Vector Measures and Control Systems (eBook)

Front Cover 1
Vector Measures and Control Systems 4
Copyright Page 5
Table of Contents 8
CHAPTER I. PRELIMINARIES 12
1. Locally Convex Cpaces 12
2. Extreme and Exposed Points 15
3. Measure Spaces 19
4. Conical Measures 21
Remarks 25
CHAPTER II. VECTOR MEASURES AND INTEGRATION 27
1. Vector Measures Variation and Semi-Variation
2. Integration 32
3. Integrability of Bounded Functions 37
4. Limit Theorems 38
5. A Sufficient Condition for Integrability 41
6. An Isomorphism Theorem 43
7. Direct Sum of Vector Measures 46
Remarks 47
CHAPTER III. FUNCTION SPACES I 49
1. Topologies 49
2. Some Relations Between Topologies 52
3. Completeness 56
4. Lattice Completeness 60
5. Weak Compactness 65
6. Completion 68
7. Extreme and Exposed Points 70
8. Vector-Valued Functions 72
Remarks 77
CHAPTER IV. CLOSED VECTOR MEASURES 78
1. Properties of the Integration Mapping 78
2. Closed Vector Measures 81
3. Closure of a Vector Measure 83
4. Completeness of L1 (m) 84
5. Lattice Completeness 85
6. Weak Compactness of the Range 86
7. Sufficient Conditions for Closedness 89
Remarks 91
CHAPTER V. LIAPUNOV VECTOR MEASURES 93
1. Liapunov Vector Measures 93
2. Consequences of the Test 96
3. Liapunov Decomposition 99
4. Moment Sequences 100
5. Liapunov Extension 104
6. Non-Atomic Vector Measures 105
7. Examples of Bang-Bang Control 109
Remarks 121
CHAPTER VI. EXTREME AND EXPOSED POINTS OF THE RANGE 123
1 . Extreme Points 123
2. Properties of the Set of Extreme Points 126
3 . Rybakov's Theorem 131
4. Exposed Points of the Range 133
Remarks 138
CHAPTER VII. THE RANGE OF A VECTOR MEASURE 139
1. The Problem 139
2. The Conical Measure Associated with a Vector Measure 141
3. The Relation Between m and .(m) 145
4. Consequences of the Test 148
Remarks 150
CHAPTER VIII. FUNCTION SPACES II 153
1. Set-Valued Functions 153
2. Measurable Selections 156
3 . Sequences of Measures 159
4. Extreme Points 163
Remarks 164
CHAPTER IX. CONTROL SYSTEMS 165
1. Attainable Set 165
2. Extreme Points of the Attainable Set 167
3. Liapunov Control Systems 169
4. Non-Atomic Control Systems 171
5. Time-optimal Control 173
Remarks 176
BIBLIOGRAPHY 180
NOTATION INDEX 188
INDEX 190