# Definition et Applications des Extensions desFonctions Reelles aux Intervalles Généralisés; reformulation de la theorie des intervalles modaux.

2 COPRIN - Constraints solving, optimization and robust interval analysis
CRISAM - Inria Sophia Antipolis - Méditerranée , ENPC - École des Ponts ParisTech
Abstract : The intervals theory allows constructing supersets of the range of real functions. Therefore, in a very natural way it allows constructing some outer approximation of the solution set of systems of real equations. When it is used in conjnuction to some usual existence theorems (e.g. Brouwer or Miranda theorems), the intervals theory also allows to rigorously prove the existence of solutions to such systems of equations.

The modal intervals theory proposed some richer interpretations. In particular, the construction of both subsets and supersets of the range of real functions are in the scope of extensions to modal intervals. As a consequence, the extensions of real functions to modal intervals have the intrinsic power of proving the existence of solutions to systems of equations. In spite of some recent developments that have shown the promising potential applications of these richer interpretations, the modal intervals theory remains unused by most of the interval community. This may be explained by the following arguments:

A) The modal intervals theory has an original and complicated construction. It is not similar to the construction of the classical intervals theory. This makes for example difficult the addition of new developments in the framework of modal intervals.
B) No preconditioning has yet been proposed that would be compatible with the richer interpretations provided by the modal intervals theory.
C) No linearization process has yet been proposed that would be compatible with the richer interpretations provided by the modal intervals theory.

These three points are developed in the present thesis. On one hand, a new formulation of the modal intervals theory is proposed. This new formulation uses only generalized intervals (intervals whose bounds are not constrained to be ordered) and follows the construction of the classical intervals theory. On the other hand, some new preconditioning and linearization processes are proposed which are compatible with the richer interpretations provided by the modal interval theory. The new linearization process which is proposed will have the form of a new mean-value extension to generalized intervals.

These theoretical developments lead to two applications: on one hand, the new mean-value extension to generalized intervals is used to construct an inner approximation of the range of a vector-valued function. This problem is not well solved today using the classical intervals theory. On the other hand, a generalized Hansen-Sengupta operator is proposed. It is dedicated to the outer approximation of non-linear AE-solution sets. It is much simpler and less expensive in computations than the other techniques that can solve the same problems. A comparison of these different techniques remains to be conducted, and will need the integration of the Hansen-Sengupta operator within some bisection algorithm.
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Cited literature [95 references]

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Submitted on : Friday, March 10, 2006 - 6:47:03 PM
Last modification on : Friday, February 4, 2022 - 3:12:38 AM
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• HAL Id : tel-00011922, version 1

### Citation

Alexandre Goldsztejn. Definition et Applications des Extensions des
Fonctions Reelles aux Intervalles Généralisés; reformulation de la theorie des intervalles modaux.. Modélisation et simulation. Université Nice Sophia Antipolis, 2005. Français. ⟨tel-00011922⟩

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