Filipkovska M. Global solvability of differential-algebraic equations and mathematical modelling of the dynamics of nonlinear radio engineering circuits

The object of research is semilinear differential-algebraic equations and mathematical models of nonlinear radio engineering circuits. The purpose of research is obtaining conditions of the global solvability, the Lagrange stability and instability of semilinear differential-algebraic equations without using global Lipschitz conditions, development of the numerical method for solving them on any given period of time and study of the global dynamics of the mathematical models of nonlinear radio engineering circuits. The methods of research are the extending solution method with the use of differential inequalities and the functions of Lyapunov and La Salle type, implicit function theorems, the special block representations of a singular operator pencil and its components, the method of spectral Riesz projectors, methods for constructing difference schemes. The main theoretical results are the obtained conditions of the unique global solvability and the Lagrange stability of semilinear differential-algebraic equations (DAEs) and the developed numerical method for solving the semilinear differential-algebraic equation (DAE). The main practical results are the obtained conditions of the smooth determinate evolution and boundedness of the states for the mathematical models of nonlinear radio engineering circuits on an infinite time interval. Scientific novelty of the results. The extending solution method for an ordinary differential equation with the use of differential inequalities and the functions of Lyapunov and La Salle type has been further developed that allowed to weaken the restrictions on the nonlinear part of the equation. The new block structure of the operator coefficients of singular semilinear DAEs, which allows to reduce the original equation to the system of purely differential and purely algebraic equations, is proposed. The new theorems on the existence and the uniqueness of global solutions for an ordinary differential equation and semilinear DAEs with regular and singular characteristic pencils, including theorems, which take into account the specificity of the equations, are obtained. The theorems contains no constraints of the global Lipschitz condition type that allows to obtain the conditions for the global solvability of dynamic equations for wider classes of applied problems. The theorems on the Lagrange stability and instability of semilinear DAEs with regular and singular characteristic pencils are obtained for the first time. These theorems give the conditions of the existence and the uniqueness of the bounded global solution and the solution with finite escape time. The new numerical method for finding the solutions of the semilinear DAE on any given period of time taking into account the obtained conditions of the existence and the uniqueness of the global solution is developed. The results of the thesis have been used in the scientific research work "Analysis of evolution problems with the equations of Sobolev type" (State registration number 0111U010369, 2012-2014) and have been introduced into the educational process of V.N. Karazin Kharkiv National University (the act of introduction of 27/05/2015). Fields of application are economics, mechanics, control theory, radio engineering, robotics and other fields of science and engineering in which mathematical models with semilinear differential-algebraic equations are used.