The paper is dedicated to the 80th birthday and the memory and honoring of late Professor V. Matrosov. He was an outstanding scientist, who contributed much to development of the stability and control theories as well as to development of different applications. In particular, he developed in algorithmic form a comparison method with vector Lyapunov function as rather unified and efficient method of nonlinear analysis and applications in systems dynamics and control theory, and we consider here mostly these results.