Kupenko O. Approximation of optimization problems for ill-posed elliptic systems.

The thesis is devoted to the investigation of qualitative properties for a wide class of optimization problems associated with ill-posed elliptic systems whose degenerate or singular coefficients in the main part of corresponding differential operators are adopted as controls. The main purpose is to study solvability and stability with respect to domain perturbations for considered class of nonlinear elliptic optimization problems as well as attainability of some its optimal pairs and minimal cost values via similar characteristics to certain sequences of regularized optimization problems. In order to reach the goals methods of regularization, approximation and relaxation were applied. The following problems were solved: sufficient conditions that guarantee solvability and attainability of solutions to optimization problem for systems containing nonlinear equations of Hammerstein type, nonlinear elliptic equations and variational inequalities, whose coefficients of generalized р-Laplacian are adopted as controls; constructive procedures for approximation of optimal solutions to considered problems via solutions to the sequence of regularized optimization problems are developed; the relaxation procedure for state-constrained optimization problems associated with nonlinear elliptic systems, whose non-smooth coefficients are adopted as controls, is proposed; the question about stability with respect to domain perturbations for optimization problems associated with systems described by degenerate nonlinear elliptic equations and coupled systems containing nonlinear equation of Hammerstein type along with nonlinear Dirichlet problem, whose non-smooth coefficients are adopted as controls, is studied; optimality systems for optimization problems in coefficients of nonlinear elliptic equations with Dirichlet boundary conditions, whose non-smooth or degenerate coefficients of the main part of the corresponding differential operators are adopted as controls, are derived and substantiated. The proposed approximation approach can be used in research and design organizations for the design of new technology and the creation of new materials and design.