Instability of Closed Invariant Sets of Semidynamical Systems. Method of Sign-Constant Lyapunov Functions

In the paper, a reduction principle for the instability property of a closed positively invariant set $M$ for semidynamical systems is proved. The fact that the result is untraditional is stressed by the assumption on the existence of a closed positively invariant set with respect to which the set $M$ has the attraction property. The corresponding instability theorem of the method of sign-constant Lyapunov functions is presented. The assertion thus obtained generalizes the well-known Chetaev and Krasovskii theorems for systems of ordinary differential equations, theorems on the instability with respect to some of the variables, and also the Shimanov and Hale theorems for systems with retarded argument. Illustrating examples are presented.