Sur quelques problèmes d'observateurs, Applications à certains modèles d'écosystème aquatique

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Résumé

The goal of this thesis is twofold. First we show that some tools from control theory are helpful to adress the stock estimation problem for an exploited sh population. The stock estimation is one of the most important problem in shery science. To make a policy decision about the exploitation of renewable ressources, it is necessary to take into account the state of the resource stocks. This implies the need of a good estimate of the available resource. Mathematical models are more and more used to describe the evolution of biological systems. In the rst part of this work, we consider two mathematical models for shery resources. The rst one is a " stage structured " model that describes the dynamics of a population divided in stage-classes (according to age, length or weight) and submitted to the shing action. The second model is a " global " model that describes the evolution of a sh population that can move between an area where it can be harvested and a reserve area where no shing is allowed. Both models are given by systems of ordinary di erential equations. For our work, we show that a tool from nonlinear control theory called " observer " can be helpul to deal with the resource stock estimation in the eld of renewable resource management. More precisely we built a dynamical systems that uses the available data ( shing efort and the total or partial of caught sh) and which produces a dynamical estimation of the stock state. In the second part of this thesis, an adaptive observer for continuous-discrete systems is proposed. It is well known that adaptive observer is very important for join state and parameters estimation ; most of known results concern continuous systems with continuous-output or discrete-time systems with discrete output. In our work, we consider a linear system (obtained after some output injection transformation) with discrete outputs and we propose an algorithm to estimate both the state and the parameter.We use rst the persistence conditions to show the convergence of the proposed algorithm, after, we propose an algorithm for the join state and parameter estimation without any persistance condition. The proposed adaptive observer are shown to quite promising due to the exponentieal error convergence.

Mots-clés

adaptive observer, state estimation, fisheries, stage-structured population model, fish population dynamics, reserve area, continuous-discrete systems, state and parameter estimation, persistence conditions, nonlinear control, Parameters estimation, High Gain, Fishery, Population dynamic, Adapative observer, Nonlinear observer, Nonlinear systems, Systèmes non-linèaires, Systèmes continus-discrets, Observateurs non-linéaires, Observateurs adaptatifs, Estimations d'états, Estimation de paramètres, Grand Gain, Halieutique et pêche, Dynamique de population