Représentation Microlocale de Solutions de Systèmes Hyperboliques, Application à l'Imagerie, et Contributions au Contrôle et aux Problèmes Inverses pour des Equations Paraboliques

Abstract : In a first part, we consider Cauchy problems for first-order hyperbolic equations and systems. A representation of the solution operator of these equations is given as an infinite product of Fourier integral operators with complex phase. The convergence in Sobolev spaces of this representation is proven as well as the convergence of the wavefront set. In the case of systems, symmetric and symmetrizable systems are considered. The proposed representation naturally yields numerical schemes for the resolution of the Cauchy problems. We present applications of this method to the field of seismic imaging. There, approximations are performed to obtain efficient schemes. Other applications of microlocal analysis to seismology are presented.
In a second part, we study the controllability to the trajectories for linear and semilinear parabolic equations. We focus on the case of operators in divergence form where the coefficient in the principal part is non continuous. We first derive a Carleman estimate, in one space dimension, for a piecewise-$C^1$ coefficient. This result is extended to a $BV$ coefficient by passing to the limit in the Carleman estimate. With these results, we prove that the controllability for parabolic equations is achievable in one space dimension without assuming any compatibility between the control region and the signs of the jumps of the discontinuous coefficient. We further exhibit a case in higher dimension for which the same conclusion holds. Finally, we make use of a Carleman estimate to identify the discontinuous coefficient from measurements of the solution.
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Submitted on : Tuesday, April 22, 2008 - 4:25:41 PM
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Jérôme Le Rousseau. Représentation Microlocale de Solutions de Systèmes Hyperboliques, Application à l'Imagerie, et Contributions au Contrôle et aux Problèmes Inverses pour des Equations Paraboliques. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2007. ⟨tel-00201887v2⟩

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