Holdings: Two-scale stochastic systems

Two-scale stochastic systems: asymptotic analysis and control

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Bibliographic Details
Main Authors:	Kabanov, Jurij M. 1948- (Author), Pergamenščikov, Sergej M. (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2003
Series:	Applications of mathematics 49
Subjects:	Stochastische Differentialgleichung - Singuläre Störung Stochastische Differentialgleichung Singuläre Störung
Online Access:	Inhaltsverzeichnis
Item Description:	Literaturverz. S. 259 - 263
Physical Description:	XIV, 266 S. 24 cm
ISBN:	3540653325

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Record in the Search Index

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adam_text	TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX 0 WARM-UP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 PROCESSES WITH FAST MARKOV MODULATIONS . . . . . . . . . . . . . . . . . . 1 0.1.1 MODEL FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1.2 ASYMPTOTIC BEHAVIOR OF D ISTRIBUTIONS . . . . . . . . . . . . . . . 2 0.2 THE LI´ ENARD OSCILLATOR UNDER RANDOM FORCE . . . . . . . . . . . . . . . 6 0.3 FILTERING OF NEARLY OBSERVED PROCESSES . . . . . . . . . . . . . . . . . . . . 9 0.4 STOCHASTIC APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 TOOLBOX: MOMENT BOUNDS FOR SOLUTIONS OF STABLE SDES . . . 19 1.1 MOMENT BOUNDS FOR NONLINEAR EQUATIONS . . . . . . . . . . . . . . . . . . 20 1.1.1 KEY LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.1.2 BOUNDS EFFICIENT ON SMALL INTERVALS . . . . . . . . . . . . . . . . . 22 1.1.3 BOUNDS EFFICIENT ON LARGE INTERVALS . . . . . . . . . . . . . . . . . 24 1.2 BOUNDS FOR LINEAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2.1 ASSUMPTION ON THE FUNDAMENTAL MATRIX. . . . . . . . . . . . . 26 1.2.2 DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS . . . . . 27 1.2.3 THE CONTINUITY THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.2.4 LINEAR SDES WITH UNBOUNDED COEFFICIENTS . . . . . . . . . . . 31 1.3 ON THE GROWTH RATE OF THE MAXIMAL FUNCTION . . . . . . . . . . . . . 34 1.3.1 LAPEYRES INEQUALITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.3.2 ORNSTEINUHLENBECK PROCESS . . . . . . . . . . . . . . . . . . . . . . . 37 1.3.3 SAMPLE PATH GROWTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.3.4 FERNIQUES LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 THE TIKHONOV THEORY FOR SDES . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.1 THE STOCHASTIC TIKHONOV THEOREM . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.1 SETTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.2 BOUNDARY LAYER BEHAVIOR . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1.3 LARGE SCALE BEHAVIOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.4 CONCLUDING STEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 VI TABLE OF CONTENTS 2.2 THE FIRST-ORDER ASYMPTOTICS FOR FAST VARIABLES . . . . . . . . . . . . 56 2.2.1 BASIC HYPOTHESES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2.2 THE FIRST-ORDER CORRECTION . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2.3 THE FIRST-ORDER APPROXIMATION OF THE REST POINT . . . . 59 2.2.4 NORMAL APPROXIMATION RESULT . . . . . . . . . . . . . . . . . . . . . 62 2.3 HIGHER-ORDER EXPANSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.3.1 FORMAL EXPANSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.3.2 CONVERGENCE OF THE REMAINDER . . . . . . . . . . . . . . . . . . . . . 65 2.3.3 EXPANSION AROUND THE REST POINT . . . . . . . . . . . . . . . . . . 69 2.4 STOCHASTIC APPROXIMATION: PROOFS . . . . . . . . . . . . . . . . . . . . . . . . 70 2.4.1 ASYMPTOTIC EXPANSION FOR THE OUTPUT SIGNAL . . . . . . . . 70 2.4.2 THE ASYMPTOTIC EXPANSION AT THE ROOT . . . . . . . . . . . . . 78 2.4.3 AVERAGING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.4.4 PROOF OF THEOREM 0.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3 LARGE DEVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.1 D EVIATIONS IN THE UNIFORM METRIC . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1.1 FORMULATION OF THE RESULT . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1.2 A LOWER EXPONENTIAL BOUND FOR THE NON-EXIT PROBABILITY . . . . . . . . . . . . . . . . . . . . . . . 89 3.1.3 AN UPPER BOUND FOR THE PROBABILITY OF DEVIATION OF A TRAJECTORY FROM THE LEBESGUE SETS OF S T . . . . . . . . 91 3.1.4 PROOF OF THEOREM 3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.1.5 EXAMPLE: THE ORNSTEINUHLENBECK PROCESS . . . . . . . . . . . 