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Problèmes inverses de sources dans des équations de transport à coefficients variables

Abstract : The thesis deals with the two main issues identifiability and identification related to a nonlinear inverse source problem. This problem consists in the identification of a time-dependent point source occurring in the right hand-side of an advection-dispersion-reaction equation with spatially varying coefficients. Starting from the stationnary case in the one-dimensional model, we derived theoritical results defining the necessary number of sensors and their positions that enable to uniquely determine the sought source. Those results gave us a good visibility on how to proceed in order to obtain similar results for the time-dependent (evolution) case. As far as the two-dimensional evolution model is concerned, using some boundary null controllability results and the records of the generated state on the inflow boundary and its flux on the outflow boundary of the monitored domain, we established a constructive identifiability theorem as well as an identification method that localizes the two coordinates of the sought source position as the unique solution of a nonlinear system of two equations and transforms the identification of its time-dependent intensity function into solving a deconvolution problem. The last part of this thesis highlights the main difficulty encountred in such inverse problems namely the nonidentifiabilityof a source in its abstract form, proposes a method that enables to overcome this difficulty in the particular case where the aim is to identify the time active limit of the involved source. And thus, this last part opens doors on new horizons and prospects.
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Submitted on : Tuesday, April 8, 2014 - 9:47:29 AM
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  • HAL Id : tel-00975168, version 1


Imed Mahfoudhi. Problèmes inverses de sources dans des équations de transport à coefficients variables. Mathématiques générales [math.GM]. INSA de Rouen, 2013. Français. ⟨NNT : 2013ISAM0032⟩. ⟨tel-00975168⟩



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