Asymptotically-stable solution

A solution of a differential system that is stable according to Lyapunov (cf. Lyapunov stability) and that attracts all the other solutions with sufficiently close initial values. Thus, the solution

$$x ( \tau , \xi _ {0} ),\ x( \alpha , \xi _ {0} ) = \xi _ {0} ,$$

of the system

$$\tag{* } \frac{dx}{d \tau } = f ( \tau , x )$$

with a right-hand side $f( \tau , \xi )$, given for all $\tau \geq \alpha$, $\xi \in \mathbf R ^ {n}$, and which is such that solutions of (*) exist and are unique, will be an asymptotically-stable solution if, together with all its sufficiently close solutions

$$x ( \tau , \xi ),\ | \xi - \xi _ {0} | \langle h ,\ h \rangle 0,$$

it is defined for all $\tau \geq \alpha$ and if for an arbitrary $\epsilon > 0$ there exists a $\delta$, $0 < \delta < h$, such that $| \xi - \xi _ {0} | < \delta$ implies

$$\| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \| < \epsilon$$

for all $\tau \geq \alpha$ and

$$\| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \| \rightarrow 0$$

as $\tau \rightarrow + \infty$.

The concept of an asymptotically-stable solution was introduced by A.M. Lyapunov [1]; it, together with various special types of uniform asymptotic stability, is extensively used in the theory of stability [2], [3], [4].

References

 [1] A.M. Lyapunov, "Problème général de la stabilité du mouvement" , Ann. of Math. Studies , 17 , Princeton Univ. Press (1947) [2] N.N. Krasovskii, "Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay" , Stanford Univ. Press (1963) (Translated from Russian) [3] W. Hahn, "Theorie und Anwendung der direkten Methode von Ljapunov" , Springer (1959) [4] N. Rouche, P. Habets, M. Laloy, "Stability theory by Liapunov's direct method" , Springer (1977)