Lyapunov Functionals And Stability Of Stochastic Difference Equations

Author : Leonid Shaikhet
Publisher : Springer Science & Business Media
Page : 374 pages
File Size : 48,5 Mb
Release : 2011-06-02
Category : Technology & Engineering
ISBN : 9780857296856

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Lyapunov Functionals and Stability of Stochastic Difference Equations by Leonid Shaikhet Pdf

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.

Author : Leonid Shaikhet
Publisher : Springer Science & Business Media
Page : 352 pages
File Size : 51,8 Mb
Release : 2013-03-29
Category : Technology & Engineering
ISBN : 9783319001012

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Lyapunov Functionals and Stability of Stochastic Functional Differential Equations by Leonid Shaikhet Pdf

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.

Author : Rafail Khasminskii
Publisher : Springer Science & Business Media
Page : 353 pages
File Size : 53,5 Mb
Release : 2011-09-20
Category : Mathematics
ISBN : 9783642232800

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Stochastic Stability of Differential Equations by Rafail Khasminskii Pdf

Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography. This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

Author : V. Lakshmikantham,S. Leela,A. A. Martynyuk
Publisher : World Scientific
Page : 228 pages
File Size : 44,5 Mb
Release : 1990
Category : Computers
ISBN : 981020356X

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Practical Stability of Nonlinear Systems by V. Lakshmikantham,S. Leela,A. A. Martynyuk Pdf

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.

Author : V. Lakshmikantham,V.M. Matrosov,S. Sivasundaram
Publisher : Springer Science & Business Media
Page : 182 pages
File Size : 40,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401579391

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Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems by V. Lakshmikantham,V.M. Matrosov,S. Sivasundaram Pdf

One service mathematics has rendered the 'Et moi, "', si j'avait su comment en revenir, je n'y serais point all".' human race. It has put common sense back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics . .'; 'One service logic has rendered com puter science . .'; 'One service category theory has rendered mathematics . .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Author : Saber Elaydi,Mustafa R. S. Kulenović,Senada Kalabušić
Publisher : Springer Nature
Page : 534 pages
File Size : 41,6 Mb
Release : 2023-03-25
Category : Mathematics
ISBN : 9783031252259

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Advances in Discrete Dynamical Systems, Difference Equations and Applications by Saber Elaydi,Mustafa R. S. Kulenović,Senada Kalabušić Pdf

​This book comprises selected papers of the 26th International Conference on Difference Equations and Applications, ICDEA 2021, held virtually at the University of Sarajevo, Bosnia and Herzegovina, in July 2021. The book includes the latest and significant research and achievements in difference equations, discrete dynamical systems, and their applications in various scientific disciplines. The book is interesting for Ph.D. students and researchers who want to keep up to date with the latest research, developments, and achievements in difference equations, discrete dynamical systems, and their applications, the real-world problems.

Author : Leonid Shaikhet
Publisher : Springer
Page : 220 pages
File Size : 44,8 Mb
Release : 2014-11-27
Category : Technology & Engineering
ISBN : 9783319132396

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Optimal Control of Stochastic Difference Volterra Equations by Leonid Shaikhet Pdf

This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equations commences with an historical introduction to the emergence of this type of equation with some additional mathematical preliminaries. It then deals with the necessary conditions for optimality in the control of the equations and constructs a feedback control scheme. The approximation of stochastic quasilinear Volterra equations with quadratic performance functionals is then considered. Optimal stabilization is discussed and the filtering problem formulated. Finally, two methods of solving the optimal control problem for partly observable linear stochastic processes, also with quadratic performance functionals, are developed. Integrating the author’s own research within the context of the current state-of-the-art of research in difference equations, hereditary systems theory and optimal control, this book is addressed to specialists in mathematical optimal control theory and to graduate students in pure and applied mathematics and control engineering.

