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Differential Inclusions: Set-Valued Maps and Viability Theory

Differential Inclusions: Set-Valued Maps and Viability Theory

A. Cellina
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A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems x' = f(t, x, u), x=xo "controlled" by parameters u . Indeed, if we introduce the set-valued map F= {f}ueu then solutions to the differential equations are solutions to the "differen tial inclusion" x'EF(t, x), x=xo in which the controls do not appear explicitely. Systems Theory provides dynamical systems of the form d x'=A(x) dt (B(x)+ C(x); x=xo in which the velocity of the state of the system depends not only upon the x of the system at time t, but also on variations of observations state B(x) of the state. This is a particular case of an implicit differential equation f(t, x, x') = 0 which can be regarded as a differential inclusion , where the right-hand side F is defined by F= {vlf=O}. During the 60's and 70's, a special class of differential inclusions was thoroughly investigated: those of the form X'E - A(x), x =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' = -VV(x), x=xo when V is a differentiable "potential." 2 Introduction There are many instances when potential functions are not differentiable.
Language
English
Pages
342
Format
Paperback
Publisher
Springer
Release
January 25, 2012
ISBN
3642695140
ISBN 13
9783642695148

Differential Inclusions: Set-Valued Maps and Viability Theory

A. Cellina
5/5 ( ratings)
Read Download
A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems x' = f(t, x, u), x=xo "controlled" by parameters u . Indeed, if we introduce the set-valued map F= {f}ueu then solutions to the differential equations are solutions to the "differen tial inclusion" x'EF(t, x), x=xo in which the controls do not appear explicitely. Systems Theory provides dynamical systems of the form d x'=A(x) dt (B(x)+ C(x); x=xo in which the velocity of the state of the system depends not only upon the x of the system at time t, but also on variations of observations state B(x) of the state. This is a particular case of an implicit differential equation f(t, x, x') = 0 which can be regarded as a differential inclusion , where the right-hand side F is defined by F= {vlf=O}. During the 60's and 70's, a special class of differential inclusions was thoroughly investigated: those of the form X'E - A(x), x =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' = -VV(x), x=xo when V is a differentiable "potential." 2 Introduction There are many instances when potential functions are not differentiable.
Language
English
Pages
342
Format
Paperback
Publisher
Springer
Release
January 25, 2012
ISBN
3642695140
ISBN 13
9783642695148

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