Introduction To The Theory Of Differential Inclusions

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Introduction to the Theory of Differential Inclusions

Georgi V. Smirnov
Introduction to the Theory of Differential Inclusions Book PDF

Author : Georgi V. Smirnov
Publisher : American Mathematical Society
Page : 226 pages
File Size : 44,6 Mb
Release : 2022-02-22
Category : Mathematics
ISBN : 9781470468545

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Introduction to the Theory of Differential Inclusions by Georgi V. Smirnov Pdf

A differential inclusion is a relation of the form $dot x in F(x)$, where $F$ is a set-valued map associating any point $x in R^n$ with a set $F(x) subset R^n$. As such, the notion of a differential inclusion generalizes the notion of an ordinary differential equation of the form $dot x = f(x)$. Therefore, all problems usually studied in the theory of ordinary differential equations (existence and continuation of solutions, dependence on initial conditions and parameters, etc.) can be studied for differential inclusions as well. Since a differential inclusion usually has many solutions starting at a given point, new types of problems arise, such as investigation of topological properties of the set of solutions, selection of solutions with given properties, and many others. Differential inclusions play an important role as a tool in the study of various dynamical processes described by equations with a discontinuous or multivalued right-hand side, occurring, in particular, in the study of dynamics of economical, social, and biological macrosystems. They also are very useful in proving existence theorems in control theory. This text provides an introductory treatment to the theory of differential inclusions. The reader is only required to know ordinary differential equations, theory of functions, and functional analysis on the elementary level. Chapter 1 contains a brief introduction to convex analysis. Chapter 2 considers set-valued maps. Chapter 3 is devoted to the Mordukhovich version of nonsmooth analysis. Chapter 4 contains the main existence theorems and gives an idea of the approximation techniques used throughout the text. Chapter 5 is devoted to the viability problem, i.e., the problem of selection of a solution to a differential inclusion that is contained in a given set. Chapter 6 considers the controllability problem. Chapter 7 discusses extremal problems for differential inclusions. Chapter 8 presents stability theory, and Chapter 9 deals with the stabilization problem.

Theory of Control Systems Described by Differential Inclusions

Zhengzhi Han,Xiushan Cai,Jun Huang
Theory of Control Systems Described by Differential Inclusions Book PDF

Author : Zhengzhi Han,Xiushan Cai,Jun Huang
Publisher : Springer
Page : 344 pages
File Size : 47,9 Mb
Release : 2016-06-15
Category : Technology & Engineering
ISBN : 9783662492451

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Theory of Control Systems Described by Differential Inclusions by Zhengzhi Han,Xiushan Cai,Jun Huang Pdf

This book provides a brief introduction to the theory of finite dimensional differential inclusions, and deals in depth with control of three kinds of differential inclusion systems. The authors introduce the algebraic decomposition of convex processes, the stabilization of polytopic systems, and observations of Luré systems. They also introduce the elemental theory of finite dimensional differential inclusions, and the properties and designs of the control systems described by differential inclusions. Addressing the material with clarity and simplicity, the book includes recent research achievements and spans all concepts, concluding with a critical mathematical framework. This book is intended for researchers, teachers and postgraduate students in the area of automatic control engineering.

Differential Inclusions

J.-P. Aubin,A. Cellina
Differential Inclusions Book PDF

Author : J.-P. Aubin,A. Cellina
Publisher : Springer Science & Business Media
Page : 353 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642695124

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Differential Inclusions by J.-P. Aubin,A. Cellina Pdf

A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems x'(t) = f(t, x(t), u(t)), x(O)=xo "controlled" by parameters u(t) (the "controls"). Indeed, if we introduce the set-valued map F(t, x)= {f(t, x, u)}ueu then solutions to the differential equations (*) are solutions to the "differen tial inclusion" (**) x'(t)EF(t, x(t)), x(O)=xo in which the controls do not appear explicitely. Systems Theory provides dynamical systems of the form d x'(t)=A(x(t)) dt (B(x(t))+ C(x(t)); x(O)=xo in which the velocity of the state of the system depends not only upon the x(t) of the system at time t, but also on variations of observations state B(x(t)) of the state. This is a particular case of an implicit differential equation f(t, x(t), x'(t)) = 0 which can be regarded as a differential inclusion (**), where the right-hand side F is defined by F(t, x)= {vlf(t, x, v)=O}. During the 60's and 70's, a special class of differential inclusions was thoroughly investigated: those of the form X'(t)E - A(x(t)), x (0) =xo where A is a "maximal monotone" map. This class of inclusions contains the class of "gradient inclusions" which generalize the usual gradient equations x'(t) = -VV(x(t)), x(O)=xo when V is a differentiable "potential". 2 Introduction There are many instances when potential functions are not differentiable

