Work

The cycle of papers of V.N. Nespirnyi, O.S. Afanas’eva, Т.V. Lomako is devoted to the development of the methods of the mapping theory 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 is devoted to solving problems connected with the investigation of the mappings with finite distortion, namely, developing of extremal problems method and the theory of the boundary behavior for the mapping classes with finite distortion. It is well-known that these aspects are very important in the function theory. Thus, the actuality of the given topic is clear. In particular, theorems on compactness are proved, variations are constructed and necessary conditions of extremum for new classes of regular solutions to the degenerate Beltrami equations with constraints of the integral and set-theoretic types for the complex coefficient are obtained. Also here it is obtained series of new criteria about continuous or homeomorphic extension of the given mappings to the boundary among which it should be noted generalizations of the well-known Gering-Martio theorem about homeomorphic extension to the boundary of quasiconformal mappings. The second part is related to 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, and the relationship between different properties of dynamical systems in classes of bounded and impulsive controls, as well as stability properties investigation for dynamical systems using functions with semidefinite derivative. In particular, impulsive stabilizability problem is solved for Brockett’s integrator and Artstein’s circle, a theorem on nesting of the class of impulsive controllable systems in the class of approximately controllable systems is proved. In a constructive way an existence of functions with semidefinite derivative for autonomous and nonautonomous systems. 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 with methods of the mappings theory” are presented in more than 30 papers and 20 theses of scientific conferences.