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Introduction to the Theory of Differential Inclusions
 
Georgi V. Smirnov University of Porto, Porto, Portugal
Introduction to the Theory of Differential Inclusions
Softcover ISBN:  978-1-4704-6854-5
Product Code:  GSM/41.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-2092-5
Product Code:  GSM/41.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6854-5
eBook: ISBN:  978-1-4704-2092-5
Product Code:  GSM/41.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
Introduction to the Theory of Differential Inclusions
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Introduction to the Theory of Differential Inclusions
Georgi V. Smirnov University of Porto, Porto, Portugal
Softcover ISBN:  978-1-4704-6854-5
Product Code:  GSM/41.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-2092-5
Product Code:  GSM/41.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-6854-5
eBook ISBN:  978-1-4704-2092-5
Product Code:  GSM/41.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 412002; 226 pp
    MSC: Primary 34; 49; Secondary 93;

    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.

    Readership

    Graduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization.

  • Table of Contents
     
     
    • Part 1. Foundations
    • Chapter 1. Convex analysis
    • Chapter 2. Set-valued analysis
    • Chapter 3. Nonsmooth analysis
    • Part 2. Differential inclusions
    • Chapter 4. Existence theorems
    • Chapter 5. Viability and invariance
    • Chapter 6. Controllability
    • Chapter 7. Optimality
    • Chapter 8. Stability
    • Chapter 9. Stabilization
  • Additional Material
     
     
  • Reviews
     
     
    • The material of the book may very well be used for an introductory lecture on differential inclusions.

      Jahresbericht der DMV
    • The book is well written and contains a number of excellent problems.

      Zentralblatt MATH
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 412002; 226 pp
MSC: Primary 34; 49; Secondary 93;

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.

Readership

Graduate students and research mathematicians interested in ordinary differential equations, calculus of variations, optimal control, and optimization.

  • Part 1. Foundations
  • Chapter 1. Convex analysis
  • Chapter 2. Set-valued analysis
  • Chapter 3. Nonsmooth analysis
  • Part 2. Differential inclusions
  • Chapter 4. Existence theorems
  • Chapter 5. Viability and invariance
  • Chapter 6. Controllability
  • Chapter 7. Optimality
  • Chapter 8. Stability
  • Chapter 9. Stabilization
  • The material of the book may very well be used for an introductory lecture on differential inclusions.

    Jahresbericht der DMV
  • The book is well written and contains a number of excellent problems.

    Zentralblatt MATH
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.