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Liapunov Functions

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

Abstract

We shall investigate whether differential inclusions
$$x'\left( t \right) \in F\left( {x\left( t \right)} \right),{\text{ }}x\left( 0 \right) = {x_0}$$
(1)
do have trajectories satisfying the property
$$\forall t > s,{\text{ }}V\left( {x\left( t \right)} \right) - V\left( {x\left( s \right)} \right) + \int\limits_s^1 {W\left( {x\left( \tau \right),x'\left( \tau \right)} \right)} dx \leqslant 0$$
(2)
where
$$\left\{ {\begin{array}{*{20}{c}} {i)\,V\,is\,function\,from\,K\dot = Dom\,F\,to\,{R_ + }} \\ {ii)\,W\,is\,a\,function\,from\,Graph\,(F)\,to\,{R_ + }} \end{array}} \right.$$
(3)
Trajectories x(·) of differential inclusion (1) satisfying (2) will be called “monotone trajectories” (with respect to V and W).

Keywords

Function Versus Closed Subset Differential Inclusion Nonnegative Function Cluster Point 
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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