Stability of finite and infinite dimensional systems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Kluwer Acad. Publ.
1998
|
| Series: | The Kluwer international series in engineering and computer science
455 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 354 S. |
| ISBN: | 0792382218 |
Staff View
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Record in the Search Index
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| adam_text | IMAGE 1
STABILITY OF
FINITE AND INFINITE DIMENSIONAL SYSTEMS
BY
MICHAEL I. GIL BEN GURION UNIVERSITY BEER SHEVA, ISRAEL
W
KLUWER ACADEMIC PUBLISHERS BOSTON / DORDRECHT / LONDON
IMAGE 2
C O N T E N TS
PREFACE
I N T R O D U C T I ON
C H A P T ER 1. P R E L I M I N A R I ES 1
1.1. VECTOR AND MATRIX NORMS 1
1.2. DEFINITIONS OF STABILITY 2
1.3. CLASSES OF MATRICES 4
1.4. EIGENVALUES OF MATRICES 5
1.5. MATRIX-VALUED FUNCTIONS 6
1.6. EVOLUTION OPERATORS 8
1.7. INTEGRAL INEQUALITIES 10
1.8. PERTURBATIONS OF EVOLUTION OPERATORS 11
1.9. LYAPUNOV EXPONENTS 14
1.10. ESTIMATES FOR CONTOUR INTEGRALS 16
1.11. ALGEBRAIC EQUATIONS 17
1.12. NOTES 18
C H A P T ER 2. E S T I M A T ES F OR M A T R I X - V A L U ED F U N C T
I O NS 21 2.1. NORM ESTIMATES FOR MATRIX-VALUED FUNCTIONS 21
2.2. ESTIMATES FOR ABSOLUTE VALUES 26
2.3. IMPULSE FUNCTIONS 28
2.4. POSITIVITY CONDITIONS FOR IMPULSE FUNCTIONS 31
2.5. THE LYAPUNOV EQUATION 34
2.6. NOTES 37
CHAPTER 3. L I N E AR F I N I TE D I M E N S I O N AL S Y S T E MS 39
3.1. GENERAL SYSTEMS 39
3.2. THE FREEZING METHOD FOR LINEAR SYSTEMS 41
3.3. SYSTEMS WITH PIECEWISE CONSTANT MATRICES 47
3.4. TRIANGULAER SYSTEMS 49
3.5. PERTURBATIONS OF TRIANGULAER SYSTEMS 51
3.6. SYSTEMS WITH THE MATRIX LIPSCHITZ PROPERTY 52
3.7. NOTES 59
**
IMAGE 3
VI
CHAPTER 4. L I N E AR F I N I TE D I M E N S I O N AL S Y S T E MS ( C O
N T I N U A T I O N) 63
4.1. THE MULTIPLICATIVE REPRESENTATION OF SOLUTIONS 63
4.2. THE LOZINSKII AND WAZEWSKI INEQUALITIES 65
4.3. LINEAR SYSTEMS WITH MAJORANTS AND MINORA.NTS 68
4.4. PERTURBATIONS OF AUTONOMOUS SYSTEMS 69
4.5. SECOND ORDER SYSTEMS 70
4.6. SCALAR LINEAR EQUATIONS WITH REAL CHARACTERISTIC ROOTS 71
4.7. INPUT-OUTPUT STABILITY OF LINEAR SYSTEMS 72
4.8. NOTES 73
CHAPTER 5. N O N L I N E AR F I N I TE D I M E N S I O N AL S Y S T E MS
W I TH A U T O N O M O US L I N E AR P A R TS 75
5.1. THE AIZERMAN CONJECTURE 75
5.2. THE GENERALIZED AIZERMAN CONJECTURE 78
5.3. REGION OF ATTRACTION AND GLOBAL STABILITY 82
5.4. STABILITY AND INSTABILITY IN THE FIRST APPROXIMATION 84
5.5. INPUT-OUTPUT STABILITY 87
5.6. THE INPUT-OUTPUT VERSION OF AIZERMAN S CONJECTURE 89
5.7. GLOBAL FEEDBACK STABILIZATION 93
5.8. NOTES 94
CHAPTER 6. N O N L I N E AR F I N I TE D I M E N S I O N AL S Y S T E MS
W I TH T I M E - V A R I A NT L I N E AR P A R TS 99
6.1. STABILITY OF SYSTEMS WITH GENERAL LINEAR PARTS 99
6.2. SYSTEMS WITH THE LIPSCHITZ PROPERTY 101
6.3. BOUNDEDNESS OF SOLUTIONSOF SYSTEMS WITH GENERAL LINEAR PARTS 104
