View source for Asymptotically-stable solution

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138101.png" /></td> </tr></table>

of the system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

with a right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138103.png" />, given for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138105.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138106.png" /></td> </tr></table>

it is defined for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138107.png" /> and if for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138108.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a0138109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381010.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381011.png" /> implies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381012.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381013.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381014.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381015.png" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   "Problème général de la stabilité du mouvement" , ''Ann. of Math. Studies'' , '''17''' , Princeton Univ. Press  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W. Hahn,   "Theorie und Anwendung der direkten Methode von Ljapunov" , Springer  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Rouche,   P. Habets,   M. Laloy,   "Stability theory by Liapunov's direct method" , Springer  (1977)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,   "Stability of motion" , Springer  (1967)  pp. 422</TD></TR></table>