Lyapunov functionals and stability of stochastic functional differential equations [electronic resource] / Leonid Shaikhet.

Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential...

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Bibliographic Details
Online Access:	Full Text (via Springer)
Main Author:	Shaĭkhet, L. E. (Leonid Efimovich)
Format:	Electronic eBook
Language:	English
Published:	Cham, Switzerland ; New York : Springer, ©2013.
Subjects:	Lyapunov functions. Stochastic differential equations.

Table of Contents:

Short Introduction to Stability Theory of Deterministic Functional Differential Equations
Stochastic Functional Differential Equations and Procedure of Constructing Lyapunov Functionals
Stability of Linear Scalar Equations
Stability of Linear Systems of Two Equations
Stability of Systems with Nonlinearities
Matrix Riccati Equations in Stability of Linear Stochastic Differential Equations with Delays
Stochastic Systems with Markovian Switching
Stabilization of the Controlled Inverted Pendulum by a Control with Delay
Stability of Equilibrium Points of Nicholson's Blowflies Equation with Stochastic Perturbations
Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator-Prey with Aftereffect and Stochastic Perturbations
Stability of SIR Epidemic Model Equilibrium Points
Stability of Some Social Mathematical Models with Delay Under Stochastic Perturbations.
Machine generated contents note: 1. Short Introduction to Stability Theory of Deterministic Functional Differential Equations
1.1. Some Peculiarities of Functional Differential Equations
1.1.1. Description of Functional Differential Equations
1.1.2. Reducing to Ordinary Differential Equations
1.2. Method of Steps for Retarded Functional Differential Equations
1.3. Characteristic Equation for Differential Equation with Discrete Delays
1.4. Influence of Small Delays on Stability
1.5. Routh-Hurwitz Conditions
2. Stochastic Functional Differential Equations and Procedure of Constructing Lyapunov Functionals
2.1. Short Introduction to Stochastic Functional Differential Equations
2.1.1. Wiener Process and Its Numerical Simulation
2.1.2. Ito Integral, Ito Stochastic Differential Equation, and Ito Formula
2.2. Stability of Stochastic Functional Differential Equations
2.2.1. Definitions of Stability and Basic Lyapunov-Type Theorems
2.2.2. Formal Procedure of Constructing Lyapunov Functionals
2.2.3. Auxiliary Lyapunov-Type Theorem
2.3. Some Useful Statements
2.3.1. Linear Stochastic Differential Equation
2.3.2. System of Two Linear Stochastic Differential Equations
2.3.3. Some Useful Inequalities
2.4. Some Unsolved Problems
2.4.1. Problem 1
2.4.2. Problem 2
3. Stability of Linear Scalar Equations
3.1. Linear Stochastic Differential Equation of Neutral Type
3.1.1. First Way of Constructing a Lyapunov Functional
3.1.2. Second Way of Constructing a Lyapunov Functional
3.1.3. Some Particular Cases
3.2. Linear Differential Equation with Two Delays in Deterministic Part
3.2.1. First Way of Constructing a Lyapunov Functional
3.2.2. Second Way of Constructing a Lyapunov Functional
3.2.3. Third Way of Constructing a Lyapunov Functional
3.2.4. Fourth Way of Constructing a Lyapunov Functional
3.2.5. One Generalization for Equation with n Delays
3.3. Linear Differential Equation of nth Order
3.3.1. Case n> 1
3.3.2. Some Particular Cases
3.4. Nonautonomous Systems
3.4.1. Equations with Variable Delays
3.4.2. Equations with Variable Coefficients
4. Stability of Linear Systems of Two Equations
4.1. Linear Systems of Two Equations with Constant Delays
4.2. Linear Systems of Two Equations with Distributed Delays
4.3. Linear Systems of Two Equations with Variable Coefficients
5. Stability of Systems with Nonlinearities
5.1. Systems with Nonlinearities in Stochastic Part
5.1.1. Scalar First-Order Differential Equation
5.1.2. Scalar Second-Order Differential Equation
5.2. Systems with Nonlinearities in Both Deterministic and Stochastic Parts
5.3. Stability in Probability of Nonlinear Systems
5.4. Systems with Fractional Nonlinearity
5.4.1. Equilibrium Points
5.4.2. Stochastic Perturbations, Centering, and Linearization