6.4. BOUNDEDNESS OF SOLUTIONS OF SYSTEMS WITH THE LIPSCHITZ PROPERTY 106
6.5. INPUT-OUTPUT STABILITY 108
6.6. GLOBAL FEEDBACK STABILIZATION 109
6.7. NOTES 112
CHAPTER 7. E S S E N T I A L LY N O N L I N E AR FINITE D I M E N S I O
N AL S Y S T E MS 115
7.1. THE FREEZING METHOD FOR NONLINEAR SYSTEMS 116
7.2. SYSTEMS WITH DIFFERENTIABLE RIGHT PARTS 119
7.3. THE GENERALIZED LOZINSKII AND WAZEWSKI INEQUALITIES 122
7.4. NONLINEAR SYSTEMS WITH LINEAR MAJORANTS 124
7.5. NONLINEAR TRIANGULAER SYSTEMS 125
7.6. PERTURBATIONS OF NONLINEAR EQUATIONS 127
7.7. NONLINEAR SYSTEMS WHICH ARE CLOSE TO TRIANGULAER ONES 128
7.8. NONLINEAR SCALAR EQUATIONS WITH REAL CHARACTERISTIC ROOTS 129
IMAGE 4
VII
7.9. INPUT-OUTPUT STABILITY OF ESSENTIALLY NONLINEAR SYSTEMS 7.10. NOTES
130 131
C H A P T ER 8. L I N E AR A U T O N O M O US S Y S T E MS W I TH D E L
AY 133
8.1. BANACH SPACES AND CHARACTERISTIC MATRICES 133
8.2. REPRESENTATION OF SOLUTIONS 138
8.3. 2 -NORM ESTIMATES FOR SOLUTIONS OF NONHOMOGENEOUS EQUATIONS 141
8.4. NORM ESTIMATES FOR THE GREEN FUNCTION 144
8.5. THE LYAPUNOV EXPONENT OF THE GREEN FUNCTION 147
8.6. STABILITY CONDITIONS 148
8.7. ESTIMATES FOR THE C-NORM OF SOLUTIONS OF NONHOMOGENEOUS EQUATIONS
150
8.8. PERTURBATIONS OF MATRIX-VALUED FUNCTIONS 151
8.9. BOUNDS FOR ROOTS OF CHARACTERISTIC FUNCTIONS OF RETARDED SYSTEMS
154
8.10. SYSTEMS WITH SMALL DELAYS 155
8.11. STABILITY WITH RESPECT TO ARBITRARY DELAY 156
8.12. NOTES 156
C H A P T ER 9. L I N E AR T I M E - V A R I A NT S Y S T E MS W I TH D
E L AY 163 9.1. DEFINITIONS 9.2. STABILITY OF RETARDED SYSTEMS WITH
BOUNDED COEFFLCIENTS 9.3. THE FREEZING METHOD FOR SYSTEMS WITH DELAY
9.4. PROOFOF THEOREM 9.3.1
9.5. INTEGRALLY SMALL PERTURBATIONS 9.6. THE GENERALIZED WAZEWSKI AND
LOZINSKII INEQUALITIES 9.7. LINEAR TIME-VARIANT SYSTEMS WITH SMALL
DELAYS
9.8. NOTES
163
164 168 171 174 179
181 183
C H A P T ER 10. N O N L I N E AR S Y S T E MS W I TH D E L AY 10 10 10
10 10 10.6. 10.7. 10.8. 10.9. 10.10
DEFINITIONS L 2 - ESTIMATES FOR SOLUTIONS ABSOLUTE STABILITY A BOUND FOR
THE REGION OF ATTRACTION STABILITY IN THE FIRST APPROXIMATION
EXPONENTIAL STABILITY OF NONLINEAR RETARDED SYSTEMS NONLINEAR SYSTEMS
WITH SMALL DELAYS PROOF OF THEOREM 10.7.2 STABILITY WITH RESPECT TO
ARBITRARY DELAY . SYSTEMS WITH NONAUTONOMOUS LINEAR PARTS 10.11. GLOBAL
FEEDBACK STABILIZATION 10.12. INPUT-OUTPUT STABILITY OF NONLINEAR
RETARDED SYSTEMS 10.13. NOTES
187 187 189 192 195 197 198 205 207 211 212 215 217 220
IMAGE 5
*
VU1
C H A P T ER 11. L I N E AR N E U T R AL T Y PE S Y S T E MS 225
11.1. CHARACTERISTIC FUNCTIONS OF AUTONOMOUS NEUTRAL SYSTEMS 225 11.2. L
2 -ESTIMATES FOR SOLUTIONS OF AUTONOMOUS SYSTEMS 229
11.3. REPRESENTATION OF SOLUTIONS OF HOMOGENEOUS SYSTEMS 232
11.4. NORM ESTIMATES FOR CHARACTERISTIC MATRICES 234
11.5. SOLUTION ESTIMATES FOR DIFFERENCE EQUATIONS 236
11.6. ESTIMATES FOR EIGENVALUES OF CHARACTERISTIC MATRICES 237
