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Uniform controllability of discrete partial differential equations

Abstract : In this thesis, we study uniform controllability properties of semi-discrete approximations for parabolic systems. In a first part, we address the minimization of the Lq-norm (q > 2) of semidiscrete controls for parabolic equation. Our goal is to overcome the limitation of [LT06] about the order 1/2 of unboundedness of the control operator. Namely, we show that the uniform observability property also holds in Lq (q > 2) even in the case of a degree of unboundedness greater than 1/2. Moreover, a minimization procedure to compute the approximation controls is provided. The study of Lq optimality in the first part is in a general context. However, the discrete observability inequalities that are obtained are not so precise than the ones derived then with Carleman estimates. In a second part, in the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi discrete version of the parabolic operator @t − @x(c@x) which allows one to derive observability inequalities that are far more precise. Here we consider in case that the diffusion coefficient has a jump which yields a transmission problem formulation. Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations.
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Thi Nhu Thuy Nguyen. Uniform controllability of discrete partial differential equations. General Mathematics [math.GM]. Université d'Orléans, 2012. English. ⟨NNT : 2012ORLE2038⟩. ⟨tel-00919255⟩

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