Lyapunov functionals and stability of stochastic difference equations

✍ Scribed by Leonid Shaikhet (auth.)

• Publisher: Springer-Verlag London
• Year: 2011
• Tongue: English
• Leaves: 384
• Edition: 1
• Category: Library

✦ Synopsis

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 coefficients. Results are then discussed for stochastic difference equations of linear, nonlinear, delayed, discrete and continuous types. Examples 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 experts in stability theory but will also be of use in the work of pure and computational mathematicians and researchers using the ideas of optimal control to study economic, mechanical and biological systems.

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✦ Table of Contents

Front Matter....Pages I-XII
Lyapunov-type Theorems and Procedure of Lyapunov Functionals Construction....Pages 1-10
Illustrative Example....Pages 11-21
Linear Equations with Stationary Coefficients....Pages 23-59
Linear Equations with Nonstationary Coefficients....Pages 61-77
Some Peculiarities of the Method....Pages 79-108
Systems of Linear Equations with Varying Delays....Pages 109-125
Nonlinear Systems....Pages 127-190
Volterra Equations of Second Type....Pages 191-225
Difference Equations with Continuous Time....Pages 227-281
Difference Equations as Difference Analogues of Differential Equations....Pages 283-353
Back Matter....Pages 355-370

✦ Subjects

Control; Difference and Functional Equations; Calculus of Variations and Optimal Control; Optimization; Mathematical and Computational Biology; Probability Theory and Stochastic Processes; Vibration, Dynamical Systems, Control