104 3.2 DEVIATIONS IN THE METRIC OF L 2 [0 ,T ] . . . . . . . . . . . . . . . . . . . . . . . 105 4 UNIFORM EXPANSIONS FOR TWO-SCALE SYSTEMS . . . . . . . . . . . . . . . 111 4.1 NO D IFFUSION AT THE FAST VARIABLE . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.1 FORMAL CALCULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.1.2 INTEGRABILITY OF COEFFICIENTS . . . . . . . . . . . . . . . . . . . . . . . . 119 4.1.3 THE BOUNDARY LAYER FUNCTION OF ZERO ORDER . . . . . . . . . 120 4.1.4 BOUNDARY LAYER FUNCTIONS OF HIGHER ORDER . . . . . . . . . . 124 4.1.5 PROOF OF THEOREM 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2 EXPANSIONS FOR THE GENERAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . 133 4.2.1 FORMULATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.2.2 GROWTH OF COEFFICIENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.3 PROOF OF THEOREM 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.3 LI´ ENARD OSCILLATOR DRIVEN BY A RANDOM FORCE . . . . . . . . . . . . . . 140 TABLE OF CONTENTS VII 5 TWO-SCALE OPTIMAL CONTROL PROBLEMS . . . . . . . . . . . . . . . . . . . . . 145 5.1 SEMILINEAR CONTROLLED SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.1.1 THE MODEL AND MAIN RESULT . . . . . . . . . . . . . . . . . . . . . . . 146 5.1.2 PROOF OF PROPOSITION 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.1.3 PROOF OF PROPOSITION 5.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.1.4 PROOF OF THEOREM 5.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.2 STRUCTURE OF THE ATTAINABILITY SETS . . . . . . . . . . . . . . . . . . . . . . . . 158 5.2.1 WEAK AND STRONG SOLUTIONS OF SD ES . . . . . . . . . . . . . . . . 158 5.2.2 CLOSED LOOP CONTROLS VERSUS OPEN LOOP . . . . . . . . . . . . 160 5.2.3 TUBES* AND ATTAINABILITY SETS FOR FEEDBACK CONTROLS . 164 5.2.4 EXTREME POINTS OF THE SET OF ATTAINABLE DENSITIES . . . . 167 5.2.5 ON THE EXISTENCE OF OPTIMAL CONTROL . . . . . . . . . . . . . . . 169 5.2.6 COMPARISON OF ATTAINABILITY SETS . . . . . . . . . . . . . . . . . . . 171 5.3 CONVERGENCE OF THE ATTAINABILITY SETS, I . . . . . . . . . . . . . . . . . . . 175 5.3.1 THE D ONTCHEV*VELIOV THEOREM . . . . . . . . . . . . . . . . . . . . . 175 5.3.2 THE FIRST STOCHASTIC GENERALIZATION . . . . . . . . . . . . . . . . . 176 5.4 CONVERGENCE OF THE ATTAINABILITY SETS, II . . . . . . . . . . . . . . . . . . 180 5.4.1 FORMULATION OF THE RESULT . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.4.2 THE FAST VARIABLE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.4.3 GENERAL CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.4.4 PROOF OF THEOREM 5.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.1 APPLICATIONS TO PD ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2 FAST MARKOV MODULATIONS REVISITED . . . . . . . . . . . . . . . . . . . . . . . 199 6.2.1 MAIN RESULT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.2.2 PRELIMINARIES FROM WEAK CONVERGENCE . . . . . . . . . . . . . . . 200 6.2.3 PROOF OF THEOREM 6.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 6.2.4 CALCULATIONS AND ESTIMATES . . . . . . . . . . . . . . . . . . . . . . . . 203 6.2.5 COX PROCESSES WITH FAST MARKOV MODULATIONS . . . . . . . . 206 6.3 ACCURACY OF APPROXIMATE FILTERS . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.4 SIGNAL ESTIMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.5 LINEAR REGULATOR WITH INFINITE HORIZON . . . . . . . . . . . . . . . . . . . . 213 6.5.1 SENSITIVE PROBABILISTIC CRITERIA . . . . . . . . . . . . . . . . . . . . . 213 6.5.2 LINEAR-QUADRATIC REGULATOR . . . . . . . . . . . . . . . . . . . . . . . . 214 6.5.3 PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.5.4 PROOF OF THEOREM 6.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.5.5 EXAMPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 VIII TABLE OF CONTENTS APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A.1 BASIC FACTS ABOUT SD ES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 A.1.1 EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR SD ES WITH RANDOM COEFFICIENTS . . . . . . . . . . . . . . . . . 223 A.1.2 EXISTENCE AND UNIQUENESS WITH A LYAPUNOV FUNCTION . 224 A.1.3 MOMENT BOUNDS FOR LINEAR SDES . . . . . . . . . . . . . . . . . . . 225 A.1.4 THE NOVIKOV CONDITION . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 A.2 EXPONENTIAL BOUNDS FOR FUNDAMENTAL MATRICES . . . . . . . . . . . . . 227 A.2.1 UNIFORM BOUND IN THE TIME-HOMOGENEOUS CASE . . . . . . 227 A.2.2 NONHOMOGENEOUS CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 A.2.3 MODELS WITH SINGULAR PERTURBATIONS . . . . . . . . . . . . . . . . . 