Download or read book Lyapunov Functionals and Stability of Stochastic Difference Equations written by Leonid Shaikhet and published by Springer Science & Business Media. This book was released on 2011-06-02 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Lyapunov Functionals and Stability of Stochastic Functional Differential Equations written by Leonid Shaikhet and published by Springer Science & Business Media. This book was released on 2013-03-29 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Stochastic Stability of Differential Equations written by Rafail Khasminskii and published by Springer Science & Business Media. This book was released on 2011-09-20 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Dynamical Systems and Applications written by Ravi P. Agarwal and published by World Scientific. This book was released on 1995 with total page 720 pages. Available in PDF, EPUB and Kindle. Book excerpt: World Scientific series in Applicable Analysis (WSSIAA) aims at reporting new developments of high mathematical standard and current interest. Each volume in the series shall be devoted to the mathematical analysis that has been applied or potentially applicable to the solutions of scientific, engineering, and social problems. For the past twenty five years, there has been an explosion of interest in the study of nonlinear dynamical systems. Mathematical techniques developed during this period have been applied to important nonlinear problems ranging from physics and chemistry to ecology and economics. All these developments have made dynamical systems theory an important and attractive branch of mathematics to scientists in many disciplines. This rich mathematical subject has been partially represented in this collection of 45 papers by some of the leading researchers in the area. This volume contains 45 state-of-art articles on the mathematical theory of dynamical systems by leading researchers. It is hoped that this collection will lead new direction in this field.Contributors: B Abraham-Shrauner, V Afraimovich, N U Ahmed, B Aulbach, E J Avila-Vales, F Battelli, J M Blazquez, L Block, T A Burton, R S Cantrell, C Y Chan, P Collet, R Cushman, M Denker, F N Diacu, Y H Ding, N S A El-Sharif, J E Fornaess, M Frankel, R Galeeva, A Galves, V Gershkovich, M Girardi, L Gotusso, J Graczyk, Y Hino, I Hoveijn, V Hutson, P B Kahn, J Kato, J Keesling, S Keras, V Kolmanovskii, N V Minh, V Mioc, K Mischaikow, M Misiurewicz, J W Mooney, M E Muldoon, S Murakami, M Muraskin, A D Myshkis, F Neuman, J C Newby, Y Nishiura, Z Nitecki, M Ohta, G Osipenko, N Ozalp, M Pollicott, Min Qu, Donal O-Regan, E Romanenko, V Roytburd, L Shaikhet, J Shidawara, N Sibony, W-H Steeb, C Stoica, G Swiatek, T Takaishi, N D Thai Son, R Triggiani, A E Tuma, E H Twizell, M Urbanski; T D Van, A Vanderbauwhede, A Veneziani, G Vickers, X Xiang, T Young, Y Zarmi.

Download or read book Stochastic Differential Equations and Applications written by X Mao and published by Elsevier. This book was released on 2007-12-30 with total page 445 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book 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 published by . This book was released on 1968 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book In Stability of Differential Inclusions written by Philipp Braun and published by Springer Nature. This book was released on 2021-07-12 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems written by V. Lakshmikantham and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Advances in Discrete Dynamical Systems Difference Equations and Applications written by Saber Elaydi and published by Springer Nature. This book was released on 2023-03-25 with total page 534 pages. Available in PDF, EPUB and Kindle. Book excerpt: ​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.

Download or read book Stochastic Stability of Differential Equations in Abstract Spaces written by Kai Liu and published by Cambridge University Press. This book was released on 2019-05-02 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a unified treatment of stochastic differential equations in abstract, mainly Hilbert, spaces.

Download or read book Stability of Stochastic Dynamical Systems written by R. F. Curtain and published by Springer. This book was released on 2006-11-15 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Functional Differential Equations written by S. E. A. Mohammed and published by Pitman Advanced Publishing Program. This book was released on 1984 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations written by Anatoliy M. Samoilenko and published by World Scientific. This book was released on 2011 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Download or read book Stochastic Stability of Differential Equations written by Rafail Khasminskii and published by Springer. This book was released on 2013-11-27 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Optimal Control of Stochastic Difference Volterra Equations written by Leonid Shaikhet and published by Springer. This book was released on 2014-11-27 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Stability of Dynamical Systems written by Anthony N. Michel and published by Springer. This book was released on 2015-03-30 with total page 669 pages. Available in PDF, EPUB and Kindle. Book excerpt: The second edition of this textbook provides a single source for the analysis of system models represented by continuous-time and discrete-time, finite-dimensional and infinite-dimensional, and continuous and discontinuous dynamical systems. For these system models, it presents results which comprise the classical Lyapunov stability theory involving monotonic Lyapunov functions, as well as corresponding contemporary stability results involving non-monotonic Lyapunov functions. Specific examples from several diverse areas are given to demonstrate the applicability of the developed theory to many important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, and artificial neural networks. The authors cover the following four general topics: - Representation and modeling of dynamical systems of the types described above - Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces involving monotonic and non-monotonic Lyapunov functions - Specialization of this stability theory to finite-dimensional dynamical systems - Specialization of this stability theory to infinite-dimensional dynamical systems Replete with examples and requiring only a basic knowledge of linear algebra, analysis, and differential equations, this book can be used as a textbook for graduate courses in stability theory of dynamical systems. It may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, economics, and the physical and life sciences. Review of the First Edition: “The authors have done an excellent job maintaining the rigor of the presentation, and in providing standalone statements for diverse types of systems. [This] is a very interesting book which complements the existing literature. [It] is clearly written, and difficult concepts are illustrated by means of good examples.” - Alessandro Astolfi, IEEE Control Systems Magazine, February 2009

