Lyapunov Functionals And Stability Of Stochastic Difference Equations
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Lyapunov Functionals And Stability Of Stochastic Difference Equations
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Author : Leonid Shaikhet
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-06-02
Lyapunov Functionals And Stability Of Stochastic Difference Equations written by Leonid Shaikhet and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-06-02 with Technology & Engineering categories.
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 a Lyapunov functional. Lyapunov Functionals and Stability of Stochastic Difference Equations describes a general method of Lyapunov functional 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 functional 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 systems including inverted pendulum control, study of epidemic development, 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.
Lyapunov Functionals And Stability Of Stochastic Functional Differential Equations
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Author : Leonid Shaikhet
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-29
Lyapunov Functionals And Stability Of Stochastic Functional Differential Equations written by Leonid Shaikhet and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-29 with Technology & Engineering categories.
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 equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for difference equations with discrete and continuous time. The text begins with both a description and a delineation of the peculiarities of deterministic and stochastic functional differential equations. There follows basic definitions for stability theory of stochastic hereditary systems, and the formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of different mathematical models such as: • inverted controlled pendulum; • Nicholson's blowflies equation; • predator-prey relationships; • epidemic development; and • mathematical models that describe human behaviours related to addictions and obesity. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations is primarily addressed to experts in stability theory but will also be of interest to professionals and students in pure and computational mathematics, physics, engineering, medicine, and biology.
Practical Stability Of Nonlinear Systems
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Author : V. Lakshmikantham
language : en
Publisher: World Scientific
Release Date : 1990
Practical Stability Of Nonlinear Systems written by V. Lakshmikantham and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Computers categories.
This is the first book that deals with practical stability and its development. It presents a systematic study of the theory of practical stability in terms of two different measures and arbitrary sets and demonstrates the manifestations of general Lyapunov's method by showing how this effective technique can be adapted to investigate various apparently diverse nonlinear problems including control systems and multivalued differential equations.
Qualitative And Asymptotic Analysis Of Differential Equations With Random Perturbations
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Author : Anatoliy M. Samoilenko
language : en
Publisher: World Scientific
Release Date : 2011
Qualitative And Asymptotic Analysis Of Differential Equations With Random Perturbations written by Anatoliy M. Samoilenko and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
1. Differential equations with random right-hand sides and impulsive effects. 1.1. An impulsive process as a solution of an impulsive system. 1.2. Dissipativity. 1.3. Stability and Lyapunov functions. 1.4. Stability of systems with permanently acting random perturbations. 1.5. Solutions periodic in the restricted sense. 1.6. Periodic solutions of systems with small perturbations. 1.7. Periodic solutions of linear impulsive systems. 1.8. Weakly nonlinear systems. 1.9. Comments and references -- 2. Invariant sets for systems with random perturbations. 2.1. Invariant sets for systems with random right-hand sides. 2.2. Invariant sets for stochastic Ito systems. 2.3. The behaviour of invariant sets under small perturbations. 2.4. A study of stability of an equilibrium via the reduction principle for systems with regular random perturbations. 2.5. Stability of an equilibrium and the reduction principle for Ito type systems. 2.6. A study of stability of the invariant set via the reduction principle. Regular perturbations. 2.7. Stability of invariant sets and the reduction principle for Ito type systems. 2.8. Comments and references -- 3. Linear and quasilinear stochastic Ito systems. 3.1. Mean square exponential dichotomy. 3.2. A study of dichotomy in terms of quadratic forms. 3.3. Linear system solutions that are mean square bounded on the semiaxis. 3.4. Quasilinear systems. 3.5. Linear system solutions that are probability bounded on the axis. A generalized notion of a solution. 3.6. Asymptotic equivalence of linear systems. 3.7. Conditions for asymptotic equivalence of nonlinear systems. 3.8. Comments and references -- 4. Extensions of Ito systems on a torus. 