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# Algorithmes numériques pour l'analyse topologique : Analyse par intervalles et théorie des graphes.

Abstract : This dissertation proposes, in a first place, numerical methods to compute qualitative properties of sets and, in a second place, algorithms to estimate the attraction domain of an equilibrium state. All the proposed approaches combine interval analysis and graph theory.

Many problems, as studing the configuration space of a robot, amount to analysing topological properties of a given set. In this thesis, we define methods to compute topological invariants of a given set. Those sets are defined by $\mathcal{C}^{\infty}$ inequalities. The main
idea is to decompose the given set into subsets that are proven contractible and to use \v Cech homology.

The second part of the thesis presents a method to estimate the attraction domain of an asymptotically stable equilibrium state $x_{\infty}$. First, one uses interval analysis and Lyapunov theory to compute a neighboorhood of $x_{\infty}$ included in the attraction
domain. Then, we combine graph theory and inclusion methods of O.D.E. to improve this neighboorhood and estimate the attraction domain.
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https://tel.archives-ouvertes.fr/tel-00340999
Contributor : Anne-Marie Plé <>
Submitted on : Monday, November 24, 2008 - 11:27:01 AM
Last modification on : Tuesday, February 11, 2020 - 1:12:01 PM
Long-term archiving on: : Thursday, October 11, 2012 - 11:55:31 AM

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• HAL Id : tel-00340999, version 1

### Citation

Nicolas Delanoue. Algorithmes numériques pour l'analyse topologique : Analyse par intervalles et théorie des graphes.. Automatique / Robotique. Université d'Angers, 2006. Français. ⟨tel-00340999⟩

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