Kogut O. Qualitative analysis of one class of optimization problems for nonlinear elliptic systems

The thesis is devoted to the investigation of qualitative properties such as solvability and stability with respect to domain perturbations for one class of nonlinear elliptic optimization problems. It is shown that the fulfillment of the given properties depends essentially on the topology on the parameter space. In this connection for the case of elliptic operators when the matrix of coefficients in its main part is taken as a parameter, it is proposed to consider the set of so-called generalized solenoidal matrixes as a set of admissible parameters. Two problems are considered: the problems of optimal control in coefficients of the main part of nonlinear elliptic equation with Dirichlet and Neumann boundary conditions, respectively. For each of considered problems the solvability within the class of generated solenoidal controls is proved. Furthermore, for the problem of optimal bounded generalized solenoidal control in coefficients of the main part of nonlinear elliptic equation with Dirichlet boundary conditions its stability with respect of two types of initial domain perturbations is studied. It is shown that optimal controls and minimal values of objective functionals of perturbed problems converge in the corresponding topologies to the similar characteristics of unperturbed problem.