈ IN, k can be represented in the form k = mp with m ∈ IN and p > 2 being a prime number. This implies that f m is p−periodic. By the Theorem of Sarkovski it follows that f m has a period point of period 2 · 3 and hence f 2m has a period point of period 3. By Theorem 3.18 (which is also true without the assumption a < b < c) f 2m is chaotic in the sense of Li and Yorke. This implies that f is also chaotic in the sense of Li and Yorke. u t


179

With the aid of Theorem 3.23 one can also gain a connection between disorderchaos and chaos in the sense of Li and Yorke. For that purpose we need the statement that a continuous mapping f of a finite closed interval J into itself with positive topological entropy has a period point of period k with k 6= 2i , i ∈ IN. This was proved by M. Misiurewicz in [29] and [30], however, with a formally different definition of entropy. An immediate consequence of this statement and Theorem 3.23 is the statement that disorder chaos (for one-dimensional mappings) implies chaos in the sense of Li and Yorke.

180


3.5 Strange (or Chaotic) Attractors This section is essentially based on representations in the book [5] of Devaney. In Section 1.3.3 we have introduced the concept of an attractor which will be slightly modified here. We assume a continuous mapping f : D → IRn , D ⊆ IRn to be given. Definition. A subset A ⊆ D is called an attractor of f , if there exists a subset V ⊆ D with f (V ) ⊆ V and A ⊆ V such that ∞ \

A=

f n (V )

n=0

where f n (V ) = {f n (x) | x ∈ V }

and

A = closure of A.

¡ ¢ Lemma 3.24. If A ⊆ D is an attractor of f , then A is invariant, i.e., f A = A. Proof. On the one hand we have Ã∞ ! ∞ \ \ ¡ ¢ n f A =f f (V ) = f n (V ) ⊇ A, n=0

n=1

on the other hand it follows that ∞ ∞ ∞ \ \ \ ¡ ¢ f A = f n−1 (f (V )) ⊆ f n−1 (V ) = f n (V ) = A. n=1

n=1

n=0

¡ ¢ hence f A = A.

u t

Lemma 3.25. Let A ⊆ D be an attractor of f such that there is some n0 ∈ IN such that f n (V ) is closed for all n ≥ n0 . Then it follows that Lf (x) ⊆ A

for all

x∈V

where Lf (x) is the limit set of the orbit γf (x) (1.42) given by (1.43). Proof. By Proposition 1.16 it follows that ¯ ½ ¾ ¯ n¯ ni Lf (x) = y ∈ IR ¯ ∃ monotone sequence ni → ∞ with y = lim f (x) . ni →∞

3.5 Strange (or Chaotic) Attractors

181

Now let x ∈ V and y ∈ Lf (x) be given. Let ni → ∞ be a monotone sequence with y = lim f ni (x). For every i ∈ IN we have i→∞

xni = f ni (x) ∈ f ni (V ) and it follows that xni → y. By assumption f ni (V ) is closed for sufficiently large i. Because of f n+1 (V ) ⊆ f n (V )

for all

n ∈ IN0

it therefore follows that A=A=

\

f ni (V ).

i∈IN

Further it follows that y ∈ f ni (V ) for all sufficiently large i and therefore y∈

\

f ni (V ) = A = A.

i∈IN

u t

Definition. An attractor A ⊆ D of f is called strange (or chaotic), if f : A → A is topologically conjugated to the shift-mapping σ : Σ → Σ of the metric space Σ of the 0 − 1−sequences (see Section 3.1) into itself. Since the shift-mapping is chaotic in the sense of Devaney (see Theorem 3.6, the Corollary 3.8 leads to the following

Theorem 3.26. If A ⊆ D is a strange attractor of f , then f : A → A is chaotic in the sense of Devaney. Examples: (a) Smale’s Horse-Shoe-Mapping: A classical example of a chaos generating mapping in the plane is Smale’s horse-shoe-mapping fH which can be illustrated by the following picture:

182


6

1

H1 1−α

V0

V1

α

H0 β

1−β

1

The mapping fH maps the square Q = [0, 1]×[0, 1] onto the horse-shoe in such a way that the horizontal strips H0 = [0, 1] × [0, α] and H1 = [0, 1] × [1 − α] are mapped onto the vertical strips V0 = [0, β] × [0, 1] and V1 = [1 − β, 1] × [0, 1] with 0 < β ≤ α < 1/2 by virtue of ¶ ¶  µ µ     f (x, y) = βx, 1 y   f (x, y) = 1 − βx, 1 − 1 y  H H α α α and ,     for 0 ≤ x ≤ 1, 0 ≤ y ≤ α for 0 ≤ x ≤ 1, 1 − α ≤ y ≤ 1 respectively. The horizontal strip between H0 and H1 is mapped in a suitable way onto the circular strip which together with V0 and V1 forms the horseshoe. Obviously we have −1 fH (Q) = Q \ [0, 1] × [α, 1 − α] = H0 ∪ H1 , ¢ ¡ ¢ ¡ −2 −1 fH (Q) = fH (H0 ∪ H1 ) = H01 ∪ H02 ∪ H11 ∪ H12

and in general for every n ∈ IN it follows that   n−1 n−1 2[ 2[ −n (Q) =  H0i ∪ H1i  fH i=1

i=1

where every H0i and H1i is a horizontal strip in H0 and H1 , respectively, which is obtained by removing a horizontal strip of relative length 1 − 2α from H0i−1 and H1i−1 , respectively.


183

If one defines ∞ \

V =

−n fH (Q),

n=1

then V is representable in the form V = [0, 1] × Cα where Cα is a Cantor set which is obtained from [0, 1] by permanently removing open intervals of relative length 1 − 2α. −n (Q) are closed subsets of Q and therefore even compact. This All sets fH implies that V is closed and therefore also compact. Further we have

fH (V ) =

∞ \

∞ ∞ \ \ ¡ −n ¢ 1−n −m fH fH (Q) = fH (Q) = fH (Q) ⊆ V.

n=1

If one defines A =

n=1

∞ T n=0

m=0

n n fH (V ), then it follows that A = A, since all sets fH (V )

are compact and therefore closed. Therefore A ⊆ Q is an attractor of the mapping fH : Q → IR2 . Lemma 3.25 implies LfH (x) ⊆ A

for all

x∈V

where LfH (x) is the limit set of the orbit γfH (x) (1.42) given by (1.43) with fH : V → V . Now let Σ2 be the metric space of twofold infinite 0 − 1−sequences, i.e., n o Σ2 = s = (sj )j∈Z | sj = 0 or 1 with metric d(s, t) =

X 1 |sj − tj |, 2|j| j∈Z

s, t ∈ Σ2 .

In Section 3.2 we have shown that the shift-mapping σ : Σ2 → Σ2 given by σ(s)j = sj+1

for all

j ∈ Z and all s ∈ Σ2

(which is a homeomorphism) is chaotic in the sense of Devaney. Now let us define a mapping S : A → Σ2 by s = S(x), x ∈ A, where for every j ∈ Z we define ( j (x) ∈ V0 , 0, if fH sj = j 1, if fH (x) ∈ V1 .

184


From ( S(x)j+1 =

)

j+1 (x) ∈ V1 , 1, if fH

( =

j+1 (x) ∈ V0 , 0, if fH

j (fH (x)) ∈ V0 , 0, if fH

)

j (fH (x)) ∈ V1 1, if fH

= S (fH (x))j

it follows that σ(S(x)) = S (fH (x))

for all

x ∈ A.

Lemma 3.27. S : A → Σ2 is a homeomorphism. Proof. At first we observe that A can be represented in the form A = V ∩ (Cβ × [0, 1]) where Cβ is a Cantor set which is obtained from [0, 1] by permanently removing open intervals of relative length 1 − 2β. In particular we have A ⊆ V0 ∪ V1 so that because of fH (A) = A the mapping S : A → Σ2 is well defined. Next we prove the injectivity of S : A → Σ2 . Let S(x) = S(y) for x, y ∈ A. Then it follows that for every j ∈ Z that ´ ³ ´ ³ j j j j or fH fH (x) ∈ V0 and fH (y) ∈ V0 (x) ∈ V1 and fH (y) ∈ V1 j Because of the linearity of fH : A → A for all j ∈ Z and the convexity of V0 and V1 it follows for every j ∈ Z and λ ∈ [0, 1] that

f j (λx + (1 − λ)y) ∈ V0

f j (λx + (1 − λ)y) ∈ V1 .

or

However, this is only possible, if λ = 0 or 1 or x = y, since [x, y] 6⊆ A, if x 6= y. This follows from the fact that A = V ∩ (Cβ × [0, 1]) and that Cβ does not contain intervals. In order to prove surjectivity of S : A → Σ2 we consider some s = (. . . , s−n , . . . , s0 , . . . , sn , . . .) in Σ2 . For every n ∈ IN0 we define In (s) =

n \

¢ −j ¡ Vsj . fH

j=−n

Every In , n ∈ IN0 , is non-empty and closed. Further we have In+1 ⊆ In for all n ∈ IN0 which implies that \ I∞ (s) = In (s) ⊆ A n∈IN0

is non-empty, since all In are contained in the compact set Q. Let x ∈ I∞ (s).


185

Then it follows that j (x) ∈ Vsj for all j ∈ IN0 fH

⇐⇒

s = S(x).

Because of the injectivity of S : A → Σ2 it follows that I∞ (s) = {x}. Finally we show the continuity of S. For that purpose we choose some x ∈ A arbitrarily and put (. . . , s−n , . . . , s0 , . . . , sn , . . .) = S(x). Let ε > 0 be given. 1 Then we choose n ∈ IN0 such that n−1 ≤ ε. Now we consider the set of 2 all In (t) for all 22n+1 combinations of t−n , t−n+1 , . . . , tn−1 , tn . These are all different and In (s) belongs to this set. Now we choose δ > 0 such that kx − yk2 < δ and y ∈ A

=⇒

y ∈ In (s).

Then it follows that S(x)i = S(y)i

for

i = −n, . . . , n.

1 This implies d(S(x), S(y)) ≤ n−1 ≤ ε (see Section 3.2) which shows the 2 continuity of S. u t This implies the continuity of the inverse mapping S −1 and shows that σ : Σ2 → Σ2 and fH : A → A are topologically conjugated. The Corollary 3.8 then implies that fH is chaotic in the sense of Devaney. If we replace in the definition of a strange attractor Σ by Σ2 which is not an essential change, then it also follows that A=

∞ \

n fH (V )

with

V =

n=0

∞ \

−n fH (Q)

n=1

is a strange attractor. (b) The Solenoid: We consider the unit circle © ª S 1 = e it | t ∈ [0, 2π] ,

i=

√

−1,

and the unit circle disk © ª B 2 = (x, y) ∈ IR2 | x2 + y 2 ≤ 1 . With these two we define a torus D = S 1 × B 2 . On D we define a continuous mapping f : D → D by µ ¶ ¢ ¡ 1 1 f e it , p = e 2it , p + e 2it for e it ∈ S 1 and p ∈ B 2 . 10 2

186

3 Chaotic Behavior of Autonomous Time-Discrete Systems ∗

Geometrically f can be described as follows: Let e it ∈ S 1 be given. Then we denote by B (t∗ ) the disk o n³ ∗ ´ B (t∗ ) = e it , p | p ∈ B 2 . With this notation we have f (B(t)) ⊆ B(2t). The exact image f (B(t)) is a disk with the center µ ¶ 1 1 . e 2it , e 2it and the radius 2 10 On the whole we obtain the picture

f (B(t))

'$ Il Jl f (B(t + π)) &%

'$ '$ &% &%

B(t)

B(t + π)

The image of D consists of two tori in the interior of D. The image f 2 (D) of f (D) consists of four tori, two of them, respectively, contained in the two tori 1 . In of which f (D) consists. The radii of this these four tori are equal to 100 general we obtain f n (D) ⊆ f n−1 (D) ⊆ . . . ⊆ f (D) ⊆ D

for every n ∈ IN

and every set f k (D) is closed. This implies that \ f n (D) A= n∈IN0

is non-empty and is a closed attractor. It is called a solenoid. For this attractor one can prove the following Theorem 3.28. (a) The set Per(f ) of the period points of f : A → A is dense in A. (b) f is topologically transitive on a suitable compact subset of A.