Resolution of Singularities

Steven Dale Cutkosky
Resolution of Singularities Book PDF

Author : Steven Dale Cutkosky
Publisher : American Mathematical Soc.
Page : 198 pages
File Size : 43,6 Mb
Release : 2004
Category : Singularities (Mathematics).
ISBN : 9780821835555

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Resolution of Singularities by Steven Dale Cutkosky Pdf

The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D $-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several proofs given for resolution of curves and surfaces in characteristic zero and arbitrary characteristic. Besides explaining the tools needed for understanding resolutions, Cutkosky explains the history and ideas, providing valuable insight and intuition for the novice (or expert). There are many examples and exercises throughout the text. The book is suitable for a second course on an exciting topic in algebraic geometry. A core course on resolutions is contained in Chapters 2 through 6. Additional topics are covered in the final chapters. The prerequisite is a course covering the basic notions of schemes and sheaves.

Stochastic Differential Inclusions and Applications

Michał Kisielewicz
Stochastic Differential Inclusions and Applications Book PDF

Author : Michał Kisielewicz
Publisher : Springer Science & Business Media
Page : 295 pages
File Size : 53,7 Mb
Release : 2013-06-12
Category : Mathematics
ISBN : 9781461467564

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Stochastic Differential Inclusions and Applications by Michał Kisielewicz Pdf

​This book aims to further develop the theory of stochastic functional inclusions and their applications for describing the solutions of the initial and boundary value problems for partial differential inclusions. The self-contained volume is designed to introduce the reader in a systematic fashion, to new methods of the stochastic optimal control theory from the very beginning. The exposition contains detailed proofs and uses new and original methods to characterize the properties of stochastic functional inclusions that, up to the present time, have only been published recently by the author. The work is divided into seven chapters, with the first two acting as an introduction, containing selected material dealing with point- and set-valued stochastic processes, and the final two devoted to applications and optimal control problems. The book presents recent and pressing issues in stochastic processes, control, differential games, optimization and their application in finance, manufacturing, queueing networks, and climate control. Written by an award-winning author in the field of stochastic differential inclusions and their application to control theory, This book is intended for students and researchers in mathematics and applications; particularly those studying optimal control theory. It is also highly relevant for students of economics and engineering. The book can also be used as a reference on stochastic differential inclusions. Knowledge of select topics in analysis and probability theory are required.

New Perspectives and Applications of Modern Control Theory

Julio B. Clempner,Wen Yu
New Perspectives and Applications of Modern Control Theory Book PDF

Author : Julio B. Clempner,Wen Yu
Publisher : Springer
Page : 538 pages
File Size : 43,6 Mb
Release : 2017-09-30
Category : Technology & Engineering
ISBN : 9783319624648

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New Perspectives and Applications of Modern Control Theory by Julio B. Clempner,Wen Yu Pdf

This edited monograph contains research contributions on a wide range of topics such as stochastic control systems, adaptive control, sliding mode control and parameter identification methods. The book also covers applications of robust and adaptice control to chemical and biotechnological systems. This collection of papers commemorates the 70th birthday of Dr. Alexander S. Poznyak.

Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

Mikhail I. Kamenskii,Valeri V. Obukhovskii,Pietro Zecca
Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces Book PDF

Author : Mikhail I. Kamenskii,Valeri V. Obukhovskii,Pietro Zecca
Publisher : Walter de Gruyter
Page : 245 pages
File Size : 43,6 Mb
Release : 2011-07-20
Category : Mathematics
ISBN : 9783110870893

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Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces by Mikhail I. Kamenskii,Valeri V. Obukhovskii,Pietro Zecca Pdf

The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of his own and as an approach to control theory. The book deals with the theory of semilinear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. These assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented.

