• Jean-Pierre Aubin
  • Arrigo Cellina
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 264)


A great impetus to study differential inclusions came from the development of Control Theory, i.e. of dynamical systems
$${\text{x'(t) = f}}\left( {{\text{t,x}}\left( t \right){\text{,u}}\left( t \right)} \right){\text{, x(0) = }}{{\text{x}}_0}$$
“controlled” by parameters u(t) (the “controls”). Indeed, if we introduce the set-valued map
$$F(t,x)\dot = {\left\{ {f\left( {t,x,u} \right)} \right\}_{u \in U}}$$
then solutions to the differential equations (*) are solutions to the “differential inclusion”
$$x'\left( t \right) \in F\left( {t,x\left( t \right)} \right),{\mkern 1mu} x\left( 0 \right) = {x_0}$$
in which the controls do not appear explicitely.


Regulatory Control Differential Inclusion Maximal Monotone Contingent Cone Contingent Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Jean-Pierre Aubin
    • 1
  • Arrigo Cellina
    • 2
  1. 1.CEREMADEUniversité de Paris-DauphineParis Cedex 16France
  2. 2.S.I.S.S.A.TriesteItaly

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