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Construction of a control and reconstruction of a source for linear and nonlinear heat equations

Abstract : My thesis focuses on two main problems in studying the heat equation: Control problem and Inverseproblem.Our first concern is the null controllability of a semilinear heat equation which, if not controlled, can blow up infinite time. Roughly speaking, it consists in analyzing whether the solution of a semilinear heat equation, underthe Dirichlet boundary condition, can be driven to zero by means of a control applied on a subdomain in whichthe equation evolves. Under an assumption on the smallness of the initial data, such control function is builtup. The novelty of our method is computing the control function in a constructive way. Furthermore, anotherachievement of our method is providing a quantitative estimate for the smallness of the size of the initial datawith respect to the control time that ensures the null controllability property.Our second issue is the local backward problem for a linear heat equation. We study here the followingquestion: Can we recover the source of a linear heat equation, under the Dirichlet boundary condition, from theobservation on a subdomain at some time later? This inverse problem is well-known to be an ill-posed problem,i.e their solution (if exists) is unstable with respect to data perturbations. Here, we tackle this problem bytwo different regularization methods: The filtering method and The Tikhonov method. In both methods, thereconstruction formula of the approximate solution is explicitly given. Moreover, we also provide the errorestimate between the exact solution and the regularized one.
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Submitted on : Wednesday, March 27, 2019 - 11:18:07 AM
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  • HAL Id : tel-02081052, version 1


Thi Minh Nhat Vo. Construction of a control and reconstruction of a source for linear and nonlinear heat equations. Mathematical Physics [math-ph]. Université d'Orléans, 2018. English. ⟨NNT : 2018ORLE2012⟩. ⟨tel-02081052⟩



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