## On the second Lyapunov method in the stability theory of dynamical systems.(Russian)Zbl 0568.34037

Let (X,$$\rho)$$ be a complete metric space. In the paper, necessary and sufficient conditions such that one point $$p\in X$$ and the corresponding trajectory $$\Pi$$ (p,t) (respectively a set $$A\subseteq X)$$ is stable in the Lyapunov sense with respect to a set $$B\subseteq X$$, are established.
The main results are contained in the theorems 1-4. It is indicated that the set $$A\subseteq \bar B$$ is stable with respect to the set B only if there exists a neighborhood U of A and a corresponding Lyapunov function for the couple $$(A,B_ U)(B_ U=B\cap U)$$. If there exists a spherical neighborhood U of the set A and a Lyapunov function $$V: A\times B_ U\to R$$ then the set A is stable with respect to the set B in the Lyapunov sense.
Necessary and sufficient conditions for the uniform stability of the set A with respect to the set B are established, and analogous conditions for uniform asymptotic stability are established. All conditions in the theorems are expressed in terms of Lyapunov functions (defined in a suitable sense).
Reviewer: N.Luca

### MSC:

 34D20 Stability of solutions to ordinary differential equations 37-XX Dynamical systems and ergodic theory

### Keywords:

uniform asymptotic stability
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