4.1. Stability of invariant tori. 4.2. Random invariant tori for linear extensions. 4.3. Smoothness of invariant tori. 4.4. Random invariant tori for nonlinear extensions. 4.5. An ergodic theorem for a class of stochastic systems having a toroidal manifold. 4.6. Comments and references -- 5. The averaging method for equations with random perturbations. 5.1. A substantiation of the averaging method for systems with impulsive effect. 5.2. Asymptotics of normalized deviations of averaged solutions. 5.3. Applications to the theory of nonlinear oscillations. 5.4. Averaging for systems with impulsive effects at random times. 5.5. The second theorem of M.M. Bogolyubov for systems with regular random perturbations. 5.6. Averaging for stochastic Ito systems. An asymptotically finite interval. 5.7. Averaging on the semiaxis. 5.8. The averaging method and two-sided bounded solutions of Ito systems. 5.9. Comments and references
Stability Of Stochastic Dynamical Systems
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Author : R. F. Curtain
language : en
Publisher: Springer
Release Date : 2006-11-15
Stability Of Stochastic Dynamical Systems written by R. F. Curtain and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
Stochastic Differential Equations And Applications
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Author : X Mao
language : en
Publisher: Elsevier
Release Date : 2007-12-30
Stochastic Differential Equations And Applications written by X Mao and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-12-30 with Mathematics categories.
This advanced undergraduate and graduate text has now been revised and updated to cover the basic principles and applications of various types of stochastic systems, with much on theory and applications not previously available in book form. The text is also useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists. Has been revised and updated to cover the basic principles and applications of various types of stochastic systems Useful as a reference source for pure and applied mathematicians, statisticians and probabilists, engineers in control and communications, and information scientists, physicists and economists
Stochastic Stability Of Differential Equations In Abstract Spaces
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Author : Kai Liu
language : en
Publisher: Cambridge University Press
Release Date : 2019-05-02
Stochastic Stability Of Differential Equations In Abstract Spaces written by Kai Liu and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-02 with Mathematics categories.
Presents a unified treatment of stochastic differential equations in abstract, mainly Hilbert, spaces.
Lyapunov Stability For Partial Differential Equations Part 1 Lyapunov Stability Theory And The Stability Of Solutions To Partial Differential Equations Part 2 Contraction Groups And Equivalent Norms
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Author :
language : en
Publisher:
Release Date : 1968
Lyapunov Stability For Partial Differential Equations Part 1 Lyapunov Stability Theory And The Stability Of Solutions To Partial Differential Equations Part 2 Contraction Groups And Equivalent Norms written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1968 with categories.
In Stability Of Differential Inclusions
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Author : Philipp Braun
language : en
Publisher: Springer Nature
Release Date : 2021-07-12
In Stability Of Differential Inclusions written by Philipp Braun and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-12 with Mathematics categories.
Lyapunov methods have been and are still one of the main tools to analyze the stability properties of dynamical systems. In this monograph, Lyapunov results characterizing the stability and stability of the origin of differential inclusions are reviewed. To characterize instability and destabilizability, Lyapunov-like functions, called Chetaev and control Chetaev functions in the monograph, are introduced. Based on their definition and by mirroring existing results on stability, analogue results for instability are derived. Moreover, by looking at the dynamics of a differential inclusion in backward time, similarities and differences between stability of the origin in forward time and instability in backward time, and vice versa, are discussed. Similarly, the invariance of the stability and instability properties of the equilibria of differential equations with respect to scaling are summarized. As a final result, ideas combining control Lyapunov and control Chetaev functions to simultaneously guarantee stability, i.e., convergence, and instability, i.e., avoidance, are outlined. The work is addressed at researchers working in control as well as graduate students in control engineering and applied mathematics.
Modern Nonlinear Equations
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Author : Thomas L. Saaty
language : en
Publisher: Courier Corporation
Release Date : 2012-04-26
Modern Nonlinear Equations written by Thomas L. Saaty and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-04-26 with Mathematics categories.
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." — Math Reviews. 1964 edition.