Control and geometric inverse problems for some linear and nonlinear partial differential systems

Abstract

In this Thesis we present results for control and geometric inverse problems associated with certain linear and non-linear PDEs. First, in Chapter 1 we perform a detailed analysis of the geometric inverse problem that consists to identify, from boundary measurements, an unknown obstacle to passage of a fluid governed by a system of linear elliptic equations. Then, by using the so-called local Carleman estimates, we get a uniqueness result, that is, we show that if two obstacles leading to the same boundary measurements are, necessarily, equals. Moreover, by applying some techniques of differentiation with respect to domains, we can obtain a stability result and then apply a reconstruction algorithm. In Chapter 2, we analyze the controllability properties of the so-called inviscid and viscous Burgers-𝛼 equations. More specifically, in the first part of the chapter we can get, by applying the so-called return method, time-reversibility and scale change arguments, a global exact controllability result for the inviscid Burgers-𝛼 system. Then, in the second part, we prove that the viscous Burgers-𝛼 equation is globally exactly controllable to constant trajectories following three steps: (1) We apply a smoothing effect result for parabolic PDEs; (2) We use a controllability result for the inviscid Burgers-𝛼 system to deduce an approximate controllability result for the viscous system; (3) We prove a local exact controllability result for regular time-dependent trajectories. In Chapter 3, we deal with a two-phase free-boundary problem associated with the heat equation. Then, by using a classical technique that reduces controllability to minimization of an appropriated functional, parabolic regularity and the Schauder Fixed-Point Theorem, we prove that it is possible to drive both temperatures and the interface to desired targets in an arbitrary small time, as long as the initial data are small enough in a suitable norm.