187

Proof. ¢ ¡ (a) Let x = e it , p0 ∈ A be given. Further let U ⊆ D be any open neighborhood of x. Then there is some δ > 0 and some n ∈ IN0 such that the tube ¢ ª ©¡ C = e it , p ∈ f n (A) | |t − t0 | < δ is contained in U . Since the period points of the mapping e it → e 2it , t ∈ 2kπ for k = [0, 2π], are of the form e itmk with m ∈ IN and tmk = m 2 −1 m 1 0, . . . , 2 − 1, they are lying dense in S . Therefore there is some t∗ ∈ {t ∈ [0, 2π] | |t − t0 | < δ} with f m (B (t∗ )) ⊆ B (t∗ ). We can assume¡ m ∈¢ IN to be so large that f m (B (t∗ ) ∩ C) ⊆ B (t∗ ) ∩ C. Since f m = f m e it , p is ¡ ∗ ¢ contractive with respect to p, there exists a point e it , p∗ ∈ B (t∗ ) ∩ C with ¡ ¢ ¡ ¢ ∗ ∗ f m e it , p∗ = e it , p∗ ∈ A which completes the proof. We will not give a direct proof of (b). The statement (b) will be the result of the following considerations: Let g : S 1 → S 1 be defined by ¡ ¢ g e it = e 2it ,

t ∈ [0, 2π].

Further we define the sequence space Σ=

¾ µ ¶¯ ½ ¯ t t Θt = e it , e i 2 , . . . , e i 2n , . . . ¯¯ t ∈ [0, 2π] .

If we define a metric in Σ by d (Θt , Θs ) =

¯ ¯ ∞ X s ¯ 1 ¯¯ i tj 2 − e i 2j ¯ , e ¯ 2j ¯ j=0

Θt , Θs ∈ Σ,

then Σ becomes a metric space. On Σ there exists a natural mapping ¶ µ µ ¶ ¶ µ t t t t i4 i2 it i 2 it i 2 σ e ,e ,e ,... = g e ,e ,e ,... ,

t ∈ [0, 2π],

which maps Σ continuously into itself and whose inverse mapping is given by ¶ µ ¶ µ t t t t t σ −1 e it , e i 2 , e i 4 , . . . = e i 2 , e i 4 , e i 8 , . . . , and is also continuous. Therefore σ : Σ → Σ is a homeomorphism.

t ∈ [0, 2π],

188


If we define a continuous mapping ϕ : S 1 → Σ by µ ¶ t t ¡ ¢ t ∈ [0, 2π] ϕ e it = e it , e i 2 , e i 4 , . . . , then it follows that ϕ is surjective. Further it follows that µ ¶ µ ¶ t t t ¡ it ¢ i4 it i 2 2it it i 2 and σ ◦ ϕ e = σ e ,e ,e ,... = e ,e ,e ,... µ ¶ t ¡ ¢ ¡ ¢ ¡ ¢ for all t ∈ [0, 2π], ϕ ◦ g e it = ϕ e 2it = e 2it , e it , e i 2 , . . . = σ ◦ g e it hence ϕ◦g = σ ◦ϕ. Since g is chaotic in the sense of Devaney (see Section 3.2), it follows from Theorem 3.7 that σ : Σ → Σ is also chaotic in the sense of Devaney. Now let π : D → S 1 be the natural projection, i.e., ¢ ¡ it ¢ ¡ for all e , p ∈ D. π e it , p = e it The mapping f : D → D is a homeomorphism of D on f (D) which implies, because of f (A) = A, that f : A → A is a homeomorphism of A on A. Therefore the mapping S : A → Σ which is defined by ¢ ¡ ¢ ¢ ¡ ¡ x ∈ A, S(x) = π(x), π f −1 (x) , π f −2 (x) , . . . , is well defined and it follows that ¢ ¢ ¡ ¡ S(f (x)) = π(f (x)), π(x), π f −1 (x) , . . . = σ(S(x))

for all x ∈ A,

hence S ◦ f = σ ◦ S. In order to further investigate the mapping S we at first analyze the set A. By definition it can be represented in the form k

A=

∞ [ 2 \

Tj (k)

k=0 j=1

ª © where Tj (k), for every k ∈ IN0 and j ∈ 1, . . . , 2k is a torus in the interior 1 of D with cross section radius k . Further we have 10 for every k ∈ IN0 and j = 1, . . . , 2k ª © and for every k ∈ IN0 and j ∈ 1, . . . , 2k there exists some f (Tj (k)) ⊆ Tj (k)

© ª i = i(j, k) ∈ 1, . . . , 2k+1

with

Ti (k + 1) ⊆ Tj (k).


189

ª © Let J = (jk )k∈IN0 be a sequence with jk ∈ 1, . . . , 2k and AJ =

∞ \

Tjk (k).

k=0

This set is a (non-empty) compact subset of A and it follows that f (AJ ) =

∞ \ k=0

f (Tjk (k)) ⊆

∞ \

Tjk (k) = AJ .

k=0

Further, for every t ∈ [0, 2π] the set AJ ∩ B(t) consists of one point which implies that the mapping S : AJ → Σ is bi-unique, continuous and surjective. Because of the compactness of AJ the mapping S : AJ → Σ is a homeomorphism which implies that the mappings f : AJ → AJ and σ : Σ → Σ are topologically conjugated. From the Corollary 3.8 we infer the Theorem 3.29. The mapping f : AJ → AJ is chaotic in the sense of Devaney. This implies statement (b) of Theorem 3.28 with the subset AJ of A.

u t

190


3.6 Bibliographical Remarks The abstract definition of a dynamical system in Section 1.1 goes back to G. D. Birkhoff (see [3]). For metric spaces X it can be found in the essay “What is a Dynamical System?” by G. R. Sell in [8] and for uniform spaces X in [33]. It is also shown there how a non-autonomous system of differential equations x˙ = f (x, t)

with

f ∈ C (W × IR, IRn ) ,

W ⊆ IRn open, can be assigned a dynamical system. The Section 1.2.1 about elementary properties of time continuous dynamical systems are also taken from the above mentioned essay of G. R. Sell. This also gives a short account of the Poincaré-Bendixon theory for systems in the plane which we have extensively represented in Section 1.2.2. The Section 1.2.3 about stability of time-continuous dynamical systems by applying the direct method of Lyapunov can also be found in [14]. The application to general predator-prey models in Section 1.2.4 is based on the two papers [18] and [20]. Section 1.3 is taken from [22] with slight modifications and corrections. The localization of limit sets with the aid of Lyapunov functions in Section 1.3.2 has been first represented by J. T. La Salle in [25] and can also be found in [14] and [22]. The stability results given in Section 1.3.3 generalize a standard result on the stability of fixed points to be found in [22], [23] (see Theorem 1.3). Theorem 1.5 in Section 1.3.4 coincides with Satz 5.4 in [23], if we choose as normed linear space the n−space IRn equipped with any norm. The results on linear systems following the Corollary of Theorem 1.6 in Section 1.3.5 and also those for non-autonomous linear systems in Section 1.3.10 are based on Section 4.2 in [23]. The stability results in Section 1.3.9 generalize those of Section 1.3.3 and are taken from [16]. Lyapunov’s method has also been applied to non-autonomous systems in [1] and [24] in order to investigate stability of fixed points. Instead of one Lyapunov function a sequence of such is used. Following the example of G. R. Sell in [8] and [33] the controlled system (2.1) can also be assigned a dynamical system. For that purpose we assume that the functions fi : IRn+m → IRn for i = 1, . . . , n are continuous and bounded. ¡ Every function u ∈ C (IR, IRm ) we now assign a function f u ∈ Cb IRn+1 , IRn ) =vector space of continuous and bounded functions g : IRn × IR → IRn by defining f u (x, t) = f (x, u(t)), (x, t) ∈ IRn+1

3.6 Bibliographical Remarks

191

and put X = {f u | u ∈ C (IR, IRm )} . Then X is a metric space with metric d (f u1 , f u2 ) =

sup (x,t)∈IRn+1

kf u1 (x, t) − f u2 (x, t)k2 ,

f u1 , f u2 ∈ X,

where || · ||2 denotes the Euclidian norm in IRn . The system (2.1) can then be written in the form x(t) ˙ = f u (x(t), t), t ∈ IR. (2.1’) By the assumption made in Section 2.1.1 there exists, for every u ∈ C (IR, IRm ) and every point x0 ∈ IRn , a solution x = ϕu (x0 , t) ∈ C 1 (IR, IRn ) of (2.1’) with ϕu (x0 , 0) = x0 . With the aid of this solution we now define a mapping π : IRn × X × IR → IRn × X by x ∈ IRn , f u ∈ X, τ ∈ IR,

π (x, f u , τ ) = (ϕu (x, τ ), fτu ) where fτu (x, t) = f u (x, t + τ )

for all

(x, t) ∈ IRn+1 .

Obviously we obtain π (x, f u , 0) = (x, f u )

for all

(x, f u ) ∈ IRn × X,

i.e., the identity property holds true. If one puts ϕ(τ ) = ϕu (x, τ )

and

ψ(τ ) = ϕu (τ + σ)

n

for x ∈ IR , τ, σ ∈ IR, then it follows that ˙ ) = f u (ψ(τ ), τ ) ψ(τ σ

for τ ∈ IR and ψ(0) = ϕ(σ) = ϕi (x, σ).

This implies ¢ ¡ ¢ ¡ π (x, f u , τ + σ) = ϕu (τ + σ), fτu+σ = ψ(τ ), fτu+σ = π (ϕu (x, σ), fσu , τ ) = π (π (x, f u , σ) , τ ) , i.e., the semigroup property also holds true. For the proof of the continuity of π : IRn × X × IR → IRn × X we refer to [33]. This shows that the mapping π defines a dynamical system in the sense of Section 1.1. The Section 2.1.1–2.1.5 are taken from the book [22] and the Section 2.1.6 about time-minimal controllability of linear systems is taken from the book [13].

192


The Sections 2.2 and 2.3 about controllability of time-discrete dynamical systems are taken from the book [22] and Chapter 3 about chaotic behavior of dynamical systems can also be found in the book [14]. The paradigmatic example of a chaotic mapping is the shift-mapping in the space Σ of the infinite 0 − 1−sequences for which not only the set of periodic points is dense in the space Σ but also there exists an orbit which is dense in Σ. From this one easily derives the topological transitivity (see Theorem 3.6). In [11] U. Kirchgraber calls the chaos that is generated by the shift-mapping “chaos in the sense of coin casting”. This can be brought together with the chaotic behavior of one-dimensional continuous mappings F : IR → IR in the following way: Let us assume that there exist two closed finite intervals A0 and A1 with A0 ∩ A1 = ∅, F (A0 ) ⊇ A0 ∪ A1

and

F (A1 ) ⊇ A0 ∪ A1 .

Further we assume that F : IR → IR is differentiable and there exists a number λ > 1 such that |F 0 (x)| ≥ λ

for all

x ∈ A0 ∪ A1 .

If we put, for every k ∈ IN0 , ¯ ª © As0 s1 ...sk = x ∈ As0 ¯ F (x) ∈ As1 , . . . , F k (x) ∈ Ask where sj = 0 or 1 for j = 0, . . . , k, then it follows that As0 ⊇ As0 s1 ⊇ . . . ⊇ As0 s1 ...sk . In [11] it is shown that As =

\

As0 s1 ...sk

k∈IN0

is non-empty and consists of exactly one point x = x(s) ∈n A0 ∪ A1¯, s = ¯ (sj )j∈IN0 . This gives rise to a mapping h : Σ → A0 ∪A1 , Σ = (sj )j∈IN0 ¯ sj = o 0 or 1 , with h(s) = x(s) which can be shown to be continuous, if one introduces a metric in Σ, for instance, by defining d(s, s0 ) = 2−j

where j is the smallest index with sj 6= s0j .

If one defines in A0 ∪ A1 the set A = h(Σ), then it follows that F (A) ⊆ A and F ◦ h(s) = h ◦ σ(s)

for all

s ∈ Σ.

Theorem 3.7 therefore implies that F : A → A is chaotic in the sense of Devaney.

3.6 Bibliographical Remarks

193

An example for this situation is the quadratic mapping F (x) = µx(1 − x), x ∈ IR

for

µ>2+

√ 5

in Section 3.2 which also has been discussed in the book of Devaney. The definition of topological entropy as a measure for chaos has been taken from the paper [28] by A. Mielke. In this paper also Theorem 3.16 is proved which states that the topological entropy is invariant under topological conjugacy. The formulae which we have derived for the topological entropy of the shiftmapping σ on Σ and on a suitable subset Σ A of Σ in Section 3.3 are a special case of a general formula for so called “subshifts” which is given in [28] in Theorem 2.1. Also the connection with a one-dimensional mapping is given there a more general representation than in this book. In Section 3.4 we have shown that disorder-chaos for one-dimensional mappings implies chaos in the sense of Li and Yorke. The reverse statement is false in general. J. Smital shows in [34] that there are chaotic mappings in the sense of Li and Yorke that have a vanishing topological entropy. In order to prove that the two chaos concepts are equivalent one has to sharpen the concept of Li and Yorke in a suitable way (see the paper [10] by Janková and Smital). In Section 3.3.2 we have shown that the topological entropy of the shiftmapping σ : Σ → Σ is equal to ln 2 and therefore positive so that it generates disorder chaos. This implies that every one-dimensional continuous mapping f : J → J of a finite closed interval J into itself is chaotic in the sense of Li and Yorke, if it is topologically conjugated with the shift-mapping σ : Σ → Σ. The representation of strange attractors in Section 3.5 is closely related to that of the book [5] of Devaney. The Appendix A is the content of the paper [19].