Author : Rafail Khasminskii
Publisher : Springer
Page : 342 pages
File Size : 43,9 Mb
Release : 2013-11-27
Category : Mathematics
ISBN : 364227028X

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Stochastic Stability of Differential Equations by Rafail Khasminskii Pdf

Since the publication of the first edition of the present volume in 1980, the stochastic stability of differential equations has become a very popular subject of research in mathematics and engineering. To date exact formulas for the Lyapunov exponent, the criteria for the moment and almost sure stability, and for the existence of stationary and periodic solutions of stochastic differential equations have been widely used in the literature. In this updated volume readers will find important new results on the moment Lyapunov exponent, stability index and some other fields, obtained after publication of the first edition, and a significantly expanded bibliography. This volume provides a solid foundation for students in graduate courses in mathematics and its applications. It is also useful for those researchers who would like to learn more about this subject, to start their research in this area or to study the properties of concrete mechanical systems subjected to random perturbations.

Author : Imre Csiszar,Gy. Michaletzky
Publisher : Springer Science & Business Media
Page : 358 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461219804

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Stochastic Differential and Difference Equations by Imre Csiszar,Gy. Michaletzky Pdf

Author : Anonim
Publisher : Unknown
Page : 140 pages
File Size : 51,9 Mb
Release : 1968
Category : Electronic
ISBN : NASA:31769000484249

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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 by Anonim Pdf

Author : Philipp Braun,Lars Grüne,Christopher M. Kellett
Publisher : Springer Nature
Page : 123 pages
File Size : 48,6 Mb
Release : 2021-07-12
Category : Mathematics
ISBN : 9783030763176

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(In-)Stability of Differential Inclusions by Philipp Braun,Lars Grüne,Christopher M. Kellett Pdf

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.

Author : Kai Liu
Publisher : CRC Press
Page : 311 pages
File Size : 48,5 Mb
Release : 2005-08-23
Category : Mathematics
ISBN : 9781420034820

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Stability of Infinite Dimensional Stochastic Differential Equations with Applications by Kai Liu Pdf

Stochastic differential equations in infinite dimensional spaces are motivated by the theory and analysis of stochastic processes and by applications such as stochastic control, population biology, and turbulence, where the analysis and control of such systems involves investigating their stability. While the theory of such equations is well establ

Author : V. Lakshmikantham,S. Leela,Anatoliĭ Andreevich Martyni︠u︡k
Publisher : CRC Press
Page : 344 pages
File Size : 48,5 Mb
Release : 1988-11-29
Category : Mathematics
ISBN : 0824780671

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Stability Analysis of Nonlinear Systems by V. Lakshmikantham,S. Leela,Anatoliĭ Andreevich Martyni︠u︡k Pdf

Investigates stability theory in terms of two different measures, treats the theory of a variety of inequalities, and demonstrates manifestations of the general Lyapunov method. Also covers the importance of utilizing different forms of nonlinear variation of parametric formulae, constructive method

Author : Silviu-Iulian Niculescu,Keqin Gu
Publisher : Springer Science & Business Media
Page : 445 pages
File Size : 48,6 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642184826

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Advances in Time-Delay Systems by Silviu-Iulian Niculescu,Keqin Gu Pdf

In the mathematical description of a physical or biological process, it is a common practice \0 assume that the future behavior of Ihe process considered depends only on the present slate, and therefore can be described by a finite sct of ordinary diffe rential equations. This is satisfactory for a large class of practical systems. However. the existence of lime-delay elements, such as material or infonnation transport, of tcn renders such description unsatisfactory in accounting for important behaviors of many practical systems. Indeed. due largely to the current lack of effective metho dology for analysis and control design for such systems, the lime-delay elements arc often either neglected or poorly approximated, which frequently results in analysis and simulation of insufficient accuracy, which in turns leads to poor performance of the systems designed. Indeed, it has been demonstrated in the area of automatic control that a relatively small delay may lead to instability or significantly deteriora ted perfonnances for the corresponding closed-loop systems.

Author : Vladislav I Zhukovskiy
Publisher : CRC Press
Page : 304 pages
File Size : 46,6 Mb
Release : 2003-01-16
Category : Mathematics
ISBN : 9781482264999

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Lyapunov Functions in Differential Games by Vladislav I Zhukovskiy Pdf

A major step in differential games is determining an explicit form of the strategies of players who follow a certain optimality principle. To do this, the associated modification of Bellman dynamic programming problems has to be solved; for some differential games this could be Lyapunov functions whose "arsenal" has been supplied by stability theor