Construction of global Lyapunov functions using radial basis functions / Peter Giesl.

Format:

Book

Author/Creator:

Giesl, Peter.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1904.

Lecture notes in mathematics, 0075-8434 ; 1904

Language:

English

Subjects (All):

Lyapunov functions.

Radial basis functions.

Physical Description:

viii, 166 pages : illustrations ; 24 cm.

Place of Publication:

Berlin : Springer, [2007]

Summary:

The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. Besides an introduction to both areas and a detailed description of the method, it contains error estimates and many examples.

Contents:

1.1 An Example: Chemostat 1

1.2 Lyapunov Functions and Radial Basis Functions 6

2 Lyapunov Functions 11

2.1 Introduction to Dynamical Systems 11

2.1.1 Basic Definitions and Concepts 11

2.1.2 Lyapunov Functions 17

2.2 Local Lyapunov Functions 22

2.2.1 The Function [characters not reproducible] (Jordan Normal Form) 22

2.2.2 The Function [characters not reproducible] (Matrix Equation) 26

2.3 Global Lyapunov Functions 30

2.3.1 The Lyapunov Function T with Constant Orbital Derivative 32

2.3.2 Level Sets of Lyapunov Functions 36

2.3.3 The Lyapunov Function V Defined in A(x[subscript 0]) 41

2.3.4 Taylor Polynomial of V 48

3 Radial Basis Functions 61

3.1 Approximation 63

3.1.1 Approximation via Function Values 63

3.1.2 Approximation via Orbital Derivatives 65

3.1.3 Mixed Approximation 69

3.1.4 Wendland Functions 72

3.2 Native Space 76

3.2.1 Characterization of the Native Space 77

3.2.2 Positive Definiteness of the Interpolation Matrices 80

3.2.3 Error Estimates 87

4 Construction of Lyapunov Functions 99

4.1 Non-Local Part 101

4.2 Local Part 106

4.2.1 Local Lyapunov Basin 109

4.2.2 Local Lyapunov Function 110

4.2.3 Taylor Polynomial 113

5 Global Determination of the Basin of Attraction 115

5.1 Approximation via a Single Operator 118

5.1.1 Approximation via Orbital Derivatives 118

5.1.2 Taylor Polynomial 121

5.2 Mixed Approximation 125

5.2.1 Approximation via Orbital Derivatives and Function Values 126

5.2.2 Stepwise Exhaustion of the Basin of Attraction 131

6 Application of the Method: Examples 133

6.1 Combination of a Local and Non-Local Lyapunov Function 135

6.1.1 Description 135

6.2 Approximation via Taylor Polynomial 140

6.2.1 Description 140

6.3 Stepwise Exhaustion Using Mixed Approximation 144

6.3.1 Description 144

A Distributions and Fourier Transformation 149

A.1 Distributions 149

A.2 Fourier Transformation 152

B.1 Wendland Functions 155.

Notes:

Includes bibliographical references (pages [161]-164) and index.

ISBN:

3540699074

9783540699071

OCLC:

123407813