A A Dynamical Method for the Calculation of Nash-Equilibria in n−Person Games

A.1 The Fixed Point Theorems of Brouwer and Kakutani

The fixed point theorem of Kakutani is a generalization of Bouwer’s fixed point theorem which says that every continuous mapping of a convex and compact subset of an n−dimensional Euclidean space into itself has at least one fixed point. For the formulation of Kakutani’s fixed point theorem we need the concept of set-valued mappings. For this purpose we consider a non-empty subset K ⊆ IRn (equipped with the Euclidean norm) and a mapping F of K into the power set 2K of K which assigns to every point x ∈ K a non-empty subset F (x) ⊆ K. A point x b ∈ K is called a fixed point of F , if x b ∈ F (b x) holds true. If in particular F is a point-map, i.e., F (x), for every x ∈ K, is a one-point set, then this definition coincides with the normal definition of a fixed point. Definition. A mapping F : K → 2K is called upper semi-continuous in x ∈ K, if for every sequence (xk )k∈IN in K with xk → x and every every sequence (yk )k∈IN in K with yk ∈ F (xk ) for all k ∈ IN and yk → y it follows that y ∈ F (x). If F : K → 2K is a point-map, i.e., F : K → K, then F is upper semicontinuous in x ∈ K, if and only if it is continuous in x ∈ K. After these preparations we formulate the Fixed Point Theorem of Kakutani: Let K ⊆ IRn be a compact and convex subset and let F : K → 2K be a mapping into the set of the compact and convex subsets of K which is upper semi-continuous in every x ∈ K, then F possesses a fixed point, i.e., there is some x b ∈ K with x b ∈ F (b x). S. Pickl and W. Krabs, Dynamical Systems: Stability, Controllability and Chaotic Behavior, 195 DOI 10.1007/978-3-642-13722-8_4, © Springer-Verlag Berlin Heidelberg 2010

196

A A Dynamical Method for the Calculation of Nash-Equilibria

If in particular F is a mapping of K into K which is continuous in every x ∈ K, then F has a fixed point x b ∈ K with x b = F (b x) which shows that the fixed point theorem of Kakutani implies the fixed point theorem of Brouwer. For the proof of the fixed point theorem of Kakutani we refer to [4].

A.2 Nash Equilibria as Fixed Points of Mappings

197

A.2 Nash Equilibria as Fixed Points of Mappings The starting point of our investigation is a non-cooperative n−person game with strategy Q sets Si , i = 1, . . . , n, each for every player Pi , and pay-off n functions Φi : i=1 Si = S → IR for i = 1, . . . , n. Every player Pi endeavors to maximize the pay-off Φi (s), s ∈ S. This, however, is simultaneously in general impossible. So the players have to compromise. This gives rise to the following definition of a so called Nash equilibrium: An n−tupel (b s1 , . . . , sbn ) of strategies sbi ∈ Si for i = 1, . . . , n is called a Nash equilibrium, if for all i = 1, . . . , n it is true that s1 , . . . , sbn ) ≥ Φi (b s1 , . . . , sbi−1 , si , sbi+1 , . . . , sbn ) Φi (b

for all

si ∈ Si . (A.1)

Interpretation: If a player deviates from his strategy in a Nash equilibrium whereas the other players stick to their strategies, his pay-off can at most diminish. In the following, we are concerned with finding a method to calculate Nash equilibria. The general idea behind this is to find a mapping f : S → S such that every fixed point sb ∈ S with f (b s) = sb is a Nash equilibrium. If such a mapping has been found, it is natural to perform an iteration method by choosing s0 ∈ S and defining a sequence (sk )k∈IN0 in S via sk+1 = f (sk ) = f k+1 (s0 ) ,

k ∈ IN0 .

(A.2)

If this sequence converges to some sb ∈ S and if f : S → S is continuous, then it follows that sb = f (b s), i.e., sb is a fixed point of f and therefore a Nash equilibrium. The definition of a Nash equilibrium can be used in order to find a function of this kind. For that purpose we make the following assumption: For every n−tupel (b s∗1 , . . . , sb∗n ) ∈ S and every i = 1, . . . , n there is exactly one sei ∈ Si with ¡ ¢ ¡ ¢ Φi s∗1 , . . . , s∗i−1 , sei , s∗i+1 , . . . , s∗n ≥ Φi s∗1 , . . . , s∗i−1 , si , s∗i+1 , . . . , s∗n for all si ∈ Si . (A.3) Under this assumption we then define a mapping f = (f1 , . . . , fn ) : S → S via fi (s∗1 , . . . , s∗n ) = sei

for i = 1, . . . , n

(A.4)

with sei ∈ Si being the unique solution of (A.3). It is obvious that sb ∈ S is a fixed point of f , if and only if sb is a Nash equilibrium. In [22] we have shown that f is a continuous mapping of S into S, if in addition to the above assumption the following two assumptions are satisfied:

198

(1) (2)


The strategy sets Si , i = 1, . . . , n, are convex and compact subsets of some IRmi . The pay-off functions Φi : S → IR are continuous.

Since the set S ⊆ IRm1 +...+mn is convex and compact, Brouwer’s fixed point theorem implies that f has at least one fixed point. Further the assumptions (1) and (2) guarantee that, for every n−tupel s∗ ∈ S, there exists an n−tupel se ∈ S such that (A.3) is satisfied. This shows that the iteration (A.2) can be performed without the uniqueness assumption on se ∈ S. In this case we define the set-valued mapping f of S into the power set 2S of S which assigns to every s∗ ∈ S the set f (s∗ ) of all se ∈ S such that (A.3) is satisfied. Obviously it follows that sb ∈ S is a Nash equilibrium, if and only if sb ∈ f (b s). By the assumptions (1) and (2), for every s∗ ∈ S the set f (s∗ ) is non-empty and compact. ¡ ¢ ¡ ¢ Now let sk k∈IN and sek k∈IN be sequences with ¡ ¢ sek ∈ f sk , and

se = lim sek k→∞

for some

s∗ = lim sk k→∞

s∗ ∈ S

and se ∈ S.

Then we have, for every k ∈ IN ¡ ¢ ¡ ¢ Φi sk1 , . . . , ski−1 , seki , ski+1 , . . . , skn ≥ Φi sk1 , . . . , ski−1 , si , ski+1 , . . . , skn for all si ∈ Si and i = 1, . . . , n. This implies ¡ ¢ ¡ ¢ Φi s∗1 , . . . , s∗i−1 , sei , s∗i+1 , . . . , s∗n ≥ Φi s∗1 , . . . , s∗i−1 , si , s∗i+1 , . . . , s∗n for all si ∈ Si and i = 1, . . . , n, i.e., se ∈ F (s∗ ). This shows that the mapping f is upper semi-continuous in every s∗ ∈ S. If in addition to the (1) and (2)¢we assume that, for every s∗ ∈ S, ¡ ∗ assumptions ∗ ∗ the functions Φi s1 , . . . , si−1 , ·, si+1 , . . . , s∗n are concave, then for every s∗ ∈ S the set f (s∗ ) is convex. Summarizing we conclude that under the assumptions (1) and (2) and the last assumption the mapping f : S → 2S defined by (A.3) maps S into the set of convex and compact subsets of S and is upper semi-continuous in every s∗ ∈ S. By Kakutani’s fixed point theorem it follows that f has a fixed point, i.e., there exists some sb ∈ S with sb ∈ f (b s) which is a Nash equilibrium.


199

The sequence (sk )k∈IN0 defined by (A.2) now has to be replaced by a sequence ¡ k¢ s k∈IN in S which is defined as follows: 0

If sk ∈ S is given for some k ∈ IN0 , then we determine, for every i = 1, . . . , n, ∈ Si with some sk+1 i ¡ ¢ , ski+1 , . . . , skn Φi sk1 , . . . , ski−1 , sk+1 i ¡ ¢ for all si ∈ Si . (A.5) ≥ Φi sk1 , . . . , ski−1 , si , ski+1 , . . . , skn ¢ ¡ and put sk+1 = sk+1 . , . . . , sk+1 n 1 If then there exists some sb ∈ S such that sb = lim sk , it follows from (A.5) k→∞

that (A.1) is satisfied, i.e., sb is a Nash equilibrium.

Now we assume in addition to the assumptions (1) ¡ ¢ and (2) that for every s∗ ∈ S the functions Φi s∗1 , . . . , s∗i−1 , ·, s∗i+1 , . . . , s∗n : Si → IR, i = 1, . . . , n, are concave and Gateaux-differentiable. Then sb ∈ S is a Nash equilibrium, if and only if T

T

s) sbi ≥ ∇si Φi (b s) si ∇si Φi (b

for all si ∈ Si

and i = 1, . . . , n (A.6)

where ∇si Φi (·) is the gradient of Φi with respect to si . The iteration procedure (A.5) can therefore be replaced by the following: If sk ∈ S is given for some k ∈ IN0 , then we determine, for every i = 1, . . . , n, ∈ Si with some sk+1 i ¡ ¢T ¡ ¢T ∇si Φi sk sk+1 ≥ ∇si Φi sk si i

for all

si ∈ Si

(A.7)

¢ ¡ and put sk+1 = sk+1 . , . . . , sk+1 n 1 If, for every i = 1, . . . , n, the mapping ∇si Φi : S → IRmi is continuous and if there exists some sb = lim sk , sb ∈ S, then it follows from (A.7) that (A.6) is k→∞

satisfied, i.e., sb is a Nash equilibrium. Special cases: (a) Let, for every i = 1, . . . , n, ¯   ¯ mi   X ¯ sij = 1 . Si = si ∈ IRmi ¯¯ sij ≥ 0 for j = 1, . . . , mi and   ¯ j=1 Then, for every s ∈ S, it follows that ∇si Φi (s)T si ≤

max

j=1,...,mi

Φisij (s)

for

i = 1, . . . , n.

200


This implies that sb ∈ S is a Nash equilibrium, if and only if T

s) sbi = ∇si Φi (b

max

j=1,...,mi

Φisij (b s)

for i = 1, . . . , n.

Now we define, for every s ∈ S, ¢ ¡ ϕij (s) = max 0, Φisij (s) − ∇si Φi (s)T si , j = 1, . . . , mi , i = 1, . . . , n. Then it follows that sb ∈ S is a Nash equilibrium, if and only if s) = 0 ϕij (b

for all j = 1, . . . , mi

and i = 1, . . . , n.

(A.8)

Now we define a mapping f = (f1 , . . . , fn ) : S → S via Ã ! mi X 1 k fi (s)k = ϕij (s) ej , sik + mi P j=1 1+ ϕij (s) | {z } j=1 ϕik (s)

(

k = 1, . . . , mi , i = 1, . . . , n, )

(A.9)

0 for k = 6 j, . Since f is a continuous mapping from S 1 for k = j. into S. Brouwer’s fixed point theorem implies that there exists a fixed point sb ∈ S of f . where ekj =

Assertion: sb ∈ S is a fixed point of f , i.e., sb = f (b s), if and only if (A.8) holds true. Proof. 1) If (A.8) holds true for some sb ∈ S, then sb is a fixed point of f . 2) Conversely let sb = f (b s). For every i = 1, . . . , n there exists at least one k ∈ {1, . . . , mi } with sbik > 0. We choose k0 ∈ {1, . . . , mi } such that s) = min {Φisik (b s)| sbik > 0} Φisik0 (b

and conclude that

ϕik0 (b s) = 0.

s) = 0 for all k = 1, . . . , mi , since otherThis, however, implies that ϕik (b wise sbik0 < sbik0 would follow which is impossible. ¡ ¢ Therefore, if one defines, starting with some s0 ∈ S, a sequence sk k∈IN 0 ¢ ¡ in S via sk+1 = f sk and does this sequence converge to some sb ∈ S, then sb is a fixed point of f and hence a Nash equilibrium. u t Remark. The construction of the mapping f is the same as that which has been given by J. Nash in [31] in order to prove the existence of Nash equilibria for the mixed extension of an n−person game with finite strategy sets.


