Nonlinear Systems Analysis

Lecture 7

Sections 3.2, 3.2, 3.4

Dependence on Data, Sensitivity Analysis,

Comparison Principle

Feng-Li LianNTU-EE

Sep04 – Jan05

© Feng-Li Lian 2004

Ch3B-2

Nonlinear Systems Analysis

Outline

Introduction (L5)

Banach Space (L5)

Contraction Mapping Theorem (L5)

Existence and Uniqueness (L6)

Continuous Dependence on Initial Conditions and Parameters (L6)

Differentiability of Solutions and Sensitivity Equations (L7)

Comparison Principle (L7)

© Feng-Li Lian 2004

Ch3B-3

Nonlinear Systems Analysis

Continuous Dependence on Data - 1 (3.2)

© Feng-Li Lian 2004

Ch3B-4

Nonlinear Systems Analysis

Continuous Dependence on Data - 2

© Feng-Li Lian 2004

Ch3B-5

Nonlinear Systems Analysis

Continuous Dependence on Data - 3

© Feng-Li Lian 2004

Ch3B-6

Nonlinear Systems Analysis

Gronwall-Bellman Inequality - 1 (App. A, page 651)

© Feng-Li Lian 2004

Ch3B-7

Nonlinear Systems Analysis

Gronwall-Bellman Inequality - 2

© Feng-Li Lian 2004

Ch3B-8

Nonlinear Systems Analysis

Closeness of Solutions - 1 (3.2)

© Feng-Li Lian 2004

Ch3B-9

Nonlinear Systems Analysis

Closeness of Solutions - 2

© Feng-Li Lian 2004

Ch3B-10

Nonlinear Systems Analysis

Closeness of Solutions - 3

© Feng-Li Lian 2004

Ch3B-11

Nonlinear Systems Analysis

Closeness of Solutions - 4

© Feng-Li Lian 2004

Ch3B-12

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 1 (3.2)

© Feng-Li Lian 2004

Ch3B-13

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 2

© Feng-Li Lian 2004

Ch3B-14

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 3

© Feng-Li Lian 2004

Ch3B-15

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 4

© Feng-Li Lian 2004

Ch3B-16

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 5

© Feng-Li Lian 2004

Ch3B-17

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 6

© Feng-Li Lian 2004

Ch3B-18

Nonlinear Systems Analysis

Dependence on Initial States & Parameters - 7

© Feng-Li Lian 2004

Ch3B-19

Nonlinear Systems Analysis

Differentiability of Solutions - 1 (3.3)

© Feng-Li Lian 2004

Ch3B-20

Nonlinear Systems Analysis

Differentiability of Solutions - 2

© Feng-Li Lian 2004

Ch3B-21

Nonlinear Systems Analysis

Differentiability of Solutions - 3

© Feng-Li Lian 2004

Ch3B-22

Nonlinear Systems Analysis

Sensitivity Equation - 1 (3.3)

© Feng-Li Lian 2004

Ch3B-23

Nonlinear Systems Analysis

Sensitivity Equation - 2

© Feng-Li Lian 2004

Ch3B-24

Nonlinear Systems Analysis

Sensitivity Equation - 3

© Feng-Li Lian 2004

Ch3B-25

Nonlinear Systems Analysis

Comparison Principle - 1 (3.4)

© Feng-Li Lian 2004

Ch3B-26

Nonlinear Systems Analysis

Comparison Principle - 2

© Feng-Li Lian 2004

Ch3B-27

Nonlinear Systems Analysis

Comparison Principle - 3

© Feng-Li Lian 2004

Ch3B-28

Nonlinear Systems Analysis

Example 3.8 - 1 (3.4)

© Feng-Li Lian 2004

Ch3B-29

Nonlinear Systems Analysis

Example 3.8 - 2