Work

The cycle of papers of V.N. Nespirnyi, O.S. Afanas’eva, Т.V. Lomako and O.A. Rudakova is devoted to the further development of the methods of the mapping theory and averaging methods and to applications of these methods to the analysis of the existence of solutions for partial differential equations, in particular, for the Beltrami equations, and also for solving stabilization problems of controllable dynamic systems. The first part of the presented cycle is devoted to the development of the extremal problems theory and the theory of the boundary behavior for the mapping classes with finite distortion, in particular for some classes of solutions of the Beltrami equations. Thus, compactness theorems are proved and necessary conditions for an extremum in new classes of solutions of the degenerate Beltrami equations are obtained; some new criteria of continuous and homeomorphic extension to the boundary for some classes of mappings of finite distortion are proved. The subject of the research in the second part of the cycle is the questions of homogenization of boundary value and variational problems with degenerations in domains of composite structure (strongly perforated domains) and the questions of Г-convergence for functionals. In this part, conditions of Г-compactness and Г-convergence for sequences of certain integral functionals, as well as the conditions of convergence for solutions of some variational problems for the given functionals are established. The third part is related to the investigation of qualitative properties of control systems, stabilization of dynamical systems by discontinuous controls, solving control and stabilization problems in the class of impulsive controls, as well as stability properties investigation for dynamical systems using functions with semidefinite derivative. In particular, impulsive stabilizability problem is solved for a wide class of control systems, an existence of functions with semidefinite derivative for autonomous and nonautonomous systems is proved in a constructive way. The presented cycle of papers is a completed work in which the new scientific results are obtained. They are essential for the development of the theory of mappings with finite distortion, the control and stability theory, the theory of differential equations of elliptic types and their applications. Scientific results of the work “The investigations of the qualitative properties of the dynamic systems and partial differential equations using methods of mappings theory and calculus of variations” are presented in more than 30 papers and 40 theses of scientific conferences.