Vector Measures and Control Systems

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