We study in this thesis, subdivided into two parts, the modeling of the free surface shallow water flows with the lattice Boltzmann method and the analysis of the properties of controllability and observability of the resulting models. The first part focusses on the modeling of the shallow water flows with the lattice Boltzmann method. Using a linearization around a given equilibrium profile, we give a state space representation of the defined lattice Boltzmann models. This representation takes into account the force term, and allows a complete definition of the inputs (controls) and outputs (measures) variables. After this, we extend the model to include the phenomena of sedimentation. The defined one-dimensional model is validated numerically by comparing it with a finite volume model which solves the Saint-Venant-Exner's equations. The defined LB model is less complex (from a numerical point of view) and easier to handle. In the second part, we deal with the analysis of the properties of controllability and observability of the models obtained from the LB modeling of the shallow water flows. The first analysis, which is done with the Kalmann's algebraic criterias, leads to the establishment of the loss of controllability when the number of discretization sites increases. An extensive analysis, based on the determination of the controllability and observability gramians, allows to show that this conclusion remains with the classical methods of discretization. The determination of the gramians is done with particular methods well suited for the sparse and large matrices which are the dynamical, control and/or observation matrices of the LB models. Finally, we establish that for a classical boundary control of the irrigation canal with flow and level, the family of LB systems variables is not uniformly controllable, while using scattering variables as the control variables, the family becomes uniformly controllable.
