In a large class of modern applications, we have to estimate the initial (or final) state of an infinite-dimensional system (typically a system governed by a Partial Differential Equation) from its partial measurement over some finite time interval. This kind of identification problems arises in medical imaging. For instance, the detection of sick cells (tumor) by thermoacoustic tomography can be viewed as an initial data reconstruction problem. Some other methods need the identification of a source term, which can be rewritten, under some assumptions, under the form of an initial data reconstruction problem. In this thesis, we are dealing with the reconstruction of the initial state of a system of evolution, working as much as possible on the infinite-dimensional system, using the new algorithm developed by Ramdani, Tucsnak and Weiss (Automatica 2010). We perform in particular the numerical analysis of the algorithm in the case of Schrödinger and wave equations, with internal observation. We study the suitable functional spaces for its use in Maxwell's equations, with internal and boundary observation. In the last chapter, we try to extend the framework of this algorithm when the initial system is perturbed or when the inverse problem is ill-posed, with application to thermoacoustic tomography.