5.4.3. Stability of Equilibrium Points
5.4.4. Numerical Analysis
6. Matrix Riccati Equations in Stability of Linear Stochastic Differential Equations with Delays
6.1. Equations with Constant Delays
6.1.1. One Delay in Deterministic Part and One Delay in Stochastic Part of Equation
6.1.2. Several Delays in Deterministic Part of Equation
6.2. Distributed Delay
6.3. Combination of Discrete and Distributed Delays
6.4. Equations with Nonincreasing Delays
6.4.1. One Delay in Deterministic and One Delay in Stochastic Parts of Equation
6.4.2. Several Delays in Deterministic Part of Equation
6.5. Equations with Bounded Delays
6.5.1. First Way of Constructing a Lyapunov Functional
6.5.2. Second Way of Constructing a Lyapunov Functional
6.6. Equations with Unbounded Delays
7. Stochastic Systems with Markovian Switching
7.1. Statement of the Problem
7.2. Stability Theorems
7.3. Application to Markov Chain with Two States
7.3.1. First Stability Condition
7.3.2. Second Stability Condition
7.4. Numerical Simulation of Systems with Markovian Switching
7.4.1. System Without Stochastic Perturbations
7.4.2. System with Stochastic Perturbations
7.4.3. System with Random Delay
7.4.4. Some Generalization of Algorithm of Markov Chain Numerical Simulation
8. Stabilization of the Controlled Inverted Pendulum by a Control with Delay
8.1. Linear Model of the Controlled Inverted Pendulum
8.1.1. Stabilization by the Control Depending on Trajectory
8.1.2. Some Examples
8.1.3. About Stabilization by the Control Depending on Velocity or on Acceleration
8.1.4. Stabilization by Stochastic Perturbations
8.2. Nonlinear Model of the Controlled Inverted Pendulum
8.2.1. Stabilization of the Trivial Solution
8.2.2. Nonzero Steady-State Solutions
8.2.3. Stable, Unstable, and One-Sided Stable Points of Equilibrium
8.3. Numerical Analysis of the Controlled Inverted Pendulum
8.3.1. Stability of the Trivial Solution and Limit Cycles
8.3.2. Nonzero Steady-State Solutions of the Nonlinear Model
8.3.3. Stabilization of the Controlled Inverted Pendulum Under Influence of Markovian Stochastic Perturbations
9. Stability of Equilibrium Points of Nicholson's Blowflies Equation with Stochastic Perturbations
9.1. Introduction
9.2. Two Points of Equilibrium, Stochastic Perturbations, Centering, and Linearization
9.3. Sufficient Conditions for Stability in Probability for Both Equilibrium Points
9.4. Numerical Illustrations
10. Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator-Prey with Aftereffect and Stochastic Perturbations
10.1. System Under Consideration
10.2. Equilibrium Points, Stochastic Perturbations, Centering, and Linearization
10.2.1. Equilibrium Points
10.2.2. Stochastic Perturbations and Centering
10.2.3. Linearization
10.3. Stability of Equilibrium Point
10.3.1. First Way of Constructing a Lyapunov Functional
10.3.2. Second Way of Constructing a Lyapunov Functional
10.3.3. Stability of the Equilibrium Point of Ratio-Dependent Predator-Prey Model
11. Stability of SIR Epidemic Model Equilibrium Points
11.1. Problem Statement
11.2. Stability in Probability of the Equilibrium Point E0 = (bμ 1 -1, 0, 0)
11.3. Stability in Probability of the Equilibrium Point E* = (S*, I*, R*)
11.4. Numerical Simulation
12. Stability of Some Social Mathematical Models with Delay Under Stochastic Perturbations
12.1. Mathematical Model of Alcohol Consumption
12.1.1. Description of the Model of Alcohol Consumption
12.1.2. Normalization of the Initial Model
12.1.3. Existence of an Equilibrium Point
12.1.4. Stochastic Perturbations, Centralization, and Linearization
12.1.5. Stability of the Equilibrium Point
12.1.6. Numerical Simulation
12.2. Mathematical Model of Social Obesity Epidemic
12.2.1. Description of the Considered Model
12.2.2. Existence of an Equilibrium Point
12.2.3. Stochastic Perturbations, Centralization, and Linearization
12.2.4. Stability of an Equilibrium Point
12.2.5. Numerical Simulation.

Lyapunov functionals and stability of stochastic functional differential equations [electronic resource] / Leonid Shaikhet.

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