201

(b) Let, for every i = 1, . . . , n, Si = {si ∈ IRmi | ||si ||2 ≤ 1} where || · ||2 denotes the Euclidean norm. Then we define a mapping f = (f1 , . . . , fn ) : S → S via    if ∇si Φi (s) = Θmi ,   Θmi for i = 1, . . . , n. fi (s) = ∇si Φi (s)  if ∇si Φi (s) 6= Θmi .    k∇si Φi (s)k2 This mapping is continuous. Therefore Brouwer’s fixed point theorem implies that f has at least one fixed point sb = f (b s) and for every such it follows T

T

s) si ≤ ||∇si Φi (b s) ||2 = ∇si Φi (b s) sbi ∇si Φi (b for all si ∈ Si and i = 1, . . . , n. which shows that sb is a Nash equilibrium.

¡ ¢ Therefore, if one defines, starting with some s0 ∈ S, a sequence sk k∈IN0 ¡ ¢ in S via sk+1 = f sk and does this sequence converge to some sb ∈ S, then sb is a fixed point of f and hence a Nash equilibrium.

202


A.3 Bi-Matrix Games We consider 2−person games with finite strategy sets which are played with mixed strategies so that their strategy sets are given by ¯  ¯ m  X ¯ S1 = s ∈ IRm ¯¯ sj ≥ 0 for j = 1, . . . , m and sj = 1   ¯ j=1  

and ¯ ) n ¯ X ¯ t ∈ IR ¯ tk ≥ 0 for k = 1, . . . , n and tk = 1 . ¯

( S2 =

n

k=1

Thus S1 and S2 are convex and compact in IRm and IRn , respectively. The pay-off functions are given by Φ1 (s, t) = sT A t

and Φ2 (s, t) = sT B t

for (s, t) ∈ S = S1 × S2

where A and B are given m × n−matrices. Obviously Φ1 and Φ2 are continuous on S and for every (s∗ , t∗ ) ∈ S the functions Φ1 (s, t∗ ) = sT A t∗ , s ∈ S1 , and Φ2 (s∗ , t) = s∗ T A t, t ∈ S2 , are linear, hence concave, and Gateaux-differentiable with gradients ∇Φ1 (s, t∗ ) = A t∗ , s ∈ S1

and

∇Φ2 (s∗ , t) = B s∗ , t ∈ S2 ,

which are continuous as functions on S. So we are in the situation of special case (a) in Section A.3. the functions Φij : S → IR defined there for j = 1, . . . , mi and i = 1, . . . , m are here given by ¡ ¢ for j = 1, . . . , m ϕ1j (s, t) = max 0, eTj A t − sT A t and

¡ ¢ ϕ2k (s, t) = max 0, sT B ek − sT B t

for

k = 1, . . . , n

where ej and ek is the j−th and the k−th unit vector in IRm and IRn , respectively. ¡ ¢ Further sb, b t ∈ S is a Nash equilibrium, if and only if ¡ ¢ t = 0, j = 1, . . . , m ϕ1j sb, b

and

¡ ¢ ϕ2k sb, b t = 0, k = 1, . . . , n. (A.10)

A.3 Bi-Matrix Games

203

The mapping f defined by (A.9) here reads f = (f1 , f2 ) : S → S where f1 (s, t)j =

sj + ϕ1j (s, t) , m P 1+ ϕ1i (s, t)

j = 1, . . . , m,

tk + ϕ2k (s, t) , n P 1+ ϕ2j (s, t)

k = 1, . . . , n.

i=1

and f2 (s, t)k =

j=1

¡ ¢ As in Section A.3 this mapping has a fixed point sb, b t ∈ S which must satisfy (A.10) and therefore is a Nash equilibrium (see Section A.3). An example: Let m = 2, n = 3 and µ A=

¶ 5 3 1 , 6 4 2

µ B=

¶ 4 5 6 . 3 2 1

Then we obtain eT1 A t = 5t1 + 3t2 + t3 , eT2 A t = 6t1 + 4t2 + 2t3 , eT3 A t = 6t1 + 4t2 + 2t3 − s1 , hence eT1 A t − sT A t = −1 + s1 ≤ 0, and eT2 A t − sT A t = s1 > 0,

if s1 > 0.

This implies and ϕ12 (s, t) = s1 > 0 ϕ11 (s, t) = 0 for all (s, t) ∈ S with 0 < s1 ≤ 1 and further f1 (s, t)1 =

s1 , 1 + s1

f1 (s, t)2 =

1 = 1 − f1 (s, t)1 . 1 + s1

204


Next we obtain sT B e1 = 4s1 + 3s2 = 3 + s1 , sT B e2 = 5s1 + 2s2 = 2 + 3s1 , sT B e3 = 6s1 + s2 = 1 + 5s1 , sT B t = 3t1 + 2t2 + t3 + s1 (t1 + 3t2 + 5t3 ) which implies sT B e1 − sT B t = 3 − s1 (2t2 + 4t3 ) − 3t1 − 2t2 − t3 = 3 − 3t1 − 2 (s1 + 1) t2 − (4s1 + 1) t3 1 >0 if s1 < , 2 sT B e2 − sT B t = 2 + s1 (2t1 − 2t3 ) − 3t1 − 2t2 − t3 < 2 + t1 − t3 − 3t1 − 2t2 − t3 =0

if

t1 > t3

and s1
0 for all

and

(s, t) ∈ S

ϕ22 (s, t) = ϕ23 (s, t) = 0 1 and t1 > t3 . with s1 < 2

We therefore have t1 + ϕ21 (s, t) = 1 − f2 (s, t)2 − f2 (s, t)3 , 1 + ϕ21 (s, t) t2 , f2 (s, t)2 = 1 + ϕ21 (s, t) t3 1 for all (s, t) ∈ S with s1 < and t1 > t3 . f2 (s, t)3 = 1 + ϕ21 (s, t) 2

f2 (s, t)1 =

A.3 Bi-Matrix Games

205

¢ ¡ Hence, if one chooses ¡s0 , t0 ¢∈ U = {(s, ¡ t) ∈ S | 0¢< s1 ¡< 1/2 ¢ and t1 > t3 } and defines a sequence sk , tk k∈IN via sk+1 , tk+1 = f sk , tk then because 0 of f1 (s, t)1 < s1 ,

f2 (s, t)2 < t2 ,

f2 (s, t)3 < t3

⇓ f2 (s, t)1 = 1 − f2 (s, t)2 − f2 (s, t)3 > 1 − t2 − t3 = t1 for all (s, t) ∈ U it follows that ¡ k k¢ s ,t ∈ U

for all

k ∈ IN0 .

Further it follows that sk1 → 0,

sk2 → 1,

tk1 → 1,

tk2 → 0,

tk3 → 0,

¢ ¡ hence sk , tk → (e2 , e1 ) ∈ U and (e2 , e1 ) is a Nash equilibrium. This example is a special case of the following general situation: Let U ⊆ S be such that there exists some j0 ∈ {1, . . . , m} and some k0 ∈ {1, . . . , n} with  ϕ1j (s, t) = 0 for all j 6= j0 , ϕ1j0 (s, t) > 0,   ϕ2k (s, t) = 0 for all k 6= k0 , ϕ2k0 (s, t) > 0   for all (s, t) ∈ U. Then it follows that

f1 (s, t)j0

sj

for all j 6= j0 , 1 + ϕ1j0 (s, t) X sj + ϕ1j0 (s, t) =1− = 0 f1 (s, t)j , 1 + ϕ1j0 (s, t)

f1 (s, t)j =

j6=j0

for all k 6= k0 , 1 + ϕ2k0 (s, t) X tk + ϕ2k0 (s, t) =1− = 0 f2 (s, t)k . 1 + ϕ2k0 (s, t)

f2 (s, t)k = f2 (s, t)k0

tk

k6=k0

This implies  f1 (s, t)j < sj for all j 6= j0 ⇒ f1 (s, t)j0 > sj0 ,   6 k0 ⇒ f2 (s, t)k0 > tk0 f2 (s, t)k < tk for all k =   for all (s, t) ∈ U. Assumption: f (U ) ⊆ U .

(A.11)

206


¢ ¡¡ ¢¢ ¡ If we now define, starting with some s0 , t0 ∈ U , a sequence sk , tk k∈IN0 ¡ ¢ ¡ ¢ in U via sk+1 = f1 sk , tk , tk+1 = f2 sk , tk , then it follows from (A.11) ¡ ¢ ¡ ¢ that sk → sb and tk → b t with sb, b t = f sb, b t ∈ U. ¡ ¢ This implies that sb, b t is a Nash equilibrium. From this it follows in particular that eTj0 A b t − sT A b t=

n X

aj0 k b tk −

k=1

= =

m X

Ã sbj

j=1 m X

m X

n X

sbj

j=1 n X

k=1

aj0 k b tk −

k=1

sbj

j=1 j6=j0

n X

ajk b tk

n X

! ajk b tk

k=1

(aj0 k − ajk ) b tk ≤ 0.

k=1

Assumption 1: aj0 k − ajk > 0

for all j 6= j0

and k = 1, . . . , n.

Then it follows that sbj = 0

for all

j 6= j0 ,

hence

sbj0 = 1 and therefore

sb = ej0 .

This in turn implies eTj0 B ek0 − eTj0 B b t = bj0 k0 −

n X

bj0 k b tk =

k=1

n X

(bj0 k0 − bj0 k ) b tk ≤ 0.

k=1 k6=k0

Assumption 2: bj0 k0 − bj0 k > 0

for all

k 6= k0 .

Then it follows that b tk = 0

for all

k 6= k0 ,

hence b tk0 = 1

and therefore b t = ek0 .

In the above example Assumption 1 is satisfied for j0 = 2 and Assumption 2 for j0 = 2 and k0 = 1. Next we consider symmetric bi-matrix games ( ) in which m = n, hence S1 = n P S2 = S = s ∈ IRn | sj ≥ 0 and sj = 1 , and A = B T . Then, one can j=1

show that there exists a Nash equilibrium of the form (b s, sb) with sb ∈ S which is characterized by the condition sbT A sb = max eTj A sb. j=1,...,n

(A.12)

A.3 Bi-Matrix Games

If we define functions ϕj : S → IR by ¡ ¢ ϕj (s) = max 0, eTj A s − sT A s

for

207

j = 1, . . . , n,

then condition (A.12) is equivalent to ϕj (b s) = 0

for all

j = 1, . . . , n.

(A.13)

We further define a mapping f = (f1 , . . . , fn ) : S → S via fj (s) = 1+

1 n P

(sj + ϕj (s))

for

j = 1, . . . , n,

(A.14)

ϕk (s)

k=1

then condition (A.13) turns out to be equivalent to f (b s) = sb (see Section A.3). Thus (b s, sb) , sb ∈ S is a Nash equilibrium, if and only if sb is a fixed point of the mapping f = (f1 , . . . , fn ) defined by (A.14). An example: Let m = n = 3 and 

 7 5 3 A = 8 4 2 = B T . 9 5 1

Then we obtain, for every s ∈ S, eT1 A s eT2 A s eT3 A s sT A s

= = = =

7s1 + 5s2 + 3s3 , 8s1 + 4s2 + 2s3 , 9s1 + 5s2 + s3 , s1 (7s1 + 5s2 + 3s3 ) + s2 (8s1 + 4s2 + 2s3 ) + s3 (9s1 + 5s2 + s3 )

which implies  eT1 A s − sT A s = s22 ,   eT2 A s − sT A s = s22 − s2 ,   eT3 A s − sT A s = s22 ,

if s1 = s3 .

