Absztrakt: | As is known, Lyapunov's classical theorem on the asymptotic stability
of the zero solution of the non-autonomous system $\dot x=X(x,t)
(x\in R\sp k$, $t\in R\sb+)$ requires the existence of a scalar
function (Lyapunov function) V(x,t) which is, among others, decrescent
(admits an infinitely small upper bound). This means that V(x,t)$\to
0$ uniformly in $t\in R\sb+$ as $\vert x\vert \to 0$. In practice
to construct such a Lyapunov function is rather difficult. In
the main theorem of this paper this condition is replaced by
the existence of a further auxiliary vector function, which is
positive definite, decrescent and "cannot change too fast along
the solutions". The results concern partial stability properties
and are applied to the study of both partial and concerning all
variables stability properties of the equilibrium of the mathematical
plain pendulum of changing length. |