Uniform Controllability of Discrete Partial Dierential Equations Thèse Dirigée Par : Rapporteurs : I Am Grateful to My Close Friends Dung

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. As shown in [LT06], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, uniform observability is achieved in L2 for semidiscrete approximations for the parabolic systems. 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. An example of application is implemented for the one-dimensional heat equation with Dirichlet boundary control. The study of Lq optimality above 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. Carleman estimate are L2 weighted energy estimates. Here the weight is chosen so as to accommodate the jump of the diffusion coefficient. Moreover, there is a strong connexion between the Carleman large parameter and the (small) discretization parameter . Consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of linear and semi-linear parabolic equations. i te l-0 09 19 25 5, v er si on 1 16 D ec 2 01 3 Acknowledgements Foremost, I would like to express my sincerest gratitude to my two supervisors Professor Emmanuel Trélat and Professor Jérôme Le Rousseau for continuous support of my PhD study and research, for their patience, motivation, enthusiam, and immense knowledge. They offered the best advices as well as encouragement throughout the course of my thesis. I feel lucky for having them as supervisors. I would like to acknowledge Professor Michel Zinmeister for giving me a chance to work in good research enviroment of University of Orleans from PUF program. I thank to Region Center for the financial support, without which this work would have never been possible. Many thanks to all members of laboratory MAPMO, who have accompanied with me during a long three years of working and studing here. A special acknowledge to the secrectaries of MAPMO for their kindness. I am grateful to my close friends Dung Duong Hoang, Peipei Shang, Tan Cao Hoang who are always by my side to support and encourage me to overcome the difficult stages in the course of this research as well as my life . I also thank to Vietnamese students who have shared with me a lot of good memories during three years in France. Finally, I would like to express the profound gratitude from my deep heart to my beloved parents and little brother for their love and spiritual support throughout my life. –Thuy NGUYEN– ii te l-0 09 19 25 5, v er si on 1 16 D ec 2 01 3