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.

Differential Inclusions: Set-Valued Maps and Viability Theory (Grundlehren Der Mathematischen Wissenschaften)

Science & Math

Jean-Pierre;Cellina Arrigo Aubin

Mathematics

Springer, Verlag

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Books by Jean-Pierre;Cellina Arrigo Aubin

See the best price to sell, buy, or rent books by Jean-Pierre;Cellina Arrigo Aubin. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Differential Inclusions: Set-Valued Maps and Viability Theory (Grundlehren Der Mathematischen Wissenschaften)