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Processus de Markov diffusifs par morceaux: outils analytiques et numériques

Abstract : This thesis studies some Markovian models allowing uncertainties to be taken into account in systems having a "hybrid" dynamics. Possible sources of uncertainty are noisy inputs, poorly known dynamics or random events for instance. Such models, sometimes known as Stochastic Hybrid Systems (SHS), are used in the fields of automatic control and operation research, among others.

In the first part of the thesis, we introduce the concept of a piecewise diffusion process, which provides a theoretical framework unifying the various classes of "hybrid" models known in the literature. Various aspects of these models are then considered, including their mathematical construction (using a revival theorem for Markov processes), the study of their extended generator and Zeno's phenomenon.

The second part of the thesis focuses on the "progagation of uncertainty", i.e. on the evolution of the marginal law of the state. The well-known Fokker-Planck-Kolmogorov (FPK) equation is generalized for various classes of piecewise diffusiion processes, thanks to concepts like the average jump intensity and the probability current. These results are illustrated by mean of two multidimensional examples, for which a numerical resolution of the generalized FPK equation was carried out thanks to a finite volumes discretization. The comparison with Monte-Carlo methods is also discussed.
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Submitted on : Tuesday, September 4, 2007 - 10:45:40 PM
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Julien Bect. Processus de Markov diffusifs par morceaux: outils analytiques et numériques. Automatique / Robotique. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00169791⟩

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