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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
 
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>
+
$$
 +
x ( \tau , \xi _ {0} ),\  x( \alpha , \xi _ {0} )  = \xi _ {0} ,
 +
$$
  
 
of the system
 
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>
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$$ \tag{* }
 +
 
 +
\frac{dx}{d \tau }
 +
  = f ( \tau , x )
 +
$$
  
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
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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
  
<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>
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$$
 +
x ( \tau , \xi ),\  | \xi - \xi _ {0} |  \langle  h ,\  h \rangle 0,
 +
$$
  
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
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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
  
<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>
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$$
 +
\| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \|  < \epsilon
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381013.png" /> and
+
for all $  \tau \geq  \alpha $
 +
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>
+
$$
 +
\| x ( \tau , \xi ) -x ( \tau , \xi _ {0} ) \|  \rightarrow  0
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013810/a01381015.png" />.
+
as $  \tau \rightarrow + \infty $.
  
 
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]]].
 
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]]].
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====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,  "Stability of motion" , Springer  (1967)  pp. 422</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,  "Stability of motion" , Springer  (1967)  pp. 422</TD></TR></table>

Latest revision as of 18:48, 5 April 2020


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)

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=45235
This article was adapted from an original article by Yu.S. Bogdanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article