Optimal Control of Coefficients for the Second Order Parabolic Free Boundary Problems
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Dissertation aims to analyze inverse Stefan type free boundary problem for the second order parabolic PDE with unknown parameters based on the additional information given in the form of the distribution of the solution of the PDE and the position of the free boundary at the final moment. This type of ill-posed inverse free boundary problems arise in many applications such as biomedical engineering problem about the laser ablation of biomedical tissues, in-flight ice accretion modeling in aerospace industry, and various phase transition processes in thermophysics and fluid mechanics. The set of unknown parameters include a space-time dependent diffusion, convection and reaction coefficients, density of the sources, time-dependent boundary flux and the free boundary. New PDE constrained optimal control framework in Hilbert-Besov spaces introduced in U.G. Abdulla, Inverse Problems and Imaging, 7, 2(2013), 307-340; 10, 4(2016), 869-898 is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary, and available information on the phase transition temperature on the free boundary. The latter presents a key advantage in dealing with applications, where phase transition temperature is not known explicitly, but involve some measurement error. Another advantage of the new variational approach is based on the fact that for a given control parameter, Stefan boundary condition turns into Neumann boundary condition on the given boundary, and parabolic PDE problem is solved in a fixed domain, and therefore a perspective opens for the development of numerical methods of least computational cost. Discretization of the optimal control problem via method of finite differences is pursued and the sequence of finite-dimensional optimal control problems are introduced. The results of the dissertation are different depending on the structure of the unknown diffusion coefficient. In the case if it is only time-dependent, the well-posedness of the optimal control problem is established in Hilbert-Besov spaces. Existence of the optimal control and convergence of the sequence of the discrete optimal control problems to the continuous optimal control problem both with respect to functional and control is proved. The methods of the proof are based on uniform H 1 -energy estimates in discrete Sobolev-Hilbert norms, weak compactness argument, Weierstrass theorem in weak topology and weak convergence of the bilinear interpolations of the solutions of the discrete PDE problems to the solution of the optimal PDE problem in the class of weakly differentiable functions. To prove similar results in the case when unknown diffusion coefficient is space-time dependent, a new Banach space is introduced. The motivation for the new space is dictated with the optimal result on the convergence of the bilinear interpolations of the grid functions in the class of weakly differentiable functions, and establishment of the discrete H 1 -energy estimate under minimal assumptions on the diffusion coefficient. Existence of the optimal control and convergence of the sequence of discrete optimal control problems to the continuous optimal control problem both with respect to functional and control is proved in the setting of the new Banach space.