Lyapunov functionals and stability of stochastic difference equations / Leonid Shaikhet.

Format:

Book

Author/Creator:

Shaĭkhet, L. E. (Leonid Efimovich)

Contributor:

Class of 1932 Fund.

Language:

English

Subjects (All):

Stochastic differential equations.

Lyapunov functions.

Physical Description:

xii, 370 pages : illustrations ; 24 cm

Place of Publication:

London ; New York : Springer, 2011.

Summary:

Hereditary systems (or systems with either delay or after-effects) are widely used to model processes in physics, mechanics, control, economics and biology. An important element in their study is their stability.

Stability conditions for difference equations with delay can be obtained using Lyapunov functionals.

Lyapunov Functionals and Stability of Stochastic Difference Equations describes the general method of Lyapunov functionals construction to investigate the stability of discrete-and continuous-time stochastic Volterra difference equations. The method allows the investigation of the degree to which the stability properties of differential equations are preserved in their difference analogues.

The text is self-contained, beginning with basic definitions and the mathematical fundamentals of Lyapunov functionals construction and moving on from particular to general stability results for stochastic difference equations with constant coefficient. Result are then discussed for stochastic difference equation of linear, nonlinear, delayed, discrete and continuous types. Example are drawn from a variety of physical and biological systems including inverted pendulum control, Nicholson's blowflies equation and predator-prey relationships.

Lyapunov Functionals and Stability of Stochastic Difference Equations is primarily addressed to expert in Stability theory but will also be of use in the work of pure and computational mathematicians and researchers using the ideas of optional control to study economic, mechanical and biological systems. Book jacket.

Contents:

1 Lyapunov-type Theorems and Procedure of Lyapunov Functionals Construction 1

1.1 Definitions and Basic Lyapunov-type Theorem 1

1.2 Formal Procedure of Lyapunov Functionals Construction 3

1.3 Auxiliary Lyapunov-type Theorems 4

1.4 Some Useful Lemmas 6

2 Illustrative Example 11

2.1 First Way of Construction of Lyapunov Functional 11

2.2 Second Way of Construction of Lyapunov Functional 13

2.3 Third Way of Construction of Lyapunov Functional 14

2.4 Fourth Way of Construction of Lyapunov Functional 16

2.5 One Generalization 18

3 Linear Equations with Stationary Coefficients 23

3.1 First Way of the Construction of the Lyapunov Functional 23

3.2 Second Way of the Construction of the Lyapunov Functional 26

3.3 Third Way of the Construction of the Lyapunov Functional 29

3.4 Fourth Way of the Construction of the Lyapunov Functional 33

3.5 One Generalization 40

3.6 Investigation of Asymptotic Behavior via characteristic Equation 44

3.6.1 Statement of the Problem 44

3.6.2 Improvement of the Known Result 46

3.6.3 Different Situations with Roots of the Characteristic Equation 48

4 Linear Equations with Nonstationary Coefficients 61

4.1 First Way of the Construction of the Lyapunov Functional 61

4.2 Second Way of the Construction of the Lyapunov Functional 64

4.3 Third Way of the Construction of the Lyapunov Functional 67

4.4 Systems with Monotone Coefficients 73

5 Some Peculiarities of the Method 79

5.1 Necessary and Sufficient Condition 79

5.2 Difference Ways of Estimation 85

5.3 Volterra Equations 89

5.4 Difference Equation with Markovian Switching 101

6 System of Linear Equations with Varying Delays 109

6.1 Systems with Nonincreasing Delays 109

6.1.1 First way of the construction of the Lyapunov Functional 109

6.1.2 Second way of the construction of the Lyapunov Functional 112

6.2 Systems with Unbounded Delays 116

6.2.1 First way of the construction of the Lyapunov Functional 116

6.2.2 Second way of the construction of the Lyapunov Functional 119

7 Nonlinear System 127

7.1 Asymptotic Mean Square Stability 127

7.1.1 Stationary Systems 127

7.1.2 Nonstationary Systems with Monotone Coefficients 133

7.2 Stability in Probality 143

7.2.1 Basic Theorem 143

7.2.2 Quasilinear System with Order of Nonlinearity Higher than One 145

7.3 Fractional Difference Equations 152

7.3.1 Equilibrium Points 152

7.3.2 Stochastic Perturbations, Centering and Linearization 153

7.3.3 Stability of Equilibrium points 155

7.3.4 Examples 157

7.4 Almost Sure Stability 175

7.4.1 Auxiliary Statements and Definitions 176

7.4.2 Stability Theorems 180

8 Volterra Equations of Second Type 191

8.1 Statement of the Problem 191

8.2 Illustrative Example 193

8.2.1 First Way of the Construction of the Lyapunov Functional 193

8.2.2 Second Way of the Construction of the Lyapunov Functional 194

8.2.3 Third Way of the Construction of Lyapunov Functional 195

8.2.4 Fourth Way of the Construction of the Lyapunov Functional 196

8.3 Linear Equations with Constant Coefficients 197

8.3.1 First Way of the Construction Lyapunov Functional 197

8.3.2 Second Way of the Construction of the Lyapunov Functional 198

8.4 Linear Equations with Variable Coefficients 201

8.4.1 First Way of the Construction of the Lyapunov Functional 202

8.4.2 Second Way of the Construction of the Lyapunov Functional 203

8.4.3 Resolvent Representation 205

8.5 Nonlinear Systems 211

8.5.1 Stationary Systems 212

8.5.2 Nonstationary Systems 214

8.5.3 Nonstationary Systems with Monotone Coefficients 218

8.5.4 Resolvent Representation 223

9 Difference Equations with Continuous Time 227

9.1 Preliminaries and General Statements 227

9.1.1 Notations, Definitions and Lyapunov Type theorem 227

9.1.2 Formal procedure of the construction of the Lyapunov Functionals 233

9.1.3 Auxiliary Lyapunov type Theorems 234

9.2 Linear Volterra equations with constant coefficients 240

9.2.1 First Way of the Construction of the Lyapunov Functional 240

9.2.2 Second way of the Construction of the Lyapunov Functional 245

9.2.3 Particular Cases and Examples 246

9.3 Nonlinear Difference Equation 260

9.3.1 Nonstationary systems with Monotone Coefficients 260

9.3.2 Stability in Probability 268

9.4 Volterra Equatiobn of second Type 278

10 Difference equations as Difference Analogues of Differential Equations 283

10.1 Stability Condition for Stochastic Differential Equations 283

10.2 Difference Analogue of the mathematical model of the Controlled inverted pendulum 285

10.2.1 Mathematical Model of the controlled Inverted pendulum 285

10.2.2 Construction of a Difference Analogue 287

10.2.3 Stability Conditions for the Auxiliary Equation 288

10.2.4 Stability Conditions for the Difference Analogue 290

10.2.5 Nonlinear Model of the Controlled Inverted Pendulum 294

10.3 Difference Analogue of Nocholson's Blowflies Equation 294

10.3.1 Nocholson's Blowflies Equation 295

10.3.2 Stability Condition for the Positive Equilibrium Point 296

10.3.3 Stability of Difference Analogue 297

10.3.4 Numerical Analysis in the Deterministic Case 305

10.3.5 Numerical Analysis in the Stochastic Case 308

10.4 Difference Analogue of the Predator-Prey Model 310

10.4.1 Positive Equilibrium point, Stochastic Perturbations, Centering and Linearization 311

10.4.2 Stability of the Difference Analogue 315

10.5 Difference Analogues of an Integro-Differential Equation Of Convolution Type 338

10.5.1 Some Difference Analogues with Discrete Time 340

10.5.2 The Construction of the Lyapunov Functionls 342

10.5.3 Proof of Asymptotic Stability 345

10.5.4 Difference Analogue with Continuous Time 349.

Notes:

Includes bibliographical references (pages 355-366) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1932 Fund.

ISBN:

0857296841

9780857296849

OCLC:

731920957

Publisher Number:

99946075174