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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Vector & tensor analysis
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Differential Inclusions - Set-Valued Maps and Viability Theory (Paperback, Softcover reprint of the original 1st ed. 1984)
Loot Price: R3,630
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Differential Inclusions - Set-Valued Maps and Viability Theory (Paperback, Softcover reprint of the original 1st ed. 1984)
Series: Grundlehren der mathematischen Wissenschaften, 264
Expected to ship within 10 - 15 working days
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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."
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