230 A.3 TOTAL VARIATION DISTANCE AND HELLINGER PROCESSES . . . . . . . . . . . 234 A.3.1 TOTAL VARIATION DISTANCE AND HELLINGER INTEGRALS . . . . . . 234 A.3.2 THE HELLINGER PROCESSES . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 A.3.3 EXAMPLE: D IFFUSION-TYPE PROCESSES . . . . . . . . . . . . . . . . . 238 A.4 HAUSDORFF METRIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 A.5 MEASURABLE SELECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 A.5.1 AUMANN THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 A.5.2 FILIPPOV IMPLICIT FUNCTION LEMMA . . . . . . . . . . . . . . . . . . 241 A.5.3 MEASURABLE VERSION OF THE CARATH´ EODORY THEOREM . . . . 241 A.6 COMPACT SETS IN P ( X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 A.6.1 NOTATIONS AND PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . 243 A.6.2 INTEGRATION OF STOCHASTIC KERNELS . . . . . . . . . . . . . . . . . . . 245 A.6.3 D ISTRIBUTIONS OF INTEGRALS . . . . . . . . . . . . . . . . . . . . . . . . . . 246 A.6.4 COMPACTNESS OF THE LIMIT OF ATTAINABILITY SETS . . . . . . . 248 A.6.5 SUPPORTS OF CONDITIONAL DISTRIBUTIONS . . . . . . . . . . . . . . . 250 A.7 THE KOML´ OS THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 HISTORICAL NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
any_adam_object	1
author	Kabanov, Jurij M. 1948- Pergamenščikov, Sergej M.
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author_facet	Kabanov, Jurij M. 1948- Pergamenščikov, Sergej M.
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id	DE-604.BV014862163
illustrated	Not Illustrated
indexdate	2024-07-09T19:08:04Z
institution	BVB
isbn	3540653325
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-010049558
oclc_num	248761056
open_access_boolean
owner	DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-83 DE-188
owner_facet	DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-824 DE-83 DE-188
physical	XIV, 266 S. 24 cm
publishDate	2003
publishDateSearch	2003
publishDateSort	2003
publisher	Springer
record_format	marc
series	Applications of mathematics
series2	Applications of mathematics
spelling	Kabanov, Jurij M. 1948- Verfasser (DE-588)124079199 aut Two-scale stochastic systems asymptotic analysis and control Yuri Kabanov ; Sergei Pergamanshchikov Berlin [u.a.] Springer 2003 XIV, 266 S. 24 cm txt rdacontent n rdamedia nc rdacarrier Applications of mathematics 49 Literaturverz. S. 259 - 263 Stochastische Differentialgleichung - Singuläre Störung Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Singuläre Störung (DE-588)4055100-3 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 s Singuläre Störung (DE-588)4055100-3 s DE-604 Pergamenščikov, Sergej M. Verfasser (DE-588)120524562 aut Applications of mathematics 49 (DE-604)BV000895226 49 SWB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010049558&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Kabanov, Jurij M. 1948- Pergamenščikov, Sergej M. Two-scale stochastic systems asymptotic analysis and control Applications of mathematics Stochastische Differentialgleichung - Singuläre Störung Stochastische Differentialgleichung (DE-588)4057621-8 gnd Singuläre Störung (DE-588)4055100-3 gnd
subject_GND	(DE-588)4057621-8 (DE-588)4055100-3
title	Two-scale stochastic systems asymptotic analysis and control
title_auth	Two-scale stochastic systems asymptotic analysis and control
title_exact_search	Two-scale stochastic systems asymptotic analysis and control
title_full	Two-scale stochastic systems asymptotic analysis and control Yuri Kabanov ; Sergei Pergamanshchikov
title_fullStr	Two-scale stochastic systems asymptotic analysis and control Yuri Kabanov ; Sergei Pergamanshchikov
title_full_unstemmed	Two-scale stochastic systems asymptotic analysis and control Yuri Kabanov ; Sergei Pergamanshchikov
title_short	Two-scale stochastic systems
title_sort	two scale stochastic systems asymptotic analysis and control
title_sub	asymptotic analysis and control
topic	Stochastische Differentialgleichung - Singuläre Störung Stochastische Differentialgleichung (DE-588)4057621-8 gnd Singuläre Störung (DE-588)4055100-3 gnd
topic_facet	Stochastische Differentialgleichung - Singuläre Störung Stochastische Differentialgleichung Singuläre Störung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010049558&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV000895226
work_keys_str_mv	AT kabanovjurijm twoscalestochasticsystemsasymptoticanalysisandcontrol AT pergamenscikovsergejm twoscalestochasticsystemsasymptoticanalysisandcontrol

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