Invariance and asymptotic stability

The stability of the trivial solution of a nonlinear autonomous system is considered. The problem of finding the variables with respect to which the solution is asymptotically stable is formulated and solved for the case when a Lyapunov function with the semidefinite derivative is known. The ivestigation is based on the additional functions method and shows interconnection between the invariance and asymptotic stability properties of dynamical systems. The constructibility of the results obtained is demonstrated by an illustrative example.