11.7. 2 -ESTIMATES FOR DERIVATIVES OF SOLUTIONS 238
11.8. STABILITY OF AUTONOMOUS SYSTEMS 239
11.9. STABILITY OF LINEAR TIME-VARIANT SYSTEMS 240
11.10. NOTES 244
CHAPTER 12. N O N L I N E AR N E U T R AL T Y PE F U N C T I O N AL D I
F F E R E N T I AL S Y S T E MS 247
12.1. PRELIMINARIES AND DEFINITIONS 247
12.2. ABSOLUTE STABILITY 249
12.3. PROOFS OF LEMMA 12.2.1 AND THEOREM 12.2.3 252
12.4. STABILITY IN THE FIRST APPROXIMATION 254
12.5. NOTES 258
C H A P T ER 13. S T R O N G LY C O N T I N U O US S E M I G R O U PS
261 13.1. UNBOUNDED LINEAR OPERATORS 262
13.2. LINEAR OPERATORS IN A HUBERT SPACE 263
13.3. DISSIPATIVE OPERATORS IN A HUBERT SPACE 265
13.4. SPECTRAL RESOLUTIONS OF SELFADJOINT AND NORMAL OPERATORS 266 13.5.
VECTOR-VALUED FUNCTIONS OF NORMAL OPERATORS 268
13.6. FUNCTIONS WITH VALUES IN A BANACH SPACE 269
13.7. STRONGLY CONTINUOUS SEMIGROUPS 271
13.8. SECTORIAL OPERATORS, ANALYTIC SEMIGROUPS AND FRACTIONAL POWERS OF
OPERATORS 272
13.9. THE CAUCHY PROBLEM FOR LINEAR EQUATIONS WITH CONSTANT OPERATORS
274
13.10. NORM ESTIMATES FOR A CLASS OF SEMIGROUPS IN A HUBERT SPACE 275
13.11. STABILITY OF LINEAR EQUATIONS WITH CONSTANT OPERATORS 277
13.12. PARABOLIC SYSTEMS WITH CONSTANT COEFFICIENTS 279
13.13. TIME-INVARIANT PARABOLIC SYSTEMS WITH COEFFICIENTS DEPENDING ON
SPATIAL VARIABLES 280
13.14. NOTES 281
C H A P T ER 14. L I N E AR T I M E - V A R I A NT E Q U A T I O NS IN B
A N A CH S P A C ES 285
14.1. EVOLUTION OPERATORS AND SOLUTION EXISTENCE 286
14.2. STABILITY DEFINITIONS FOR LINEAR EVOLUTION EQUATIONS 287
IMAGE 6
IX
14.3. THE MULTIPLICATIVE REPRESENTATION OF SOLUTIONS 288
14.4. THE LOZINSKII AND WAZEWSKI INEQUALITIES FOR EVOLUTION EQUATIONS
290 14.5 LOWER ESTIMATES FOR SOLUTIONS 291
14.6. EQUATIONS WITH RELATIVELY BOUNDED OPERATORS 293
14.7. THE FREEZING METHOD FOR EVOLUTION EQUATIONS 296
14.8. PROOFS OF THEOREMS 14.7.1 AND 14.7.2 300
14.9. SYSTEMS OF LINEAR EQUATIONS WITH MAJORANTS 304
14.10. SYSTEMS OF LINEAR EQUATIONS WITH UNBOUNDED OFF-DIAGONAL OPERATORS
309
14.11. PROOF OF THEOREM 14.10.1 311
14.12. NOTES 312
C H A P T ER 15. S E M I L I N E AR E Q U A T I O NS IN B A N A CH S P A
C ES W I TH C O N S T A NT L I N E AR P A R TS 315
15.1. DEFINTIONS 316
15.2. STABILITY OF EQUATIONS IN A BANACH SPACE 317
15.3. PROOF OF THEOREM 15.2.1 318
15.4. ABSOLUTE STABILITY OF EQUATIONS WITH CONSTANT OPERATORS IN A
HUBERT SPACE 320
15.5. THE REGION OF ATTRACTION FOR EQUATIONS IN HUBERT SPACES 323
15.6. THE AIZERMAN HYPOTHESIS FOR EQUATIONS IN A HUBERT SPACE 324
15.7. PASSAGE TO AN IMBEDDED NORM 328
15.8. NOTES 330
C H A P T ER 16. S E M I L I N E AR E Q U A T I O NS IN B A N A CH S P A
C ES W I TH T I M E - V A R I A NT L I N E AR P A R TS 333
16.1. GENERAL EQUATIONS IN A BANACH SPACE 333
16.2. EQUATIONS IN BANACH SPACES WITH THE LIPSCHITZ CONTINUOUS LINEAR
PARTS 336
16.3. EQUATIONS IN A HUBERT SPACE 337
16.4. EXAMPLE 339
16.5. NOTES 341
C H A P T ER 17. A P P E N D IX 1 343
