PDF - Universiteit Twente

More documents

Recommendations

Info

Summary The relation between continuous time systems and discrete time systems is the main topic of this research. A continuous time system can be transformed into a discrete time system using the Cayley transform. In this process, the solutions of the continuous time system, given by a semigroup, are mapped to the solutions of the discrete time system, given by a power sequence. To be more precise, the generator of the semigroup is mapped to a difference operator, the cogenerator. Stability analysis plays a central role in the study of the relation between continuous and discrete time systems. Do stable continuous time systems correspond to stable discrete time systems? This is the main question of this dissertation. For many stable continuous time systems the corresponding discrete time systems are stable as well. In Banach spaces however, several examples are known of stable semigroups where the corresponding cogenerators have unstable power sequences. In Hilbert spaces no such examples are known. It remains an open problem whether for every stable continuous time system the corresponding discrete time system is stable as well. This dissertation addresses the main question in three ways. First, a growth bound for the cogenerator is provided. This bound holds for exponentially stable semigroups in Hilbert spaces. Using Lyapunov equations it is shown that for such semigroups the power sequence of the corresponding cogenerator cannot grow faster than ln(n). Second, we extend the class of stable continuous time systems for which the corresponding discrete time systems are stable as well. For this, the notion of Bergman distance is introduced. The Bergman distance defines a metric for semigroups and a metric for power sequences. In other words, the Bergman distance is a distance between two semigroups or two power sequences. If the Bergman distance is finite, the two semigroups have the same stability behaviour. This holds for two power sequences as well. Furthermore, the Bergman distance is preserved by the Cayley transform. This means that the Bergman distance of two semigroups is equal to the Bergman distance 103
Summary of the power sequences of their corresponding cogenerators. Combined with the stability properties of the Bergman distance, this enables us to identify new stable continuous time systems for which the corresponding discrete time systems are stable as well. Third, the inverse of the generator is taken into account in the stability analysis. For stable semigroups the growth of the power sequence of the cogenerator is related to the growth of the semigroup generated by the inverse of the generator. For exponentially stable semigroups on Banach spaces similarity is shown between the growth of the semigroup of the inverse and the growth of the power sequence of the cogenerator. Bounded semigroups on Hilbert spaces are examined as well. It is shown that if the semigroup generated by the inverse is bounded, the growth of the cogenerator is bounded as well. 104
Page 2 and 3:
Stability analysis in continuous an
Page 4 and 5:
STABILITY ANALYSIS IN CONTINUOUS AN
Page 6:
This thesis is dedicated to my pare
Page 9 and 10:
Preface side(s) of the North Sea an
Page 11 and 12:
Contents 5 Extension Bergman distan
Page 13 and 14:
Chapter 1. Introduction This family
Page 15 and 16:
Chapter 1. Introduction In this the
Page 17 and 18:
Chapter 1. Introduction 600 500 400
Page 19 and 20:
Chapter 1. Introduction Lemma 1.3 L
Page 21 and 22:
Chapter 1. Introduction A contracti
Page 23 and 24:
Chapter 1. Introduction 1.5 General
Page 25 and 26:
Chapter 1. Introduction we find tha
Page 28 and 29:
Chapter 2 Stability In this chapter
Page 30 and 31:
is strongly stable. However its adj
Page 32 and 33:
2.1. Continuous-time case For furth
Page 34 and 35:
2.2 Discrete-time case Next, we def
Page 36 and 37:
2.2. Discrete-time case Remark 2.16
Page 38 and 39:
2.2. Discrete-time case To prove th
Page 40 and 41:
2.3. From continuous to discrete ti
Page 42 and 43:
2.3. From continuous to discrete ti
Page 44 and 45:
Chapter 3 Log estimate using Lyapun
Page 46 and 47:
3.2. Estimates on operators Definit
Page 48 and 49:
3.2. Estimates on operators where w
Page 50 and 51:
3.3. Proof of Theorem 3.1 Proof: If
Page 52 and 53:
Chapter 4 Bergman distance 4.1 Intr
Page 54 and 55:
4.1. Introduction � So two semigr
Page 56 and 57:
4.2. Properties of semigroups with
Page 58 and 59:
4.3. Properties of cogenerators wit
Page 60 and 61:
4.4. Equivalence of the Bergman dis
Page 62 and 63:
4.4. Equivalence of the Bergman dis
Page 64 and 65: 4.5. Proof of Theorem 4.7 In partic
Page 66 and 67: 4.6. Applications The first time in
Page 68 and 69: 4.6. Applications In [16, Theorem I
Page 70: 4.7. Conclusions stability properti
Page 73 and 74: Chapter 5. Extension Bergman distan
Page 87 and 88: Chapter 6. Norm relations using Lag
Page 96 and 97: Chapter 7 Growth relation cogenerat
Page 98 and 99: we have for t ≥ 1 �e (A0)dt ∞
Page 100 and 101: 7.1. Exponentially stable semigroup
Page 102 and 103: 7.2. Bounded semigroups on a Hilber
Page 104 and 105: Substituting this in (7.11), we fin
Page 106 and 107: 7.2. Bounded semigroups on a Hilber
Page 108: 7.3 Conclusions 7.3. Conclusions In
Page 111 and 112: Bibliography [10] R. F. Curtain and
Page 113: Bibliography [35] M. Tucsnak and G.
Page 117 and 118: Samenvatting machtreeksen. Dit bete
show all

PDF - Universiteit Twente

Create successful ePaper yourself

Delete template?

Save as template?