From this it follows that ϕ1 (s) = ϕ3 (s) = s22 > 0 and ϕ2 (s) = 0

for all

s∈U

with U = {s ∈ S | s1 = s3 , s2 > 0} . The functions (A.14) for s ∈ U are given by f1 (s) =

s1 + s22 , 1 + 2s22

This shows that f (U ) ⊆ U .

f2 (s) =

s2 , 1 + 2s22

f3 (s) =

s1 + s22 . 1 + 2s22

208


¡ ¢ If we now define, starting with some s0 ∈ U , a sequence sk k∈IN0 in U via ¢ ¡ sk+1 = f sk , k ∈ IN0 , then it follows from f2 (s) < s2 that sk2 → sb2 ≥ 0 and

for all

s∈U

sb2 = sb2 ⇔ 2b s22 = 0 ⇔ sb2 = 0. 1 + 2b s2

Because of f1 (s) = (1 − f2 (s))/2 this implies ¡ ¢ 1 ¡ ¡ ¢¢ 1 ¡ ¢ 1 1 − f2 sk = 1 − sk+1 → . = f1 sk = sk+1 1 2 2 2 2 µ ¶T 1 1 , 0, Hence sk → sb = and (b s, sb) is a Nash equilibrium. 2 2

A.4 Evolution Matrix Games

209

A.4 Evolution Matrix Games An evolution game (see [26]) can be considered as a 1−person game whose player is a population of individuals that have a finite number of strategies I1 , . . . , In (n > 1) in order to survive in the struggle of life. Let si ∈ [0, 1], for every i = 1, . . . , n, be the probability for the strategy Ii to be chosen in the population. Then the corresponding state of the population is defined by the n P si = 1. The space of all population states is vector s = (s1 , . . . , sn ) where i=1

given by the simplex ( S=

¯ ) n ¯ X ¯ si = 1 . s = (s1 , . . . , sn ) ¯ 0 ≤ si ≤ 1, ¯ i=1

Every vector ei = (0, . . . , 0, 1i , 0, . . . , 0) , i = 1, . . . , n, denotes a so called pure population state where all individuals choose the strategy Ii . All the other states are called mixed states. If an individual that chooses strategy Ii meets an individual that chooses strategy Ij , we assume that the Ii −individual is given a pay-off aij ∈ IR by the Ij −individual. All the pay-offs then form a matrix A = (aij )i,j=1,...,n , the so called pay-off matrix which defines a matrix game. The expected pay-off of an Ii −individual in the population state s ∈ S is defined by n X aij sj = ei A tT . j=1

Definition. A population state s∗ ∈ S is called a Nash equilibrium, if s A s∗ T ≤ s∗ A s∗ T

for all

s ∈ S.

(A.15)

In words this means that the average pay-off of s∗ to s∗ is always not smaller than its average pay-off to any other state s ∈ S. It is easy to see that condition (A.15) is equivalent to s∗ A s∗ T = max ei A s∗ T . i=1,...,n

Now let us define, for every i = 1, . . . , n and s ∈ S, ¢ ¡ ϕi (s) = max 0, ei A sT − s A sT and fi (s) = 1+

1 n P j=1

(si + ϕi (s)) . ϕj (s)

(A.16)

210


Then we obtain a continuous mapping f = (f1 , . . . , fn ) : S → S which, by Brouwer’s fixed point theorem, has at least one fixed point s∗ ∈ S, i.e., f (s∗ ) = s∗ . This implies ϕi (s∗ ) = 0

for all

i = 1, . . . , n

(see Section A.3)

and in turn (A.16) which shows that, if s∗ ∈ S is a fixed point of S (existence being insured), s∗ is a Nash equilibrium. Condition (A.16) implies ei A s ∗ T = s∗ A s ∗ T

for all

i ∈ {1, . . . , n}

s∗i > 0.

with

Now let J be a non-empty subset of {1, . . . , n} such that ei A s∗ T < s∗ A s∗ T

for all

i ∈ J.

Further, there exists an open subset U ⊆ S with s∗ ∈ U such that ei A sT < s A sT

for all

i∈J

and all

s∈U

which implies ϕi (s) = 0

for all

i∈J

and all

s ∈ U,

hence fi (s) =

1+

s Pi

for all

ϕj (s)

i∈J

and all

s ∈ U.

j6∈J

Now let us assume that for every s ∈ U there is some i ∈ J with si > 0. Further, we assume that for every s ∈ U with s 6= s∗ there is some j 6∈ J with ϕj (s) > 0. Then it follows that fi (s) ≤ si

for all i ∈ J and all s ∈ U with s 6= s∗ and

fi (s) < si

for some i = i(s) ∈ J.

If we define a Lyapunov function V : U → IR by X si , s ∈ U, V (s) = i∈J

then it follows that V (s∗ ) = 0

and

V (s) > 0

for all

s∈U

with s 6= s∗

for all

s ∈ U, s 6= s∗ .

and V (f (s)) =

X i∈J

fi (s)
0

for all

i 6= i0

and j = 1, . . . , n.

(A.18)

Then it follows that s∗i = 0

for all

hence s∗i0 = 1 and therefore s∗ = ei0 .

i 6= i0 ,

Let us demonstrate this by an example: Let n = 3 and   9 6 3 A = 8 5 2 . 7 4 1 Then we calculate e 1 A sT e2 A sT e3 A sT s A sT

= = = =

9s1 8s1 7s1 9s1

+ + + +

6s2 5s2 4s2 6s2

+ 3s3 , + 2s3 , + s3 , + 3s3 −s2 − 2s3 .

From this we obtain e 1 A s T − s A sT =

s2 + 2s3 > 0

for all s ∈ S with s2 + 2s3 > 0 ⇐⇒ s1 − s3 < 1,

e2 A sT − s A sT = −1 + s2 + 2s3 < 0

for all s ∈ S with s2 + 2s3 < 1 ⇐⇒ s3 < s1 ,

e3 A sT − s A sT = −2 + s2 + 2s3 < 0

for all s ∈ S with s2 + 2s3 < 2 ⇐⇒ 0 ≤ 2s1 + s2 .

This implies ϕ1 (s) > 0 ϕ2 (s) = ϕ3 (s) = 0

for all for all

s ∈ U = {s ∈ S | 0 < s3 < s1 < 1} , s∈U

and in turn, for every s ∈ U , fi (s) < si

for all

i = 2, 3 and f1 (s) > s1

which shows that (A.17) is satisfied for i0 = 1. Further, it follows that f (U ) ⊆ U . The assumption (A.18) is also fulfilled. In general, there exists a subsequence (kl )l∈IN0 of IN0 such that every sequence ³ ´ ski l converges to some s∗i ∈ [0, 1] for every i 6∈ J. This implies l∈IN0

s∗i =

s∗ Pi 1+ ϕj (s∗ ) j6∈J

for all

i ∈ J.


213

If there exists some i0 ∈ J with s∗i0 > 0, then it follows that ϕj (s∗ ) = 0

for all

j 6∈ J

and hence for all j ∈ {1, . . . , n}

which implies that s∗ is a Nash equilibrium. Definition. A Nash equilibrium s∗ ∈ S is called evolutionarily stable if s A s∗ T = s∗ A s∗ T for some s ∈ S with s 6= s∗ implies that s A sT < s∗ A sT . In words this means: If the average pay-off of s∗ to s∗ is the same as that of s∗ for some s ∈ S with s 6= s∗ , then s cannot be a Nash equilibrium. Assertion: If, for some i0 ∈ {1, . . . , n}, the condition (A.18) is satisfied, then ei0 is an evolutionarily stable Nash equilibrium. Condition (A.18) implies that ai0 i0 > aii0

for all

i 6= i0 .

From this we infer, for every s ∈ S with s 6= ei0 , that s A eTi0

=

n X

si aii0 < ai0 i0 = ei0 A eTi0

i=1

which implies that ei0 is an evolutionarily stable Nash equlibrium. In [22] we have shown that for every evolutionarily stable Nash equilibrium s∗ ∈ S there exists some ε∗ > 0 such that s A sT < s∗ A sT

for all

s∈S

with

s 6= s∗

and

ks − s∗ k2 < ε∗ .

Now let ei0 be an evolutionarily stable Nash equilibrium. Then there is a relatively open set U ⊆ S with ei0 ∈ U and s A s T < e i0 A s T

for all

ϕi0 (s) > 0

for all s ∈ U

with

for all

and i 6= i0 .

s∈U

with s 6= ei0 .

This implies s 6= ej0 .

Assumption: ϕi (s) = 0

s∈U

Then it follows that fi (s) < si fi0 (s) > si0

for all for all

s∈U s∈U

and i 6= i0 , and s = 6 e i0 .

Now let us define a Lyapunov function V : S → IR by V (s) = 1 − si0 ,

s ∈ S.

214


Then V (ei0 ) = 0

and

V (s) > 0

for all

s∈S

with s 6= ei0 ,

V (f (s)) − V (s) = 1 − fi0 (s) − 1 + si0 < 0

for all

s ∈ U, s 6= ei0 .

By Theorem 1.3 it therefore follows that ei0 is an asymptotically stable fixed point of f . In [22] we also consider the following dynamical treatment of the game. We assume that for all s ∈ S s A sT > 0 and define a mapping f : S → S by putting fi (s) =

ei A s T si s A sT

i = 1, . . . , n and s ∈ S.

Obviously s∗ ∈ S is a fixed point of f , i.e., f (s∗ ) = s∗ , if and only if ei A s∗ T = s∗ A s∗ T where

for all

i ∈ S (s∗ )

S (s∗ ) = {i ∈ {1, . . . , n} | s∗i > 0} .

From Lemma 1.1 in Section 1.2.5 it therefore follows that s∗ is a fixed point of f , if s∗ is a Nash equilibrium. Now the question arises under which conditions this Nash equilibrium is an asymptotically stable fixed point of f . In order to answer this question we make use of Theorem 1.3. For this purpose we put X (s∗ ) = {s ∈ S | S(s) ⊇ S (s∗ )} and define a function X

V (s) =

s∗i (ln s∗i − ln si ) ,

s ∈ X (s∗ ) .

i∈S(s∗ )

Then it follows that V (s) ≥ 0

for all

s ∈ X (s∗ )

and

V (s) = 0 ⇔ s = s∗

(see Section 1.2.5). Now let us assume that s∗ is evolutionarily stable. Then, by Lemma 1.15 in Section 1.2.5, there exists some ε > 0 such that s A sT < s∗ A sT Let us put

for all

s ∈ S with ||s − s∗ ||2 < ε and s 6= s∗ .

U (s∗ ) = {s ∈ X (s∗ ) | ||s − s∗ || < ε} .


215

Further we put © ª Y (s∗ ) = s ∈ X (s∗ ) | ei A sT = ej A sT for all i, j ∈ S (s∗ ) . Then it follows that s A sT < ei A sT

for all

i ∈ S (s∗ ) and all s ∈ Y (s∗ )∩U (s∗ ) and s 6= s∗ .

Further we obtain X ¡ ¢ s∗i ln s∗i − ln ei A sT + ln s A sT − ln si V (f (s)) = i∈S(s∗ )

= V (s) +

X

µ s∗i

ln

i∈S(s∗ )

s A sT ei A sT

¶ ,

s ∈ Y (s∗ ) ∩ U (s∗ ) ,

which implies V (f (s)) < V (s)

for all

s ∈ Y (s∗ ) ∩ U (s∗ ) with s 6= s∗ .

From Theorem 1.3 with X = S and G = Y (s∗ ) ∩ U (s∗ ) it then follows that s∗ is an asymptotically stable fixed point of f .

B Optimal Control in Chemotherapy of Cancer

B.1 The Mathematical Model and Two Control Problems We describe the time dependent size of the tumor by a real valued function T = T (t), t ∈ IR (t denotes the time), which we assume to be differentiable. The temporal development of this size (without treatment) we assume to be governed by the differential equation T˙ (t) = f (T (t)) T (t),

t ∈ IR,

where the function f : IR+ → IR+ is the growth rate of the tumor which is assumed to be in C 1 (IR+ ) and to satisfy the condition f 0 (T ) < 0

for all

T ≥ 0.

Let [0, tf ] for some tf > 0 be the time interval of treatment. The controlled growth of the tumor within this time interval is assumed to be described by the differential equation ³ ´ (B.1) T˙ (t) = f (T (t)) − ϕ(t) L(M (t)) T (t), t ∈ (0, tf ) , with the initial condition T (0) = T0 > 0

(B.2)

at the beginning of the treatment. The function L = L(M ), M ≥ 0, denotes the destruction rate of the drug amount M and is assumed to satisfy the conditions L ∈ C 1 (IR+ ) , L(0) = 0 and L0 (M ) > 0

for all

M ≥ 0.

S. Pickl and W. Krabs, Dynamical Systems: Stability, Controllability and Chaotic Behavior, 217 DOI 10.1007/978-3-642-13722-8_5, © Springer-Verlag Berlin Heidelberg 2010

218


The function ϕ = ϕ(t), t ∈ [0, tf ] is a so called “resistance factor” that takes care of a possible resistance of the cancer cells against the drugs that is developed in the course of the treatment. It is assumed to be in C 1 [0, tf ] and to satisfy the conditions ϕ(0) = 1, ϕ(t) ˙ ≤0

for all

t ∈ [0, tf ] and ϕ (tf ) > 0.