Download or read book Stability Elements of the Theory and Application with Examples written by Anatoliy A Martynyuk and published by Sciendo. This book was released on 2020-12-20 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to familiarize the readers with basic concepts, and classic results of stability theory stated in a way as required by the rigorous rules of contemporary mathematics and, simultaneously, to introduce the learners to broad elds of not only the stability theory but also applications involved. The emphasis is put on various dynamical systems which are defined by different branches of science and through diverse areas of human activity but always with care not to exceed the basic classical approach in the presentation. All in all, the authors plan to combine the textbook-like with encyclopaedia-like content. Another special goal of the authors is to attract the reader's attention to those aspects of theories whose incomplete understanding may lead to inaccuracies or errors. Sometimes, anyway just as designed, the offered information is limited to the pure statements of facts without any proofs. The reader should consult the references to find out missing pieces of information. This book also makes use of numerical (computer) computations. Most of the material contained in the book has already been published, a large part in various works of the authors. Fragments of several chapters come from published works of other authors - some excerpts, particularly relating to basic concepts, and some classic results are taken from outside sources. The book is offered as a textbook for the college-level students or as an aid to the PhD students interested in practical problems of the stability theory. The prerequisites are not demanding - the basic knowledge of calculus, complex functions, and linear algebra which are covered in the suitable, elementary courses is required. The first two chapters include what is typically covered in most introductory courses for students. The first chapter contains definitions of various types of stability; the second commences classic stability theorems regarding ordinary differential equations, but the most basic, applicable in technical sciences. The linear equations are treated more broadly, which creates a foundation for the linear approximation of differential equations in the stability research. Chapter three deals with integral inequalities and their application to the stability studies. Integral inequalities, both linear and nonlinear, are effectively applied in the development of the direct Lyapunov method when the boundedness and stability of motion of nonlinear weakly coupled systems are studied. Chapter four is predominantly dedicated to the Lyapunov direct method. Still, some attention is also paid to the method of limiting equations because it can be used to study motion stability even in hopeless cases when other methods fail. The issue of constructing of the Lyapunov function is a key element in applications of the direct method, and this chapter provides several methods of constructing the function. In the end, a string of examples illustrating the use of the Lyapunov direct method is posted. Chapter five contains a detailed presentation of the comparison method and its use in the stability research. This method, being is essential part of the qualitative theory of equations, is particularly central in studies of largescale systems. In the method, some differential inequalities and Lyapunov functions allow nonlinear transformations of the original system to an equation (a system or a matrix system) of a lower dimension. The idea of delimiting and estimating so-called stability domains is developed in chapter six, where also a qualitative comparison of different stability procedures is made. The evaluation of the efficiency of various methods is conducted by applying, in each case, the same vector norm as a measure of the distance between solutions - no surprise the Lyapunov direct method wins the competition. The contrast between various method results is shown using an example of a simple second-order differential equation. Moreover, for linear systems, the notion of the best Lyapunov function is made. Manifolds of non-holonomic equations of motion are in the focus of chapter seven. Application of topological manifolds and mapping techniques prove to be effective tools in the stability research that extends more and more to advanced fields of mathematics. The chapter reviews specific applications of the Lyapunov direct method to investigations of invariant manifolds and some practical results of the topological fixed point theory. Chapter eight deals with recurrence equations, difference equations, and difference inequalities that mainly are associated with discrete dynamic systems. These types of models are usually obtained by converting the time-continuous dynamics into discrete-time dynamics by employing the Poincare-type mappings. The main objective is the stability investigation of solutions and its estimates. Chapter nine is limited to a short overview of some stability issues for delay differential equations modelling some practical processes and systems with aftereffect phenomena - the main worry is about the compensation for the loss of stability due to delay in the system. Linear models are discussed, but the emphasis is put on Lyapunov functionals for nonlinear equations. Chapter ten on partial differential equations, not including the means of discretization to the stability analysis, uses an approach based on the utilization Lyapunov functionals. The Lyapunov theory is exercised here in relation to a particular class of continuous models - it is an outline of some techniques rather than the methodology. The presented here approach is anecdotal, and it is based on specific cases and examples. Chapter eleven presents some samples of the probabilistic approach to stability matters. This category of problems is necessary when in the modelling process, it turns out that the excitations are not clear, not defined, or not repeatable. In the present considerations, the stability study is reduced to examining the stability of the trivial solution, and the focus is on the almost-sure probability. The last chapter provides a brief introduction to themes of chaos, focusing on the dependence of chaos on the Lyapunov exponent. The irregular behaviour of solutions of motion which is identified with chaos is not due to stochastic forcing or sensitive dependence on initial conditions. The real reason for it is the exponential rate of the distance between the trajectories due to nonlinearities of the system - the Lyapunov exponent is a measure of it.