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Viability Theory and Regulation of Controled Systems: The Convex Case

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

Abstract

When we assume that the viability subset K is convex and compact, we obtain many more properties. The most striking one is that the tangential condition
$$\forall x \in K,\,F\left( x \right) \cap {T_K}\left( x \right) \ne \phi $$
(1)
which is necessary and sufficient when F has convex values for the differential inclusion
$$\begin{array}{*{20}{c}} {i)x\prime \left( t \right) \in F\left( {x\left( t \right)} \right),} \\ {ii)x\left( 0 \right) = {x_0},{x_0}{\text{given in K}},} \end{array}{\text{ }}$$
(2)
to have viable trajectories for all initial states x0 in K, is also a sufficient condition for F to have an equilibrium state in K.

Keywords

Convex Subset Differential Inclusion Price System Tangent Cone Closed Convex Cone 
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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