The function M = M (t), t ∈ [0, tf ], denotes the drug amount at the time t and is assumed to satisfy the differential equation M˙ (t) = −δ M (t) + V (t),

t ∈ [0, tf ] ,

(B.3)

and the initial condition M (0) = 0

(B.4)

where V (t) denotes the dosis that is administered per time unit at the time t and δ ≥ 0 is the decay rate of the drug. We assume the function V = V (t) to be measurable and bounded and to satisfy the condition for almost all t ∈ [0, tf ]

0 ≤ V (t) ≤ A

(B.5)

where A > 0 is a given constant. In the sequel we consider the following two control problems: Problem 1: Determine a control function V ∈ L∞ [0, tf ] such that under the conditions (B.1)–(B.5) and the end condition T (tf ) = tf

(B.6)

for some Tf ∈ (0, T0 ) the total amount of drugs administered which is given by Z tf V (t) dt 0

is minimized . Problem 2: Determine a control function V ∈ L∞ [0, tf ] such that under the conditions (B.1)–(B.5) and Z

t

V (t) dt = B 0

for some constant B > 0 the end value T (tf ) is minimized.

(B.7)

B.1 The Mathematical Model and Two Control Problems

219

If we define y = ln T

for T > 0,

then the differential equation (B.1) is transformed into ´ ³ t ∈ (0, tf ) , y(t) ˙ = f e y(t) − ϕ(t) L(M (t)),

(B.1’)

the initial condition (B.2) reads y(0) = y0 = ln T0

(B.2’)

and the end condition (B.6) is given by y (tf ) = yf = ln Tf .

(B.6’)

220


B.2 Solution of the First Control Problem Let us repeat the problem to be solved: Determine a control function V ∈ L∞ [0, tf ] such that under the conditions (B.1’),(B.2’),(B.3)–(B.5),(B.6’) the functional Z tf

V (t) dt

F (y, M, V ) = 0

is minimized. The Hamiltonian of this problem reads H (t, y, M, V, p1 , p2 , λ0 ) = (f (e y ) − ϕL(M )) p1 + (−δM + V )p2 − λ0 V. ³ ´ c, Vb be an optimal tripel; then, by the maximum principle (see A. D. Let yb, M Joffe and V. M. Tichomirov: Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin 1979) there exists a number λ0 ≥ 0 and two functions p1 , p2 ∈ C 1 [0, tf ] with (p1 , p2 , λ0 ) 6= (0, 0, 0) and ³ ´ t ∈ [0, tf ] , p˙1 (t) = −f 0 e yb(t) e yb(t) p1 (t), ³ ´ c(t) p1 (t) + δ p2 (t), t ∈ [0, tf ] , p2 (tf ) = 0; p˙2 (t) = ϕ(t) L0 M further we have (p2 (t) − λ0 ) Vb (t) = max (p2 (t) − λ0 ) V

for almost all t ∈ [0, tf ] .

0≤V ≤A

If we put

³ ´ g(t) = −f 0 e yb(t) e yb(t) ,

t ∈ [0, tf ] ,

then we obtain p˙1 (t) = g(t) p1 (t), and therefore p1 (t) = p1 (0) e If we put

Rt 0

g(s) ds

t ∈ [0, tf ] , ,

³ ´ c(t) p1 (t), h(t) = ϕ(t) L0 M

t ∈ [0, tf ] . t ∈ [0, tf ] ,

then it follows that p˙2 (t) = δ p2 (t) + h(t),

t ∈ [0, tf ] ,

which implies µ Z p2 (t) = e δt p2 (0) +

t 0

¶ h(s) e −δs ds ,

t ∈ [0, tf ] .

B.2 Solution of the First Control Problem

221

From p2 (tf ) = 0 we infer Z p2 (0) = −

tf

h(s) e −δs ds

0

and therefore Z p2 (t) = −e

tf

δt

h(s) e −δs ds,

t ∈ [0, tf ] .

t

Now we distinguish three cases: a) p1 (0) = 0; then p1 (t) = 0 for all t ∈ [0, tf ] and therefore h(t) = 0 for all t ∈ [0, tf ] which implies p2 (t) = 0 for all t ∈ [0, tf ] and λ0 > 0. Hence it follows that p2 (t) − λ0 < 0 for all t ∈ [0, tf ] and therefore Vb (t) = 0

t ∈ [0, tf ] .

for almost all

c(t) = 0 for all t ∈ [0, tf ] and in turn This implies M ³ ´ c(t) = 0 L M

for all

t ∈ [0, tf ] .

From this it follows that yb (tf ) > y0 > ytf contradicting the requirement y (tf ) = yf . b) p1 (0) > 0; then it follows that h(t) > 0 for all t ∈ [0, tf ] which implies p2 (t) < 0 for all t ∈ [0, tf ) and therefore p2 (t) − λ0 < 0

for all

t ∈ [0, tf )

which again leads to Vb (t) = 0

for almost all

t ∈ [0, tf ]

being impossible as seen in a). Therefore we necessarily have c) p1 (0) < 0; then it follows that h(t) < 0 for all t ∈ [0, tf ] which implies p2 (t) > 0 for all t ∈ [0, tf ). α) Let λ0 = 0; then it follows p2 (t)−λ0 > 0 for all t ∈ [0, tf ) and therefore Vb (t) = A

for almost all

t ∈ [0, tf ] .

β) Let λ0 > 0; then it follows that p2 (tf ) − λ0 = −λ0 < 0.

(B.8)

222


Assumption: p˙2 (t) < 0

for all t ∈ [0, tf ] .

(B.9)

This assumption implies that p2 = p2 (t), t ∈ [0, tf ], is a strictly decreasing function with p2 (0) > 0 and p2 (tf ) = 0. If p2 (0) − λ0 ≤ 0, then it follows that p2 (t) − λ0 < 0 for all t ∈ [0, tf ] which is impossible as shown in a). Therefore p2 (0) − λ0 > 0; since p2 (tf ) − λ0 = −λ0 < 0 there exists exactly one t0 ∈ (0, tf ) such that   > 0 for t ∈ [0, t0 ) , p2 (t) − λ0 = 0 for t = t0 ,   < 0 for t ∈ (t0 , tf ] which implies Vb (t) =

( A 0

for almost all t ∈ [0, t0 ) , for almost all t ∈ (t0 , tf ] .

(B.10)

Result. Under the assumption (B.9) an optimal control is necessarily of the form (B.8) or (B.10). Let us give a sufficient condition for (B.9) to hold true in the case λ0 > 0. From the assumption on ϕ = ϕ(t) it follows that ϕ(t) > 0

for all

t ∈ [0, tf ] .

This then implies that

³ ´ c(t) pe1 (t) < 0 h(t) = ϕ(t) L0 M

for all

t ∈ [0, tf ] ,

if λ0 > 0. Because of p˙2 (t) = δ p2 (t) + h(t),

t ∈ [0, tf ] ,

the assumption (B.9) is equivalent to δ p2 (t) + h(t) < 0

for all

t ∈ [0, tf ]

which is satisfied, if δ = 0 where no decay of the drug takes place. If δ > 0, it follows from Z

tf

p2 (t) = −e δt

h(s) e −δs ds,

t ∈ [0, tf ] ,

e δ(t−s) h(s) ds < 0,

t ∈ [0, tf ] ,

t

that

Z

tf

h(t) − δ

(B.11)

t

is necessary and sufficient for (B.9) to hold. Since this condition is satisfied for δ = 0, it is to be assumed that it is satisfied, if δ > 0 is sufficiently small.

B.2 Solution of the First Control Problem

223

This can be quantified in the special case of the Gompertz growth law f (T ) = λ ln

Θ , T

T > 0,

with constants λ > 0 and Θ > 0 and a linear destruction rate L(M ) = c M,

M ≥ 0,

with a constant c > 0. In this case we obtain ³ ´ g(t) = −f 0 e yb(t) e yb(t) = λ, and

h(t) = c ϕ(t) pe1 (0) e λt ,

t ∈ [0, tf ] ,

t ∈ [0, tf ] .

Condition (B.11) turns out to be equivalent to Z ϕ(t) e

(λ−δ)t

−δ

tf

ϕ(s) e (λ−δ)s ds > 0

for all

t ∈ [0, tf ] .

t

If we put

ϕ(t) = e −(λ−δ)t ,

t ∈ [0, tf ] ,

and assume λ > δ, then it follows that ϕ ∈ C 1 [0, tf ], ϕ(0) = 1, ϕ(t) ˙ < 0 and ϕ(t) > 0 Further (B.12) turns out to be equivalent to ´ ³ 1 − δ (tf − t) > 0 for all t ∈ [0, tf ]

for all

⇐⇒

t ∈ [0, tf ] .

δ tf < 1.

(B.12)

224


B.3 Solution of the Second Control Problem If we define

Z

t

V (s) ds,

Q(t) =

t ∈ [0, tf ] ,

0

then condition (B.7) turns out to be equivalent to ˙ Q(t) = V (t)

for almost all

t ∈ [0, tf ] ,

Q(0) = 0, Q (tf ) = B. (B.13)

Problem 2 then can be formulated in the following form: Find a function V ∈ L∞ [0, tf ] such that under the conditions (B.1’),(B.2’),(B.3)–(B.5) and (B.13) the functional Z tf ³ ³ ´ ´ f e y(t) − ϕ(t) L(M (t)) dt F (y, M, Q, V ) = y (tf ) − y0 = 0

is minimized. The Hamiltonian of this control problem is given by H (t, y, M, Q, V, p1 , p2 , p3 , λ0 ) = (f (e y ) − ϕ L(M )) (p1 − λ0 ) + (V − δ M )p2 + V p3 . ³ ´ c, Q, b Vb be an optimal quadruple. Then by the maximum prinNow let yb, M ciple there exists a number λ0 ≥ 0 and three absolutely continuous functions p1 , p2 , p3 on [0, tf ] with (λ0 , p1 , p2 , p3 ) 6= (0, 0, 0, 0) such that ³ ´ p1 (tf ) = 0, p˙1 (t) = −f 0 e yb(t) e yb(t) (p1 (t) − λ0 ) , t ∈ [0, tf ] , ³ ´ c(t) (p1 (t) − λ0 ) + δ p2 (t), t ∈ [0, tf ] , p2 (tf ) = 0, p˙2 (t) = ϕ(t) L0 M p˙3 (t) = 0

for almost all t ∈ [0, tf ] =⇒ p3 (t) = p3 = const.

for almost all t ∈ [0, tf ] .

Further it is (p2 (t) + p3 ) Vb (t) = max (p2 (t) + p3 ) V

for almost all

0≤V ≤A

t ∈ [0, tf ] .

¡ ¢ If we define pe1 (t) = p1 (t) − λ0 and g(t) = −f 0 e yb(t) e yb(t) for t ∈ [0, tf ], then it follows that for all t ∈ [0, tf ] pe˙ 1 (t) = g(t) pe1 (t) and hence pe1 (t) = pe1 (0) e

Rt 0

g(s) ds

,

t ∈ [0, tf ] .

From p1 (tf ) = 0 we obtain pe1 (tf ) = −λ0 = pe1 (0) e

R tf 0

g(s) ds

.

B.3 Solution of the Second Control Problem

If we define

³ ´ c(t) pe1 (t), h(t) = ϕ(t) L0 M

225

t ∈ [0, tf ] ,

then it follows that p˙2 (t) = δ p2 (t) + h(t)

for t ∈ (0, tf ) .

This implies µ Z p2 (t) = e p2 (0) +

¶

t

δt

e

−δs

h(s) ds ,

t ∈ [0, tf ] .

0

From p2 (tf ) = 0 we obtain Z p2 (0) = −

tf

e −δs h(s) ds

0

which implies Z p2 (t) = −e δt

tf

e −δs h(s) ds,

t ∈ [0, tf ] .

t

Now we distinguish two cases: 1) λ0 = 0; then it follows that p1 (t) = pe1 (t) = 0 for all t ∈ [0, tf ] which implies h(t) = 0 for all t ∈ [0, tf ] and in turn p2 (t) = 0


Because of (λ0 , p1 , p2 , p3 ) 6= (0, 0, 0, 0) it follows that p3 6= 0. a) Let p3 < 0; then it follows that Vb (t) = 0

for almost all

t ∈ [0, tf ]

which is impossible because of (B.7) with B > 0. b) Let p3 > 0; then it follows that Vb (t) = A

for almost all t ∈ [0, tf ]

(B.8)

which implies Atf = B. 2) λ0 > 0; then it follows that pe1 (t) < 0 for all t ∈ [0, tf ] which implies h(t) < 0 for all t ∈ [0, tf ] and in turn p2 (t) > 0 for all t ∈ [0, tf ) from which again we infer (B.8) and Atf = B, if p3 ≥ 0. If p3 < 0, we again make the assumption p˙2 (t) < 0


(B.9)

Then p2 = p2 (t) is a strictly decreasing function from p2 (0) > 0 to p2 (tf ) = 0.

