Controllability of of some kinetic equations, of parabolic degenerated equations and of the Schrödinger equation via domain transformation.

Abstract : This memoir presents the results obtained during my PhD, whose goal is the study of the controllability of some Partial Differential Equations.The first part of this thesis is concerned with the study of the controllability of some kinetic equations undergoing different regimes. Under a collisional regime, we study the controllability of the Kolmogorov equation, a particular case of kinetic Fokker-Planck equation, in the phase space $R^d times R^d$. We obtain the null-controllability of this equation thanks to the use of a spectral inequality associated to the Laplace operator in the whole space. Under a non-collisional regime, we study the controllability of two fluid-kinetic models, the Vlasov-Stokes system and the Vlasov-Navier-Stokes system, which exhibe nonlinearities due to the coupling terms. In those cases, the strategy relies on the Return method.In the second part, we study the controllability of a family of 1-D degenerate parabolic equations by the flatness method, which allows the construction of explicit controls.The third part is focused on the problem of the controllability of the Schrödinger equation via domain deformations, i.e., using the domain as a control. We obtain a result of this kind in the case of the two-dimensional unit disk, for radial data. Our methods are based on a local exact controllability result around a certain trajectory, obtained thanks to the Inverse Mapping theorem.
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Iván Moyano Garcia. Controllability of of some kinetic equations, of parabolic degenerated equations and of the Schrödinger equation via domain transformation.. Analysis of PDEs [math.AP]. Université Paris-Saclay, 2016. English. ⟨NNT : 2016SACLX062⟩. ⟨tel-01496831⟩

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