Uniform boundedness of differential equation system solutions relating to variables with partially controlled initial conditions

Background. Japanese mathematician T. Yoshizawa developed the theory of various types of boundedness of solutions of systems of differential equations. In the investigation of boundedness of solutions he used a method which is similar to the direct method of Lyapunov in the stability theory. Research of different types of boundedness of solutions of systems of differential equations with respect to variables and, in particular, of the uniform boundedness of solutions with respect to variables were first carried out by V. V. Rumyantsev and A. S. Oziraner. On the other hand, V. I. Vorotnikov and Yu. G Martyshenko developed a new direction in the theory of partial stability, namely, the theory of partial stability of the equilibrium position with a controlled part of the coordinates. Thus, there is an interesting idea to create in the theory of partial boundedness of solutions the methods that would be conceptually similar to those of the above-mentioned theory of the stability with respect to a part of the variables of partial equilibrium positions. The main research method in this paper is the method of Lyapunov functions. All the main theorems in this paper are stated and proved in terms of the Dini derivative. The author introduces the concept of the partial uniform boundedness of solutions of systems of differential equations with partially controlled initial conditions. The researcher obtains a criterion of the partial uniform boundedness of solutions with partially controlled initial conditions and a sufficient condition for partial total boundedness. The results obtained in this paper can be applied in research of real processes to different types of partial boundedness in the natural and technical problems.