Stability of Dynamical Systems (eBook)

Front cover 1
Stability of Dynamical Systems 4
Copyright page 5
Preface 6
Contents 10
Chapter 1. Fundamental Concepts and Mathematical Tools 14
1.1. Fundamental theorems of ordinary differential equations 14
1.2. Lyapunov function 17
1.3. K-class function 20
1.4. Dini derivative 23
1.5. Differential and integral inequalities 26
1.6. A unified simple condition for stable matrix, p.d. matrix and M matrix 29
1.7. Definition of Lyapunov stability 34
1.8. Some examples of stability relation 37
Chapter 2. Linear Systems with Constant Coefficients 48
2.1. NASCs for stability and asymptotic stability 48
2.2. Sufficient conditions of Hurwitz matrix 56
2.3. A new method for solving Lyapunov matrix equation: BA+ATB=C 66
2.4. A simple geometrical NASC for Hurwitz matrix 74
2.5. The geometry method for the stability of linear control systems 82
Chapter 3. Time-Varying Linear Systems 90
3.1. Stabilities between homogeneous and nonhomogeneous systems 90
3.2. Equivalent condition for the stability of linear systems 93
3.3. Robust stability of linear systems 97
3.4. The expression of Cauchy matrix solution 103
3.5. Linear systems with periodic coefficients 108
3.6. Spectral estimation for linear systems 113
3.7. Partial variable stability of linear systems 117
Chapter 4. Lyapunov Direct Method 124
4.1. Geometrical illustration of Lyapunov direct method 125
4.2. NASCs for stability and uniform stability 126
4.3. NASCs for uniformly asymptotic and equi-asymptotic stabilities 132
4.4. NASCs of exponential stability and instability 140
4.5. Sufficient conditions for stability 143
4.6. Sufficient conditions for asymptotic stability 152
4.7. Sufficient conditions for instability 165
4.8. Summary of constructing Lyapunov functions 175
Chapter 5. Development of Lyapunov Direct Method 180
5.1. LaSalle's invariant principle 180
5.2. Comparability theory 184
5.3. Lagrange stability 190
5.4. Lagrange asymptotic stability 198
5.5. Lagrange exponential stability of the Lorenz system 201
5.6. Robust stability under disturbance of system structure 209
5.7. Practical stability 213
5.8. Lipschitz stability 216
5.9. Asymptotic equivalence of two dynamical systems 221
5.10. Conditional stability 231
5.11. Partial variable stability 237
5.12. Stability and boundedness of sets 248
Chapter 6. Nonlinear Systems with Separate Variables 254
6.1. Linear Lyapunov function method 254
6.2. General nonlinear Lyapunov function with separable variable 266
6.3. Systems which can be transformed to separable variable systems 276
6.4. Partial variable stability for systems with separable variables 281
6.5. Autonomous systems with generalized separable variables 291
6.6. Nonautonomous systems with separable variables 293
Chapter 7. Iteration Method for Stability 298
7.1. Picard iteration type method 298
7.2. Gauss-Seidel type iteration method 303
7.3. Application of iteration method to extreme stability 315
7.4. Application of iteration method to stationary oscillation 320
7.5. Application of iteration method to improve frozen coefficient method 322
7.6. Application of iteration method to interval matrix 328
Chapter 8. Dynamical Systems with Time Delay 334
8.1. Basic concepts 334
8.2. Lyapunov function method for stability 337
8.3. Lyapunov function method with Razumikhin technique 343
8.4. Lyapunov functional method for stability analysis 351
8.5. Nonlinear autonomous systems with various time delays 354
8.6. Application of inequality with time delay and comparison principle 363
8.7. Algebraic method for LDS with constant coefficients and time delay 369
8.8. A class of time delay neutral differential difference systems 375
8.9. The method of iteration by parts for large-scale neural systems 379
8.10. Stability of large-scale neutral systems on C1 space 386
8.11. Algebraic methods for GLNS with constant coefficients 391
Chapter 9. Absolute Stability of Nonlinear Control Systems 402
9.1. The principal of centrifugal governor and general Lurie systems 402
9.2. Lyapunov-Lurie type V function method 407
9.3. NASCs of negative definite for derivative of Lyapunov-Lurie type function 412
9.4. Popov's criterion and improved criterion 415
9.5. Simple algebraic criterion 420
9.6. NASCs of absolute stability for indirect control systems 433
9.7. NASCs of absolute stability for direct and critical control system 447
9.8. NASCs of absolute stability for control systems with multiple nonlinear controls 455
9.9. NASCs of absolute stability for systems with feedback loops 467
9.10. Chaos synchronization as a stabilization problem of Lurie system 472
9.11. NASCs for absolute stability of time-delayed Lurie control systems 482
Chapter 10. Stability of Neural Networks 500
10.1. Hopfield energy function method 500
10.2. Lagrange globally exponential stability of general neural network 504
10.3. Extension of Hopfield energy function method 506
10.4. Globally exponential stability of Hopfield neural network 515
10.5. Globally asymptotic stability of a class of Hopfield neural networks 528
10.6. Stability of bidirectional associative memory neural network 543
10.7. Stability of BAM neural networks with variable delays 547
10.8. Exp. stability and exp. periodicity of DNN with Lipschitz type activation function 554
10.9. Stability of general ecological systems and neural networks 563
10.10. Cellular neural network 576
Chapter 11. Limit Cycle, Normal Form and Hopf Bifurcation Control 604
11.1. Introduction 604
11.2. Computation of normal forms and focus values 607
11.3. Computation of the SNF with parameters 626
11.4. Hopf bifurcation control 639
References 684
Subject index 710