# 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

of the system

(*) |

with a right-hand side , given for all , , 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

it is defined for all and if for an arbitrary there exists a , , such that implies

for all and

as .

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) |

#### Comments

#### References

[a1] | W. Hahn, "Stability of motion" , Springer (1967) pp. 422 |

**How to Cite This Entry:**

Asymptotically-stable solution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-stable_solution&oldid=15407