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Optimal control of evolution equations and its applications

Abstract : This thesis begins with a rigorous mathematical analysis of the radiative heating of a semi-transparent body made of glass, by a black radiative source surrounding it. This requires the study of the coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution, and give also uniform bounds on the solution i.e. on the absolute temperature distribution inside the body and on the radiative intensities. Now, we consider the temperature TS of the black radiative source S surrounding the semi-transparent body as the control variable. We adjust the absolute temperature distribution (x, t) 7! T(x, t) inside the semi-transparent body near a desired temperature distribution Td(·, ·) during the time interval of radiative heating ]0, tf [ by acting on TS. In this respect, we introduce the appropriate cost functional and the set of admissible controls TS, for which we prove the existence of optimal controls. Introducing the State Space and the State Equation, a first order necessary condition for a control TS : t 7! TS(t) to be optimal is then derived in the form of a Variational Inequality by using the Implicit Function Theorem and the adjoint problem. We come now to the goal problem which is the deformation of the semi-transparent body by heating it with a black radiative source surrounding it. We introduce a weak mixed formulation of this thermoviscoelasticity problem and study the existence and uniqueness of its solution, the novelty here with respect to the work of M.E. Rognes et R. Winther (M3AS, 2010) being the apparition of the viscosity in some of the coefficients of the constitutive equation, viscosity which depends on the absolute temperature T(x, t) and thus in particular on the time t. Finally, we state in this setting the related optimal control problem of the deformation of the semi-transparent body , by acting on the absolute temperature of the black 5 radiative source surrounding it. We prove the existence of an optimal control and we compute the Fréchet derivative of the associated reduced cost functional.
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Submitted on : Wednesday, November 7, 2018 - 3:53:06 PM
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Hawraa Nabolsi. Optimal control of evolution equations and its applications. Analysis of PDEs [math.AP]. Université de Valenciennes et du Hainaut-Cambresis; Université libanaise, 2018. English. ⟨NNT : 2018VALE0027⟩. ⟨tel-01915425⟩

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