Introduction to the Series |
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xi | |
Preface |
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xiii | |
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1 | (22) |
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1 | (1) |
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Random Variables and Probability Distributions |
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2 | (3) |
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A multidimensional normal (Gaussian) distribution |
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4 | (1) |
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A Poisson distribution with parameter λ > 0 |
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5 | (1) |
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Probability Processes and their Mathematical Description. Markov Processes |
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5 | (5) |
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Random Differential Equations. Modelling of Systems Exposed to Random Perturbations |
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10 | (4) |
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System with Random Structure. Examples |
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14 | (8) |
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22 | (1) |
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Stability Analysis Using Scalar Lyapunov Functions |
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23 | (50) |
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23 | (1) |
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Stability Concepts for Stochastic Systems |
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24 | (4) |
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Random Scalar Lyapunov Functions |
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28 | (6) |
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Conditions of Stability in Probability |
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34 | (12) |
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46 | (6) |
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52 | (12) |
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Stability in Mean Square of Linear Systems |
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64 | (5) |
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69 | (4) |
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Stability Analysis Using Multicomponent Lyapunov Functions |
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73 | (62) |
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73 | (1) |
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Vector Lyapunov Functions |
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74 | (9) |
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75 | (5) |
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80 | (3) |
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Stochastic Matrix-Valued Lyapunov Functions |
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83 | (10) |
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The structure of averaged derivatives of stochastic matrix-valued functions |
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87 | (6) |
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Stability Analysis in General |
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93 | (2) |
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Stability Analysis of Systems in Ito's Form |
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95 | (9) |
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Decomposition of perturbed motion equations |
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95 | (3) |
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The structure of averaged derivatives of hierarchical matrix-valued functions |
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98 | (2) |
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Sufficient conditions of stability in probability of a stochastic Ito system |
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100 | (4) |
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Stochastic Singularly Perturbed Systems |
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104 | (13) |
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Statement of the problem and main definitions |
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104 | (2) |
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SMVF for singularly perturbed systems |
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106 | (3) |
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Theorems on stability in the whole of a singularly perturbed system |
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109 | (3) |
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A stochastic linear autonomous singularly perturbed system |
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112 | (5) |
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Large-Scale Singularly Perturbed Systems |
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117 | (16) |
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Description of the systems |
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117 | (2) |
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119 | (1) |
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Stability conditions for stochastic singularly perturbed autonomous large-scale systems |
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120 | (9) |
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Analysis of linear autonomous singularly perturbed large-scale system |
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129 | (4) |
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133 | (2) |
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Stability Analysis by the First-Order Approximation |
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135 | (30) |
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135 | (1) |
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Stability Criterion by the First-Order Approximation |
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135 | (6) |
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Asymptotic stability in probability in the whole and exponential stability in mean square |
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135 | (3) |
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A systems with random structure and slowly-varying right-hand sides |
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138 | (3) |
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Stability with Respect to the First-Order Approximation (continued) |
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141 | (8) |
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The system of the first-order approximation with constant structure |
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141 | (3) |
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The system of the first-order approximation with an unstable structure |
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144 | (5) |
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Stability by First-Order Approximation of Systems with Random Delay |
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149 | (7) |
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Convergence of Stochastic Approximation Procedures |
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156 | (7) |
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The Robbins -- Monro procedure |
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156 | (1) |
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The stochastic programming procedure |
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157 | (3) |
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The procedure of establishing the root of an unknown function |
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160 | (2) |
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Time-continuous measurements in the presence of random jumps |
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162 | (1) |
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163 | (2) |
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Stabilization of Controlled Systems with Random Structure |
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165 | (36) |
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165 | (1) |
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Problems of Stabilization |
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166 | (5) |
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171 | (4) |
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Linear-Quadratic Optimal Stabilization |
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175 | (5) |
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Sufficient Stabilization Conditions for Linear Systems |
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180 | (6) |
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Optimal Solution Existence |
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186 | (5) |
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The Small Parameter Method Algorithm |
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191 | (8) |
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199 | (2) |
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201 | (18) |
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201 | (1) |
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A Stochastic Version of the Lefschetz Problem |
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202 | (2) |
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Stability in Probability of Oscillating Systems |
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204 | (4) |
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Stability in Probability of Regulation Systems |
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208 | (3) |
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Price Stability in a Stochastic Market Model |
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211 | (7) |
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A stochastic market model |
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212 | (1) |
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A management problem based on stability conditions |
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212 | (1) |
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213 | (5) |
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218 | (1) |
References |
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219 | (16) |
Index |
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235 | |