Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations
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Dissertation research is on the optimal control of systems with distributed parameters described by singular nonlinear partial differential equations (PDE) modeling multi-phase Stefan type second order parabolic free boundary problems. This type of free boundary problems arise in various applications, such as biomedical engineering problem on the laser ablation of biological tissues, aerospace engineering problem on the ice accretion in aircrafts mid-flight, biomedical problem on the growth of cancerous tumor, and many other phase transition processes in thermophysics and fluid mechanics. The aim of the optimal control of distributed free boundary systems is two fold: identification of functional parameters of the model via solving inverse free boundary problems, or optimizing the performance of such systems via optimal choice of control parameters. Ill-posed nature of inverse free boundary problems, formation of singularities by free boundaries, and irregularity of solutions are major difficulties in modeling and controlling distributed free boundary systems. Dissertation exploits a new approach introduced in U.G. Abdulla & B. Poggi, Calculus of Variations & PDEs, 59:61, 2020,which is based on the transformation of the multiphase multidimensional Stefan problem to singular PDE problem with discontinuous coefficient in a fixed domain. Optimal control of second order singular parabolic PDE with principal part in divergence form with bounded measurable coefficients is analyzed. Control parameters are boundary heat flux or density of heat sources, and cost functional is a norm difference of the trace of the solution from the available temperature measurement at the final moment. Existence of the optimal control is proved. Discretization of the optimal control problems via finite differences is pursued. Convergence of the sequence of discrete optimal control problems to continuous optimal control problem both with respect to functional and control is proved. Precisely, it is proved that the sequence of multilinear interpolations of the discrete minimizers converge to the optimal solution of the singular PDE problem in a weak topology of the Hilbert space of weakly differentiable functions. In particular, convergence of the method of finite differences, and existence, uniqueness and stability estimations are established for the singular PDE problem under minimal regularity assumptions on the coefficients expressed in terms of anisotropic Sobolev spaces setting.