Balanenko I. Qualitative analysis metods of optimization problems for degenerate parabolic equations

Object of study is a class of optimal control problems for degenerate parabolic equations with control either in the coefficients of the elliptic operator or in the right-hand side of the original equation or in the initial conditions. Subject of study is optimization problems for linear degenerate parabolic equations, optimality conditions for such problems and the conditions which guarantee the existence of program optimal controls and optimal feedback controls. Methods of study are the methods of optimal control theory for differential equations with partial derivatives, functional analysis and optimization theory in normed spaces, extremal problem in weighted Sobolev spaces, and direct methods of compactness and calculus of variations. The goal of the manuscript is the study of optimization problems by the methods of calculus of variations and degenerate parabolic equation theory, finding the solvability conditions, and deriving of optimality conditions for such problems. The manuscript pioneers the existence of optimal solutions for the wide class of optimal control problems for degenerate parabolic equations with Dirichlet boundary conditions. It has been obtained and substantiated the necessary optimality conditions in the form of Pontryagin maximum principle for the degenerate parabolic equations with the distributed and started control function. It was the first time to make use of the Hardy-Poincare inequality and the dynamic programming method lof Bellman to the study of feedback control problem for degenerate parabolic equations. It was proposed the setting of control problem with a priori feedback law for such type of equations, given its geometrical generalization, and obtained the corresponding solvability consditions. Area of applications is the optimal control theory and qualitative analysis of the dynamical systems.