226


If p2 (0) + p3 ≤ 0, then p2 (t) + p3 < 0 for all t ∈ (0, tf ] implying Vb (t) = 0

for almost all

t ∈ [0, tf ]

which is impossible (see above). If p2 (0) + p3 > 0, then there is exactly one t0 ∈ (0, tf ) with p2 (t) + p3 > 0 p2 (t) + p3 < 0

for all for all

t ∈ [0, t0 ) , t ∈ (t0 , tf ] .

p2 (t0 ) + p3 = 0,

As a consequence of this we obtain ( A for almost all t ∈ [0, t0 ) , b V (t) = 0 for almost all t ∈ (t0 , tf ] . where t0 is the solution of the equation Z t0 A dt = B ⇐⇒

(B.10)

A t0 = B.

0

Result. Under the assumption (B.9) an optimal control has necessarily the form (B.8) or (B.10). The form (B.8) is possible, if Atf = B. The form (B.10) occurs, if Atf > B. As in Section 2 one can show that the condition Z tf e δ(t−s) h(s) ds < 0, h(t) = −δ

t ∈ [0, tf ] ,

(B.11)

t

is necessary and sufficient for (B.9) to hold true in the case λ0 > 0 and p3 < 0. Again in the special case of the Gompertz growth law for the untreated tumor and a linear destruction rate we can show for ϕ(t) = e −(λ−δ)t , t ∈ [0, tf ]

with

λ>δ

that condition (B.11) is satisfied, if and only if δ tf < 1. In addition the differential equation (B.1’) reads y(t) ˙ = −λ y(t) + λ ln Θ − c ϕ(t) M (t),

t ∈ (0, tf ) .

Under the initial condition (B.2’) the unique solution is then given by ¶ µ Z t³ ´ λ ln Θ − c ϕ(s) M (s) e λs ds y(t) = e −λt y0 + 0

¡

+ λ ln Θ 1 − e

−λt

¢

Z

t

e λ(s−t) ϕ(s) M (s) ds Z t ¢ ¡ −λt −λt −λt = y0 e −ce + λ ln Θ 1 − e p(s) V (s) ds, = y0 e

−λt

−c

0

0

t ∈ [0, tf ] ,

B.3 Solution of the Second Control Problem

where

Z p(s) = e

δs

t

e (λ−δ)σ ϕ(σ) dσ,

227

s ∈ [0, tf ] ,

s

and the end value y (tf ) reads y (tf ) = y0 e

¡

−λtf

+ λ ln Θ 1 − e

−λtf

¢

Z −ce

tt

−λtf

p(t) V (t) dt. 0

Problem 2 can now be equivalently formulated as follows: Find a function V ∈ L∞ [0, tf ] with (B.5) and (B.7) such that Z

Z

tf

p(t) V (t) dt

is maximized where

p(t) = e

δt

0

tf

e (λ−δ)s ϕ(s) ds.

t

Let Vb ∈ L∞ [0, tf ] be a solution of this problem (whose existence is ensured). then there exists some λ0 ∈ IR with Z tf Z tf (p(t) − λ0 ) Vb (t) dt ≥ (p(t) − λ0 ) V (t) dt 0

0

for all This implies Vb (t) =

( A, 0,

V ∈ L∞ [0, tf ] with (B.5). if p(t) − λ0 > 0, if p(t) − λ0 < 0.

If λ0 ≤ 0, it follows that p(t) − λ0 > 0

for all

t ∈ [0, tf )

which implies Vb (t) = A for all t ∈ [0, tf )

and

A tf = B.

If λ0 > 0, we again put ϕ(t) = e −(λ−δ)t , t ∈ [0, tf ] ,

with λ > δ.

Then it follows that p(t) = (tf − t) e δt , hence

p(t) ˙ = (δ (tf − t) − 1) e δt < 0

t ∈ [0, tf ] , for all

t ∈ [0, tf ] ,

if and only if δ tf < 1. Under this condition p = p(t) is a strictly decreasing function with p(0) = tf and p (tf ) = 0.

228


If tf ≤ λ0 , then p(t) − λ0 < 0 for all t ∈ (0, tf ] which implies Vb (t) = 0

for all

t ∈ (0, tf ]

which is impossible. If tf > λ0 , then p(0) − λ0 > 0 and p (tf ) − λ0 = −λ0 < 0 and there exists exactly one t0 ∈ (0, tf ) such that p(t) − λ0 > 0 p(t) − λ0 < 0 which implies

for all for all

( A, Vb (t) = 0,

t ∈ [0, t0 ) , t ∈ (t0 , tf ]

p (t0 ) − λ0 = 0,

if t ∈ [0, t0 ) , if t ∈ (t0 , tf ] ,

and t0 is the solution of the equation Z t0 A dt = B ⇐⇒

A t0 = B.

0

In addition we obtain Z tf Z b p(t) V (t) dt = 0

Z

t0

t0

p(t) A dt = A 0

(tf − t) e δt dt.

(B.12)

0

Summarizing every solution of the above problem is the form ( A, if t ∈ [0, t0 ] , b V (t) = 0, if t ∈ (t0 , tf ] , for some t0 ∈ (0, tf ] and the extreme value is given by (B.12). Finally we consider the problem of determining a function V ∈ L∞ [0, tf ] which satisfies (B.5) and Z tf Z t0 p(t) V (t) dt = A (tf − t) e δt dt 0

0

Z

tf

and which minimizes

V (t) dt. 0

Let V ∗ ∈ L∞ [0, tf ] be a solution of this problem. Then it is necessarily of the form (see Section 2) ( A, if t ∈ [0, t∗0 ] , ∗ V (t) = 0, if t ∈ (t∗0 , tf ] , for some t∗0 ∈ (0, tf ] . This implies Z tf Z ∗ p(t) V (t) dt = A 0

t∗ 0

Z (tf − t) e

δt

t0

dt = A

0

hence t∗0 = t0 and V ∗ (t) = Vb (t) for all t ∈ [0, tf ].

0

(tf − t) e δt dt,

B.4 Pontryagin’s Maximum Principle

229

B.4 Pontryagin’s Maximum Principle The two control problems considered in Sections B.1–B.3 are special cases of the following situation: Given a system of differential equations of the form x˙ i (t) = fi (x(t), u(t)),

t ∈ (t0 , t1 ) , for i = 1, . . . , n

(B.13)

with initial conditions xi (t0 ) = x0i ,

i = 1, . . . , n,

(B.14)

and conditions xik (t1 ) = x1ik ,

k = 1, . . . , r(≤ n),

(B.15)

where x(t) = (x1 (t), . . . , xn (t)) , u(t) = (u1 (t), . . . , um (t)) , t ∈ [t0 , t1 ], fi : IRn × IRm → IR, i = 1, . . . , m, with fi ∈ C (IRn × IRm ) and fi (·, u) ∈ C 1 (IRn ) for every u ∈ IRm . The functions ui = ui (t) for t ∈ [t0 , t1 ] and i = 1, . . . , m are considered as control functions and are assumed to be in the space L∞ [t0 , t1 ]. Further we assume that u(t) ∈ U ⊆ IRm

for almost all t ∈ [t0 , t1 ]

(B.16)

where U is a non-empty subset of IRm . Finally we consider a functional Z

t1

f0 (x(t), u(t)) dt.

F (x(·), u(·)) =

(B.17)

t0

The control problem now consists of determining some u ∈ L∞ ([t0 , t1 ] , IRm ) which satisfies (B.16) such that the functional (B.17) is minimized where x = x(t), t ∈ [t0 , t1 ], is the corresponding solution of (B.13),(B.14),(B.15) in the space of absolutely continuous n−vector functions on [t0 , t1 ]. Let us define a Hamilton function (or Hamiltonian) by

H (t, x, u, p, λ0 ) =

n X

pj fj (x, u) − λ0 f0 (x, u)

j=1

where p ∈ IRn , λ0 ∈ IR+ . The we have the following

230


Maximum Principle: Let (b x, u b) be a solution of the above control problem. Then there exists a number λ0 ≥ 0 and n absolutely continuous functions p1 = p1 (t), . . . , pn = pn (t), t ∈ [t0 , t1 ], with (p1 , . . . , pn , λ0 ) 6= (0, . . . , 0) ∈ IRn+1 such that b(t), u b(t), p(t), λ0 ) (p(t) = (p1 (t), . . . , pn (t))) p˙i (t) = −Hxi (t, x n X =− pj fjxi (b x(t), u b(t)) + λ0 f0xi (b x(t), u b(t)) j=1

for almost all t ∈ (t0 , t1 ) , i = 1, . . . , n, pi (t1 ) = 0 for all i 6∈ {i1 , . . . , ik } , and b(t), u, p(t), λ0 ) H (t, x b(t), u b(t), p(t), λ0 ) = max H (t, x u∈U

for almost all

t ∈ [t0 , t1 ] .

For the proof we refer to: A. D. Joffe and V. M. Tichomirov: Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin 1979.

C List of Authors

Prof. Dr. rer.nat. Werner Krabs 1934 1954–1959 1963 1967/1968 1968 1970–1972 1971 1972 1977 1979–1981 1986–1987

born in Hamburg-Altona study of mathematics, physics and astronomy at the University of Hamburg, diploma in mathematics PhD-thesis visiting assistant professor at the University of Washington in Seattle habilitation in applied mathematics at the University of Hamburg professor at the RWTH Aachen visiting associate professor at the Michigan State University in East Lansing professor at the TH Darmstadt visiting full professor at the Oregon State University in Corvallis vice-president of the TH Darmstadt chairman of the Society for Mathematics, Economy and Operations Research.

232

Prof. Dr. Stefan Pickl

Prof. Dr. Stefan Pickl Studied mathematics, electrical engineering, pedagogy and philosophy at TU Darmstadt and EPFL Lausanne 1987–93. Dipl.-Ing. 1993, Doctorate 1998 with award. Assistant Professor at Cologne University (Dr. habil. 2005; venia legendi “Mathematics”). Chair for Operations Research at UBw Munich since 2005. Visiting Professor at University of New Mexico (USA), University Graz (Austria), Naval Postgraduate School Monterey, University of California at Berkeley. Visiting scientist at SANDIA, Los Alamos National Lab, Santa Fe Institute for Complex Systems and MIT. Associated with Centre for the Advanced Study of Algorithms (CASA, USA), with Center for Network Innovation and Experimentation (CENETIX, USA). Vice-chair of EURO working group Experimental Economics, chair of advisory board of the German Society for Operations Research (GOR), advisory board of the German Society for Disaster and Emergency Management. International Research Programs, 80 publications, international Best-Paper-Awards 2003, 2005, 2007. Foundation of COMTESSA (Competence Center for Operations Research, Management of Intelligent Engineered Secure Systems & Algorithms) and the Academy for Highly Gifted Pupils in Munich.

References

1. Agarwal, R. T.: Difference Equations and Inequalities. Marcel Dekker, Inc., New York, Basel, Hongkong (1992). 2. Banks, J., Brooks, J., Cairns, G., Davis, G., and Stacey, P.: On Devaney’s Definition of Chaos. American Mathematical Monthly 99, 332–334 (1992). 3. Birkhoff, G. D.: Dynamical Systems. American Math. Society Colloqu. Publ., Providence (1927). 4. Burger, E.: Einf¨ uhrung in die Theorie der Spiele. Walter de Gruyter & Co.,Berlin (1966). 5. Devaney, R. L.: An Introduction to Chaotic Dynamical Systems. AddisonWesley (1989). 6. Gantmacher, F. R.: Application of the Theory of Matrices. Interscience Publishers, New York-London-Sydney (1959). 7. Hale, J. K.: Ordinary Differential Equations. Wiley-Interscience (1969). 8. Hale, J. K. (editor): Studies in Ordinary Differential Equations. 14, published by the Mathematical Association of America (1977). ¨ 9. Hurwitz, A.: Uber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Teilen besitzt. Math. Ann. 46, 273–284 (1895). 10. Jankov´ a, K., and Smital, J.:A Characterization of Chaos. Bull. Austral. Math. Soc. 34, 283–292 (1986). 11. Kirchgraber, U.: Mathematik im Chaos. Ein Zugang auf dem Niveau der Sekundarstufe II. Mathematische Semesterberichte 39, 43–68 (1992). 12. Knobloch, H. W., and Kappel, F.: Gew¨ ohnliche Differentialgleichungen. Verlag B. G. Teubner, Stuttgart (1974). 13. Krabs, W.: Optimal Control of Undamped Linear Vibrations. Heldermann Verlag (1995). 14. Krabs, W.: Dynamische Systeme: Steuerbarkeit und Chaotisches Verhalten. Verlag B. G. Teubner, Stuttgart-Leipzig (1998). 15. Krabs, W.: On Local Controllability of Time-Discrete Dynamical Systems into Steady States. Journal of Difference Equations and Applications 8, 1–11 (2002). 16. Krabs, W.: Stability and Controllability in Non-Autonomous Time-Discrete Dynamical Systems. Journal of Difference Equations and Applications 8, 1107– 1118 (2002). 17. Krabs, W.: On Local Fixed Point Controllability of Nonlinear Difference Equations. Journal of Difference Equations and Applications 9, 827–832 (2003).