17.1. PROOF OF THE ESTIMATE FOR REGULAER MATRIX-VALUED FUNCTIONS 343
17.2. AN INDEPENDENT PROOF OF COROLLARY 2.1.7 350
17.3. PROOF OF THE ESTIMATE FOR THE NORM OF RESOLVENTS 353
17.4. PROOF OF PROPOSITION 2.1.1 354
LIST OF M A IN S Y M B O LS 355
INDEX
FC
|
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| id | DE-604.BV012773305 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:33:24Z |
| institution | BVB |
| isbn | 0792382218 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008686127 |
| oclc_num | 39307394 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XVIII, 354 S. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | The Kluwer international series in engineering and computer science |
| series2 | The Kluwer international series in engineering and computer science |
| spelling | Gil', Michail I. 1941- Verfasser (DE-588)128577916 aut Stability of finite and infinite dimensional systems by Michael I. Gil' Boston [u.a.] Kluwer Acad. Publ. 1998 XVIII, 354 S. txt rdacontent n rdamedia nc rdacarrier The Kluwer international series in engineering and computer science 455 Automatic control Control theory Differential equations, Partial Stability Stabilität (DE-588)4056693-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Stabilität (DE-588)4056693-6 s DE-604 The Kluwer international series in engineering and computer science 455 (DE-604)BV023545171 455 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008686127&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gil', Michail I. 1941- Stability of finite and infinite dimensional systems The Kluwer international series in engineering and computer science Automatic control Control theory Differential equations, Partial Stability Stabilität (DE-588)4056693-6 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4012249-9 |
| title | Stability of finite and infinite dimensional systems |
| title_auth | Stability of finite and infinite dimensional systems |
| title_exact_search | Stability of finite and infinite dimensional systems |
| title_full | Stability of finite and infinite dimensional systems by Michael I. Gil' |
| title_fullStr | Stability of finite and infinite dimensional systems by Michael I. Gil' |
| title_full_unstemmed | Stability of finite and infinite dimensional systems by Michael I. Gil' |
| title_short | Stability of finite and infinite dimensional systems |
| title_sort | stability of finite and infinite dimensional systems |
| topic | Automatic control Control theory Differential equations, Partial Stability Stabilität (DE-588)4056693-6 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Automatic control Control theory Differential equations, Partial Stability Stabilität Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008686127&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023545171 |
| work_keys_str_mv | AT gilmichaili stabilityoffiniteandinfinitedimensionalsystems |