Constructing Lyapunov Functions in Partial Stability Problems

This paper considers systems of nonlinear ordinary differential equations with a continuous right-hand side. For such systems, stability analysis of the trivial equilibrium with respect to part of the variables is performed in the context of the direct Lyapunov method. Lyapunov functions are constructed by choosing an appropriate pair of a main Lyapunov function and an auxiliary function: the latter function is used to correct the domain of the state space of the system where the former function is introduced and analyzed. Illustrative examples are provided in which such pairs are constructed for some nonlinear systems of two and three differential equations.