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Université de Versailles-Saint Quentin en Yvelines (05/12/2011), Michel Sorine (Dir.)
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Some inverse scattering problems on star-shaped graphs: application to fault detection on electrical transmission line networks
Filippo Visco Comandini1

In this thesis, having in mind applications to the fault-detection/diagnosis of electrical networks, we consider some inverse scattering problems for the Zakharov-Shabat equations and time-independent Schrödinger operators over star-shaped graphs. The first chapter is devoted to describe reflectometry methods applied to electrical networks as an inverse scattering problems on the star-shaped network. Reflectometry methods are presented and modeled by the telegrapher's equations. Reflectometry experiments can be written as inverse scattering problems for Schrödinger operator in the lossless case and for Zakharov-Shabat system for the lossy transmission network. In chapter 2 we introduce some elements of the inverse scattering theory for 1 d Schrödinger equations and the Zakharov-Shabat system. We recall the basic results for these two systems and we present the state of art of scattering theory on network. The third chapter deals with some inverse scattering for the Schrödinger operators. We prove the identifiability of the geometry of the star-shaped graph: the number of the edges and their lengths. Next, we study the potential identification problem by inverse scattering. In the last chapter we focus on the inverse scattering problems for lossy transmission star-shaped network. We prove the identifiability of some geometric informations by inverse scattering and we present a result toward the identification of the heterogeneities, showing the identifiability of the loss line factor.
1 :  INRIA Rocquencourt - SISYPHE (SIgnals and SYstems in PHysiology and Engineering)
Transmission Line Network – Reflectometry – Inverse scattering – Schrodinger operators – Zakharov-Shabat equations

In this thesis, having in mind applications to the fault-detection/diagnosis of electrical networks, we consider some inverse scattering problems for the Zakharov-Shabat equations and time-independent Schrödinger operators over star-shaped graphs. The first chapter is devoted to describe reflectometry methods applied to electrical networks as an inverse scattering problems on the star-shaped network. Reflectometry methods are presented and modeled by the telegrapher's equations. Reflectometry experiments can be written as inverse scattering problems for Schrödinger operator in the lossless case and for Zakharov-Shabat system for the lossy transmission network. In chapter 2 we introduce some elements of the inverse scattering theory for 1 d Schrödinger equations and the Zakharov-Shabat system. We recall the basic results for these two systems and we present the state of art of scattering theory on network. The third chapter deals with some inverse scattering for the Schrödinger operators. We prove the identifiability of the geometry of the star-shaped graph: the number of the edges and their lengths. Next, we study the potential identification problem by inverse scattering. In the last chapter we focus on the inverse scattering problems for lossy transmission star-shaped network. We prove the identifiability of some geometric informations by inverse scattering and we present a result toward the identification of the heterogeneities, showing the identifiability of the loss line factor.
Transmission Line Network – Reflectometry – Inverse scattering – Schrodinger operators – Zakharov-Shabat equations

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