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Stability of Dynamical Systems
XIAOXIN LIAO
Huazhong University of Science and Technology
Wuhan 430074
China
LIQIU WANG
The University of Hong Kong
Hong Kong
Hong Kong
PEI YU
The University of Western Ontario
London, Ontario
Canada
AMSTERDAM
Preface v
Chapter 1. Fundamental Concepts and Mathematical Tools 1
1.1. Fundamental theorems of ordinary differential equations 1
1.2. Lyapunov function 4
1.3. K-class function 7
1.4. Dini derivative 10
1.5. Differential and integral inequalities 13
1.6. A unified simple condition for stable matrix, p.d. matrix andM matrix
16
1.7. Definition of Lyapunov stability 21
1.8. Some examples of stability relation 24
Chapter 2. Linear Systems with Constant Coefficients 35
2.1. NASCs for stability and asymptotic stability 35
2.2. Sufficient conditions of Hurwitz matrix 43
2.3. A new method for solving Lyapunov matrix equation: BA+AT B =
C 53
2.4. A simple geometrical NASC for Hurwitz matrix 61
2.5. The geometry method for the stability of linear control systems 69
Chapter 3. Time-Varying Linear Systems 77
3.1. Stabilities between homogeneous and nonhomogeneous systems 77
3.2. Equivalent condition for the stability of linear systems 80
3.3. Robust stability of linear systems 84
3.4. The expression of Cauchy matrix solution 90
3.5. Linear systems with periodic coefficients 95
3.6. Spectral estimation for linear systems 100
3.7. Partial variable stability of linear systems 104
ix
Chapter 4. Lyapunov Direct Method 111
4.1. Geometrical illustration of Lyapunov direct method 112
4.2. NASCs for stability and uniform stability 113
4.3. NASCs for uniformly asymptotic and equi-asymptotic stabilities 119
4.4. NASCs of exponential stability and instability 127
4.5. Sufficient conditions for stability 130
4.6. Sufficient conditions for asymptotic stability 139
4.7. Sufficient conditions for instability 152
4.8. Summary of constructing Lyapunov functions 162
Chapter 5. Development of Lyapunov Direct Method 167
5.1. LaSalle’s invariant principle 167
5.2. Comparability theory 171
5.3. Lagrange stability 177
5.4. Lagrange asymptotic stability 185
5.5. Lagrange exponential stability of the Lorenz system 188
5.6. Robust stability under disturbance of system structure 196
5.7. Practical stability 200
5.8. Lipschitz stability 203
5.9. Asymptotic equivalence of two dynamical systems 208
5.10. Conditional stability 218
5.11. Partial variable stability 224
5.12. Stability and boundedness of sets 235
Chapter 6. Nonlinear Systems with Separate Variables 241
6.1. Linear Lyapunov function method 241
6.2. General nonlinear Lyapunov function with separable variable 253
6.3. Systems which can be transformed to separable variable systems 263
6.4. Partial variable stability for systems with separable variables 268
6.5. Autonomous systems with generalized separable variables 278
6.6. Nonautonomous systems with separable variables 280
Chapter 7. Iteration Method for Stability 285
7.1. Picard iteration type method 285
7.2. Gauss–Seidel type iteration method 290
7.3. Application of iteration method to extreme stability 302
7.4. Application of iteration method to stationary oscillation 307
7.5. Application of iteration method to improve frozen coefficient method 309
7.6. Application of iteration method to interval matrix 315
Chapter 8. Dynamical Systems with Time Delay 321
8.1. Basic concepts 321
8.2. Lyapunov function method for stability 324
8.3. Lyapunov function method with Razumikhin technique 330
8.4. Lyapunov functional method for stability analysis 338
8.5. Nonlinear autonomous systems with various time delays 341
8.6. Application of inequality with time delay and comparison principle 350
8.7. Algebraic method for LDS with constant coefficients and time delay 356
8.8. A class of time delay neutral differential difference systems 362
8.9. The method of iteration by parts for large-scale neural systems 366
8.10. Stability of large-scale neutral systems on C1 space 373
8.11. Algebraic methods for GLNS with constant coefficients 378
Chapter 9. Absolute Stability of Nonlinear Control Systems 389
9.1. The principal of centrifugal governor and general Lurie systems 389
9.2. Lyapunov–Lurie type V function method 394
9.3. NASCs of negative definite for derivative of Lyapunov–Lurie type
function 399
9.4. Popov’s criterion and improved criterion 402
9.5. Simple algebraic criterion 407
9.6. NASCs of absolute stability for indirect control systems 420
9.7. NASCs of absolute stability for direct and critical control system 434
9.8. NASCs of absolute stability for control systems with multiple nonlinear
controls 442
9.9. NASCs of absolute stability for systems with feedback loops 454
9.10. Chaos synchronization as a stabilization problem of Lurie system 459
9.11. NASCs for absolute stability of time-delayed Lurie control systems 469
Chapter 10. Stability of Neural Networks 487
10.1. Hopfield energy function method
10.2. Lagrange globally exponential stability of general neural network 491
10.3. Extension of Hopfield energy function method 493
10.4. Globally exponential stability of Hopfield neural network 502
10.5. Globally asymptotic stability of a class of Hopfield neural networks 515
10.6. Stability of bidirectional associative memory neural network 530
10.7. Stability of BAM neural networks with variable delays 534
10.8. Exp. stability and exp. periodicity of DNN with Lipschitz type activation
function 541
10.9. Stability of general ecological systems and neural networks 550
10.10. Cellular neural network 563
Chapter 11. Limit Cycle, Normal Form and Hopf Bifurcation Control 591
11.1. Introduction 591
11.2. Computation of normal forms and focus values 594
11.2.1 The Takens method 594
11.2.2 A perturbation method 597
11.2.3 The singular point value method 599
11.2.4 Applications 602
11.3. Computation of the SNF with parameters 613
11.3.1 General formulation 614
11.3.2 The SNF for single zero 622
11.3.3 The SNF for Hopf bifurcation 624
11.4. Hopf bifurcation control 626
11.4.1 Continuous-time systems 627
11.4.2 Discrete maps 640
11.4.3 2-D lifting surface 658
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