Preface | p. ix |
Notations | p. xi |
General introduction | p. xix |
Introductory comments on stability concepts | p. 1 |
Comments on Lyapunov's stability concept | p. 1 |
Lyapunov's definition of stability and the definitions of stability in the Lyapunov sense | p. 1 |
Definitions of attraction | p. 8 |
Definitions of asymptotic stability | p. 9 |
Definitions of exponential stability | p. 10 |
Definitions of absolute stability on N[subscript i](L) | p. 11 |
Definitions of attraction with finite attraction time | p. 13 |
Definitions of stability with finite attraction time | p. 14 |
Definitions of absolute stability with finite attraction time | p. 14 |
Comments on the practical stability concept | p. 15 |
Introductory comments | p. 15 |
Definition of practical stability | p. 16 |
Definition of practical contraction with settling time | p. 18 |
Definition of practical stability with settling time | p. 19 |
Stability domain concepts | p. 21 |
Introductory comments | p. 21 |
Domains of Lyapunov stability properties | p. 21 |
The notion of domain | p. 21 |
Definitions of stability domains | p. 22 |
Definitions of attraction domains | p. 28 |
Definitions of asymptotic stability domains | p. 33 |
Definitions of exponential stability domains | p. 36 |
Definitions of asymptotic stability domains on N[subscript (.)] (..) | p. 39 |
Domains of practical stability properties | p. 40 |
Definitions of domains of practical stability | p. 40 |
Definitions of domains of practical contraction with settling time | p. 41 |
Definitions of domains of practical stability with settling time | p. 41 |
Qualitative features of stability domains properties | p. 43 |
Introductory comments | p. 43 |
Definition of a motion | p. 43 |
Existence of motions | p. 45 |
Existence and uniqueness of motions | p. 47 |
Continuity of motions in initial conditions | p. 53 |
Differentiability of motions in initial conditions | p. 53 |
Generalised motions | p. 54 |
Motivation | p. 54 |
Dini derivatives | p. 54 |
Generalised motions | p. 57 |
Limit points and limit sets | p. 61 |
Limit sets, Lagrange stability, precompactness and stability domains | p. 64 |
Invariance properties of sets | p. 68 |
Invariance properties of limit sets | p. 73 |
System regimes | p. 75 |
Forced regimes and the free regime | p. 75 |
Periodic regimes | p. 75 |
Stationary regimes and stationary points | p. 76 |
Equilibrium regimes and equilibrium points | p. 77 |
Invariance properties of sets of equilibrium states | p. 79 |
Dynamical and generalised dynamical systems | p. 81 |
Definition and properties of dynamical systems | p. 81 |
Definition and properties of generalised dynamical systems | p. 86 |
Stability properties and invariance properties of sets | p. 86 |
Invariance features of stability domains properties | p. 88 |
Features of equilibrium states on boundaries of domains of stability properties | p. 90 |
Foundations of the Lyapunov method | p. 93 |
Introductory comment | p. 93 |
Sign definite functions | p. 94 |
Sign semi-definite functions | p. 94 |
Sign definite functions | p. 96 |
Comparison functions | p. 100 |
Positive definite functions and comparison functions | p. 101 |
Radially unbounded and radially increasing positive definite functions | p. 102 |
Uniquely bounded sets | p. 105 |
Definition of uniquely bounded sets | p. 105 |
Properties of uniquely bounded sets | p. 106 |
O-uniquely bounded sets and positive definite functions | p. 108 |
Definition of uniquely bounded neighbourhoods of sets | p. 108 |
Properties of uniquely bounded neighbourhoods of sets | p. 109 |
Dini derivatives and the Lyapunov method | p. 113 |
Fundamental lemmae on Dini derivatives | p. 113 |
LaSalle principle | p. 116 |
Dini derivatives, positive definiteness, positive invariance and precompactness | p. 118 |
Stability theorems | p. 119 |
Stability of a set | p. 120 |
Stability of X = 0 | p. 121 |
Comment | p. 121 |
Asymptotic stability theorems | p. 121 |
Asymptotic stability of a set | p. 122 |
Complete global asymptotic stability of sets | p. 125 |
Asymptotic stability of X = 0 | p. 126 |
Complete global asymptotic stability of X = 0 | p. 126 |
Absolute stability of X = 0 of Lurie systems | p. 127 |
Exponential stability of X = 0 | p. 134 |
Krasovskii criterion | p. 134 |
Yoshizawa criterion | p. 135 |
Stability domain estimates | p. 136 |
Definitions of stability domain estimates | p. 136 |
Estimates of the stability domain of a set | p. 136 |
Estimates of the stability domain of X = 0 | p. 138 |
Asymptotic stability domain and estimates | p. 139 |
Classical approach | p. 139 |
Definition of asymptotic stability domain estimate | p. 143 |
Estimates of the asymptotic stability domain of a set | p. 143 |
Estimates of the asymptotic stability domain of X = 0 | p. 146 |
Exponential stability domain estimate | p. 156 |
Definition of exponential stability domain estimate | p. 156 |
Estimates of the exponential stability domain of a set | p. 156 |
Estimates of the exponential stability domain of X = 0 | p. 157 |
Asymptotic stability domains on N[subscript (.)] (..) | p. 158 |
Definition of estimate of the asymptotic stability domain on N[subscript (.)] (..) | p. 158 |
Algebraic approach | p. 158 |
Frequency domain approach | p. 162 |
Novel development of the Lyapunov method | p. 165 |
Introductory comment | p. 165 |
Systems with differentiable motions | p. 166 |
Smoothness property | p. 166 |
Two-stage approach | p. 166 |
Approach via O-uniquely bounded sets | p. 181 |
General one-shot approach | p. 195 |
Exponential stability | p. 204 |
Systems with continuous motions (generalised motions) | p. 208 |
Smoothness property | p. 208 |
Approach via O-uniquely bounded sets | p. 209 |
General one-shot approach | p. 212 |
Conclusion | p. 215 |
Foundations of practical stability domains | p. 217 |
Introductory comment | p. 217 |
System aggregation function and sets | p. 217 |
System description and sets | p. 217 |
Definition of estimates of practical stability domains of systems | p. 218 |
System aggregation function extrema and sets | p. 219 |
Estimate of the system practical stability domain | p. 220 |
Estimate of the domain of practical stability with settling time [tau subscript s] | p. 233 |
Conclusion | p. 240 |
Comparison systems and vector norm-based Lyapunov functions | p. 241 |
Introductory comments and definitions | p. 241 |
Presentation | p. 241 |
Comparison systems | p. 243 |
Differential inequalities, overvaluing systems | p. 245 |
-M matrices | p. 249 |
Vector norm-based comparison systems | p. 254 |
Definition and aim of vector norms | p. 254 |
A first statement | p. 256 |
Computation of overvaluing systems | p. 258 |
Overvaluation lemma | p. 263 |
Vector norms and Lyapunov stability criteria | p. 266 |
Stability of equilibrium points | p. 266 |
Stability of sets | p. 270 |
Examples | p. 273 |
Vector norms and practical stability criteria with domains estimation | p. 279 |
Conclusions | p. 283 |
References | p. 285 |
Author index | p. 299 |
Subject index | p. 301 |
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