Gorbonos S. Qualitative analysis of some optimal control problems for parabolic equations with unbounded coefficients

The object of research is some class of optimal boundary control problems for a linear parabolic equation with unbounded coefficients in the main part of operator. The subject of research is a qualitative analysis of optimal control problems for parabolic equations with unbounded coefficients, an approximation of their solutions and optimality conditions for these problems. The thesis is aimed to provide a qualitative analysis of optimal control problems for linear parabolic equations with unbounded coefficients in the main part of differential operator, derive an optimality conditions, obtain the solvability conditions, and also to study the attainability of optimal solutions to the original problem. To do this we use the methods of functional analysis, calculus of variations, non-linear analysis, methods of optimal control problems for partial differential equations. The conditions ensuring the existence of solutions to the boundary optimal control problems with L2-skew-symmetric matrix of coefficient were obtained. The notions of variational and non- variational solutions to the above problems were introduced. It was shown that the non- variational solutions to described class of optimal control problems can be attained by the solutions of the special class of fictitious optimal control problems. The optimality conditions were derived for optimal boundary control problems of parabolic equations with unbounded coefficient. The natural application of the above obtained results is the simulation of the diffusion processes in turbulence flows and in the general theory of optimal control problems for distributed systems.