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
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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
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it is defined for all and if for an arbitrary
there exists a
,
, such that
implies
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for all and
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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 |
Asymptotically-stable solution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-stable_solution&oldid=15407