Theory of Fuzzy Differential Equations and Inclusions

V. Lakshmikantham,Ram N. Mohapatra
Theory of Fuzzy Differential Equations and Inclusions Book PDF

Author : V. Lakshmikantham,Ram N. Mohapatra
Publisher : CRC Press
Page : 192 pages
File Size : 49,5 Mb
Release : 2003-03-20
Category : Mathematics
ISBN : 0415300738

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Theory of Fuzzy Differential Equations and Inclusions by V. Lakshmikantham,Ram N. Mohapatra Pdf

Fuzzy differential functions are applicable to real-world problems in engineering, computer science, and social science. That relevance makes for rapid development of new ideas and theories. This volume is a timely introduction to the subject that describes the current state of the theory of fuzzy differential equations and inclusions and provides a systematic account of recent developments. The chapters are presented in a clear and logical way and include the preliminary material for fuzzy set theory; a description of calculus for fuzzy functions, an investigation of the basic theory of fuzzy differential equations, and an introduction to fuzzy differential inclusions.

Viability Theory

Jean-Pierre Aubin
Viability Theory Book PDF

Author : Jean-Pierre Aubin
Publisher : Springer Science & Business Media
Page : 558 pages
File Size : 52,8 Mb
Release : 2009-05-28
Category : Science
ISBN : 9780817649104

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Viability Theory by Jean-Pierre Aubin Pdf

"The book is a compendium of the state of knowledge about viability...Mathematically, the book should be accessible to anyone who has had basic graduate courses in modern analysis and functional analysis...The concepts are defined and many proofs of the requisite results are reproduced here, making the present book essentially self-contained." —Bulletin of the AMS "Because of the wide scope, the book is an ideal reference for people encountering problems related to viability theory in their research...It gives a very thorough mathematical presentation. Very useful for anybody confronted with viability constraints." —Mededelingen van het Wiskundig Genootschap

Advances in the Applications of Nonstandard Finite Difference Schemes

Ronald E Mickens
Advances in the Applications of Nonstandard Finite Difference Schemes Book PDF

Author : Ronald E Mickens
Publisher : World Scientific
Page : 664 pages
File Size : 46,5 Mb
Release : 2005-10-25
Category : Mathematics
ISBN : 9789814479868

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Advances in the Applications of Nonstandard Finite Difference Schemes by Ronald E Mickens Pdf

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic. Contents:Nonstandard Finite Difference Methods (R E Mickens)Application of Nonstandard Finite Difference Schemes to the Simulation Studies of Robotic Systems (R F Abo-Shanab et al.)Applications of Mickens Finite Differences to Several Related Boundary Value Problems (R Buckmire)High Accuracy Nonstandard Finite-Difference Time-Domain Algorithms for Computational Electromagnetics: Applications to Optics and Photonics (J B Cole)Nonstandard Finite Difference Schemes for Solving Nonlinear Micro Heat Transport Equations in Double-Layered Metal Thin Films Exposed to Ultrashort Pulsed Lasers (W Dai)Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.)Applications of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont)Finite Difference Schemes on Unbounded Domains (M Ehrhardt)Asymptotically Consistent Nonstandard Finite-Difference Methods for Solving Mathematical Models Arising in Population Biology (A B Gumel et al.)Nonstandard Finite Difference Methods and Biological Models (S R-J Jang)Robust Discretizations versus Increase of the Time Step for Chaotic Systems (C Letellier & E M A M Mendes)Contributions to the Theory of Nonstandard Finite-Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar)Frequency Accurate Finite Difference Methods (A L Perkins et al.)Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger) Readership: Applied mathematicians, and researchers in numerical & computational mathematics and analysis & differential equations. Usable as a secondary text to a standard undergraduate or graduate course on numerical methods for differential equations. Keywords:Numerical Integration Methods;Finite Differences;Nonstandard Finite Difference Schemes;Differential Equations;Discrete Models;Numerical and Computational MathematicsKey Features:A collection of papers from renowned experts in their respective fieldsProvides the most recent work on the application of NSFD schemes and some of the mathematical analysis related to these schemes

Advances in the Applications of Nonstandard Finite Diffference Schemes

Ronald E. Mickens
Advances in the Applications of Nonstandard Finite Diffference Schemes Book PDF

Author : Ronald E. Mickens
Publisher : World Scientific
Page : 665 pages
File Size : 41,7 Mb
Release : 2005
Category : Mathematics
ISBN : 9789812703316

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Advances in the Applications of Nonstandard Finite Diffference Schemes by Ronald E. Mickens Pdf

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990''s and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.