234

References

18. Krabs, W.: A General Predator-Prey Model. Mathematical and Computer Modelling of Dynamical Systems 9, 387–401 (2003). 19. Krabs, W.: A Dynamical Method for the Calculation of Nash-Equilibria in n−Person Games. Journal of Difference Equations and Applications 11, 919– 932 (2005). 20. Krabs, W.: Stability in Predator-Prey Models and Discretization of a Modified Volterra-Lotka-Model. Mathematical and Computer Modelling of Dynamical Systems 12, 577–588 (2006). 21. Krabs, W.: On Local Reachability and Controllability of Nonlinear Difference Equations. Mathematical and Computer Modelling of Dynamical Systems 12, 487–494 (2006). 22. Krabs, W., and Pickl, St.: Analysis, Controllability and Optimization of Time Discrete Systems and Dynamical Games. Lecture Notes in Economics and Mathematical Systems 529, Springer-Verlag, Berlin, Heidelberg (2003). 23. Krause, U., and Nesemann, T.: Differenzengleichungen und diskrete dynamische Systeme. B. G. Teubner, Stuttgart, Leipzig (1995). 24. Lakshmikantham, V., and Frigiante, D.: Theory of Difference Equations: Numerical Methods and Applications. Academic Press, Inc., Boston et al. (1988). 25. La Salle, J. P.: The Stability and Control of Discrete Processes. Springer Verlag, New York-Berlin-Heidelberg-London-Paris-Tokio (1986). 26. Li, J.: Das dynamische Verhalten diskreter Evolutionsspiele. Shaker Verlag, Aachen (1999). 27. Li, T. Y., and Yorke, J. A.: Period Three Implies Chaos. American Mathematical Monthly 85, 985–992 (1975). 28. Mielke, A.: Topological Methods for Discrete Dynamical Systems. GAMMMitteilungen 2, 19–37 (1990). 29. Misiurewicz, M.: Horseshoes for Mappings of the Interval. Bulletin de l’Académie Polonaise des Sciences XXVII(2), 166–168 (1979). 30. Misiurewicz, M., and Szlenk, W.: Entropy of Piecewise Monotonic Mappings. Studia Matematica LXVII, 45–63 (1980). 31. Nash, J.: Non-Cooperative Games. Annals of Mathematics 54, 286–295 (1951). 32. Pickl, St.: Der τ −value als Kontrollparameter. Modellierung und Analyse eines Joint-Implementation Programmes mithilfe der dynamischen kooperativen Spieltheorie und der diskreten Optimierung. Shaker Verlag, Aachen (1999). 33. Sell, G. R.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold Company, London (1971). 34. Smital, J.: Chaotic Functions with Zero Topological Entropy. Transactions of the American Mathematical Society 297, 269–282 (1986).

Index

Ω−controllable, 76 IRm −controllable, 77 IRm −null-controllable, 83 n−person game, 191 admissible control functions, 120, 127, 129, 131 alpha limit set, 3 approximation problem, 87 asymptotical stability, 103 asymptotically stable, 12, 13, 19, 21, 22, 33, 34, 42, 44, 46, 48–54, 56, 64–69, 121–123, 138, 139, 207 asymptotically stable equilibrium state, 24, 25 asymptotically stable fixed point, 55, 56, 58, 207, 208 asymptotically stable rest point, 17, 19 attractive, 50, 51, 66, 67, 69 attractor, 41–45, 64, 65, 100, 138, 139, 175, 178, 204 autonomous, 35, 145 autonomous system, 13 autonomous time-discrete dynamical system, 60 average pay-off, 202 bi-matrix game, 31 Bi-Matrix Games, 196 Bouwer’s fixed point theorem, 189 Brouwer’s fixed point theorem, 192, 194, 203 Cantor set, 152, 178, 179

Cayley-Hamilton, 114 Chaos in the Sense of Devaney, 145 Chaos in the Sense of Li and Yorke, 168 chaos in the sense of Li and Yorke, 174, 187 chaotic behavior of dynamical systems, 186 chaotic behavior of one-dimensional continuous mappings, 186 chaotic in the sense of Devaney, 176, 178, 180, 182, 183, 186 chaotic in the sense of Li and Yorke, 174 chaotic mapping, 186 completely IRm −controllable, 78 completely Ω−controllable, 76 completely IR4 −controllable, 79 control function, 75, 83, 84, 99, 100, 106, 109, 120, 129, 131, 133, 134, 210, 211 Controllability of Linear Systems, 76 controllability of time-discrete dynamical systems, 185 controllable set, 101, 135 controlled system, 103, 106, 107, 123, 138 cost functional, 104 cyclic, 38 decay rate, 210 destruction rate, 209, 214, 217 difference equation, 128 difference equations, 99, 131 differential equation, 79, 84, 128, 209, 210

236

Index

Direct Method of Lyapunov, 12 direct method of Lyapunov, 13, 184 Discretization, 55, 57 discretization, 54 disorder-chaos, 163, 174, 187 disorder-chaotic, 161, 162, 166, 167 Dual Problem, 91 dual problem, 91, 92, 96 dynamical system, 1, 3, 12, 17, 184 dynamical systems, 1 Emission Reduction Model, 58 emission reduction model, 107, 123 equilibrium, 6 equilibrium state, 20, 22, 23 evolution matrix game, 30 evolutionarily stable, 33, 34, 206, 207 evolutionarily stable Nash equilibrium, 32, 33 feedback controls, 138 fixed point, 38, 42, 44, 45, 49, 59, 99, 101, 103, 109, 121, 123, 129, 131, 134, 140, 189, 191, 192, 194, 197, 200, 203, 204, 207 Fixed Point Controllability, 99 fixed point controllability, 119 fixed point equation, 118 Fixed Point Theorem of Kakutani, 189 fixed point theorem of Kakutani, 189, 190 flow, 1, 6, 10–13 General Predator-Prey Model, 20 general predator-prey models, 184 global attractor, 110 globally asymptotically stable, 53 globally attractive, 86 Gompertz growth, 214, 217 gradient projection method, 117 halftrajectory, 3 Hamiltonian, 211, 215 identity property, 1, 2, 185 implicit function theorem, 102, 136, 137, 143 Interacting Logistic Growth, 57 invariant, 4, 6, 8, 36–40, 175

invariant subset, 4, 39, 44 invariantly connected, 38, 40 inverse function theorem, 102, 136, 137, 140 Kakutani’s fixed point theorem, 192 Kalman condition, 77–79, 81, 82, 84–86, 110–112, 114, 116, 124 Lagrangean multiplier rule, 92 limit point, 12 limit set, 2, 35–38, 60, 175, 178 linear system, 109 Linearisation, 45 local controllability, 83 Local Fixed Point Controllability, 131 localization of limit sets, 184 locally controllable, 83 locally restricted controllable, 84, 85 Lyapunov, 1 Lyapunov function, 11, 13, 16, 17, 27, 29, 39, 40, 42, 44, 53, 62, 64, 65, 139, 184, 203, 206 Lyapunov’s Method, 40 Lyapunov’s method, 23 Marquardt’s algorithm, 104 matrix game, 202 maximum principle, 211, 215 minimal, 6, 7 minimal subset, 6, 8 minimum norm control, 95, 96, 98 Minimum Norm Problem, 91 minimum norm problem, 91, 93, 94, 96 minimum time, 89, 95, 98 movement, 5, 10 moving linear pendulum, 78 Nash equilibrium, 31–33, 191–194, 196–200, 202, 204, 206, 207 non-autonomous linear systems, 184 non-autonomous systems, 184 non-autonomous time-discrete dynamical system, 60, 66, 99, 131, 137 non-linear planar pendulum, 128 nonlinear pendulum with movable suspension point, 84 Normality Condition, 90

Index normality condition, 91, 93, 95–98 null-controllability, 124 omega limit set, 3–5, 8, 12 optimal control, 213 Optimal Control in Chemotherapy of Cancer, 209 optimization problem, 104 orbit, 2, 11, 35 pay-off, 191 pay-off functions, 192, 196 pay-off matrix, 202 period, 38 period of the movement, 6 period point, 145–147, 166, 168, 172, 174 period points, 157 periodic, 6, 38 periodic movement, 7, 8 periodic orbit, 7, 8, 10, 12 plane pendulum, 8 Poincaré, 1 Poincaré-Bendixon theory, 184 positive definite, 13, 15, 17, 24, 41, 42, 44, 53, 63–65, 139 positive halftrajectory, 5 positive semi-orbit, 12 positively compact, 5, 10–12 positively compact movement, 5, 6 positively compact orbit, 8 positively invariant, 36, 38, 41, 44, 45 predator prey model, 103, 126 Predator-Prey Model, 10, 55 Problem of Controllability, 134 problem of fixed point controllability, 100 Problem of Null-Controllability, 109 problem of null-controllability, 110, 111, 116 Process of Hemo-Dialysis, 71 Reachability, 140 reachable, 140, 141 rest point, 6–8, 10–13, 15, 18–20, 33, 55, 76, 83, 86 Restricted Null-Controllability, 80 restricted null-controllability, 81, 87–89, 95, 98

237

restricted null-controllable, 82, 84 semigroup property, 1, 2, 185 sensitively depending on initial values, 146 set of period points, 145, 149, 150, 181 set of periodic points, 186 shift-mapping, 148, 157, 161, 178, 186, 187 Smale’s horse-shoe-mapping, 176 Solenoid, 180 stability of fixed points, 184 Stabilization, 121, 137 stabilization, 138 stable, 12, 13, 40–42, 44, 50, 53, 63, 64, 66–69, 138, 139 stable equilibrium state, 22, 25, 27 stable fixed point, 52, 53 stable rest point, 16, 17 state function, 99, 131 strange attractor, 176, 180 strange attractors, 187 strategy sets, 191, 194, 196 subset, 4 subshift, 163 symmetric bi-matrix games, 200 system of differential equations, 75 Theorem of Sarkovski, 170, 174 time-continuous, 3 time-continuous dynamical system, 13 time-continuous dynamical systems, 184 time-discrete, 1 time-discrete autonomous dynamical system, 121 time-discrete dynamical system, 35, 54, 58, 156 time-discrete flow, 35 time-minimal control, 96, 98 time-minimal controllability, 185 time-minimal null-control, 95 time-minimal restricted nullcontrollability, 90 topological conjugacy, 187 topological entropy, 159, 161, 163, 165, 174, 187 topological transitivity, 186 topologically conjugated, 150, 156, 180, 183, 187

238

Index

topologically transitive, 145, 147, 149, 150, 157, 181 trajectory, 2, 4 uncontrolled system, 75, 83, 86, 99–101, 103, 105, 109, 110, 123, 129, 131, 134, 140

unstable, 46, 48–50, 122, 123 unstable fixed point, 55, 57 upper semi-continuous, 189

Dynamical systems: Stability, controllability and chaotic behavior

Dynamical Systems: Stability, Controllability and Chaotic Behavior

Stability of Dynamical Systems

Stability of Dynamical Systems

Stability of Dynamical Systems

Stability of dynamical systems

Chaotic Transport in Dynamical Systems

Lectures on chaotic dynamical systems

Lectures on chaotic dynamical systems

Introduction to chaotic dynamical systems

Global Stability of Dynamical Systems

Stability Theory of Dynamical Systems

Global Stability of Dynamical Systems

Stability of Stochastic Dynamical Systems

Uncertain dynamical systems. Stability and motion control

Uncertain Dynamical Systems: Stability and Motion Control

Dynamical Systems: Stability Theory and Applications

Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Dynamical Systems: Stability Theory and Applications

An Introduction to Chaotic Dynamical Systems

Chaotic behavior of rapidly oscillating Lagrangian systems

Stability of Dynamical Systems, Volume 5

The stability of input-output dynamical systems

Stability of dynamical systems : continuous, discontinuous, and discrete systems

The stability of input-output dynamical systems

Stability Theory of Switched Dynamical Systems

Stability of Dynamical Systems: Continuous, Discontinuous, and Discrete Systems

Chaotic Systems

Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control

Lectures on Dynamical Systems, Structural Stability, and their Applications

Lectures on Dynamical Systems, Structural Stability and Their Applications

Dynamical systems: Stability, controllability and chaotic behavior