Approximation and Optimization of Discrete and Differential Inclusions

Elimhan N Mahmudov
Approximation and Optimization of Discrete and Differential Inclusions Book PDF

Author : Elimhan N Mahmudov
Publisher : Elsevier
Page : 396 pages
File Size : 40,5 Mb
Release : 2011-08-25
Category : Mathematics
ISBN : 9780123884336

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Approximation and Optimization of Discrete and Differential Inclusions by Elimhan N Mahmudov Pdf

Optimal control theory has numerous applications in both science and engineering. This book presents basic concepts and principles of mathematical programming in terms of set-valued analysis and develops a comprehensive optimality theory of problems described by ordinary and partial differential inclusions. In addition to including well-recognized results of variational analysis and optimization, the book includes a number of new and important ones Includes practical examples

Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control

Rafal K. Goebel
Set Valued Convex and Nonsmooth Analysis in Dynamics and Control Book PDF

Author : Rafal K. Goebel
Publisher : SIAM
Page : 234 pages
File Size : 47,7 Mb
Release : 2024-06-26
Category : Mathematics
ISBN : 9781611977981

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Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control by Rafal K. Goebel Pdf

Set-valued analysis, convex analysis, and nonsmooth analysis are relatively modern branches of mathematical analysis that have become increasingly relevant in current control theory and control engineering literature. This book serves as a broad introduction to analytical tools in these fields and to their applications in dynamical and control systems and is the first to cover these topics with this scope and at this level. Both continuous-time and discrete-time mutlivalued dynamics, modeled by differential and difference inclusions, are considered. Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction is aimed at graduate students in control engineering and applied mathematics and researchers in control engineering who have no prior exposure to set-valued, convex, and nonsmooth analysis. The book will also be of interest to advanced undergraduate mathematics students and mathematicians with no prior exposure to the topic. The expected mathematical background is a course on nonlinear differential equations / dynamical systems and a course on real analysis. Knowledge of some control theory is helpful, but not essential.

Topological Methods for Differential Equations and Inclusions

John R. Graef,Johnny Henderson,Abdelghani Ouahab
Topological Methods for Differential Equations and Inclusions Book PDF

Author : John R. Graef,Johnny Henderson,Abdelghani Ouahab
Publisher : CRC Press
Page : 360 pages
File Size : 53,7 Mb
Release : 2018-09-25
Category : Mathematics
ISBN : 9780429822629

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Topological Methods for Differential Equations and Inclusions by John R. Graef,Johnny Henderson,Abdelghani Ouahab Pdf

Topological Methods for Differential Equations and Inclusions covers the important topics involving topological methods in the theory of systems of differential equations. The equivalence between a control system and the corresponding differential inclusion is the central idea used to prove existence theorems in optimal control theory. Since the dynamics of economic, social, and biological systems are multi-valued, differential inclusions serve as natural models in macro systems with hysteresis.

Impulsive Differential Inclusions

John R. Graef,Johnny Henderson,Abdelghani Ouahab
Impulsive Differential Inclusions Book PDF

Author : John R. Graef,Johnny Henderson,Abdelghani Ouahab
Publisher : Walter de Gruyter
Page : 412 pages
File Size : 48,9 Mb
Release : 2013-07-31
Category : Mathematics
ISBN : 9783110295313

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Impulsive Differential Inclusions by John R. Graef,Johnny Henderson,Abdelghani Ouahab Pdf

Differential equations with impulses arise as models of many evolving processes that are subject to abrupt changes, such as shocks, harvesting, and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. There are also many different studies in biology and medicine for which impulsive differential equations provide good models. During the last 10 years, the authors have been responsible for extensive contributions to the literature on impulsive differential inclusions via fixed point methods. This book is motivated by that research as the authors endeavor to bring under one cover much of those results along with results by other researchers either affecting or affected by the authors' work. The questions of existence and stability of solutions for different classes of initial value problems for impulsive differential equations and inclusions with fixed and variable moments are considered in detail. Attention is also given to boundary value problems. In addition, since differential equations can be viewed as special cases of differential inclusions, significant attention is also given to relative questions concerning differential equations. This monograph addresses a variety of side issues that arise from its simpler beginnings as well.