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[[Lyapunov stability|Lyapunov stability]], uniform with respect to the initial time. A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952602.png" />, of a system of differential equations
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<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/u/u095/u095260/u0952603.png" /></td> </tr></table>
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is called uniformly stable if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952604.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952605.png" /> such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952606.png" /> and every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u0952607.png" /> of the system satisfying the inequality
+
[[Lyapunov stability|Lyapunov stability]], uniform with respect to the initial time. A solution $  x _ {0} ( t) $,
 +
$  t \in \mathbf R  ^ {+} $,
 +
of a system of differential equations
  
<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/u/u095/u095260/u0952608.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = f ( t, x),\ \
 +
x \in \mathbf R  ^ {n} ,
 +
$$
 +
 
 +
is called uniformly stable if for every  $  \epsilon > 0 $
 +
there is a  $  \delta > 0 $
 +
such that for every  $  t _ {0} \in \mathbf R  ^ {+} $
 +
and every solution  $  x ( t) $
 +
of the system satisfying the inequality
 +
 
 +
$$
 +
| x ( t _ {0} ) - x _ {0} ( t _ {0} ) |  < \delta ,
 +
$$
  
 
the inequality
 
the inequality
  
<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/u/u095/u095260/u0952609.png" /></td> </tr></table>
+
$$
 +
| x ( t) - x _ {0} ( t) |  < \epsilon
 +
$$
  
holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526010.png" />.
+
holds for all $  t \geq  t _ {0} $.
  
A Lyapunov-stable fixed point of an autonomous system of differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526012.png" />, is uniformly stable, but, in general, a Lyapunov-stable solution need not be uniformly stable. For example, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526014.png" />, of the equation
+
A Lyapunov-stable fixed point of an autonomous system of differential equations $  \dot{x} = f ( x) $,  
 +
$  x \in \mathbf R  ^ {n} $,  
 +
is uniformly stable, but, in general, a Lyapunov-stable solution need not be uniformly stable. For example, the solution $  x ( t) = 0 $,  
 +
$  t \in \mathbf R  ^ {+} $,  
 +
of the equation
  
<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/u/u095/u095260/u09526015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x}  = [ \sin  \mathop{\rm ln}  ( 1 + t) - \alpha ] x
 +
$$
  
is stable for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526016.png" /> but is not uniformly stable for such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526017.png" />.
+
is stable for each $  \alpha \in ( 1/ \sqrt 2 , 1) $
 +
but is not uniformly stable for such $  \alpha $.
  
 
Suppose one is given a linear system of differential equations
 
Suppose one is given a linear system of differential equations
  
<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/u/u095/u095260/u09526018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot{x}  = A ( t) x,\ \
 +
x \in \mathbf R  ^ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526019.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526020.png" /> that is summable on each interval.
+
where $  A ( \cdot ) $
 +
is a mapping $  \mathbf R  ^ {+} \rightarrow  \mathop{\rm Hom} ( \mathbf R  ^ {n} , \mathbf R  ^ {n} ) $
 +
that is summable on each interval.
  
In order that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526021.png" /> of (2) be uniformly stable, it is necessary that the upper singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526022.png" /> of (2) be less than or equal to zero (cf. also [[Singular exponents|Singular exponents]]). For example, in the case of equation (1), the upper singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526023.png" />, and the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526024.png" />. For the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526025.png" /> such that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526026.png" /> of any system
+
In order that the solution $  x = 0 $
 +
of (2) be uniformly stable, it is necessary that the upper singular exponent $  \Omega  ^ {0} ( A) $
 +
of (2) be less than or equal to zero (cf. also [[Singular exponents|Singular exponents]]). For example, in the case of equation (1), the upper singular exponent $  \Omega  ^ {0} ( A) = 1 - \alpha $,  
 +
and the [[Lyapunov characteristic exponent|Lyapunov characteristic exponent]] $  \lambda _ {1} ( A) = ( 1/ \sqrt 2 ) - \alpha $.  
 +
For the existence of a $  \delta > 0 $
 +
such that the solution $  x = 0 $
 +
of any 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/u/u095/u095260/u09526027.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = A ( t) x + g ( t, x),\ \
 +
x \in \mathbf R  ^ {n} ,
 +
$$
  
 
that satisfies the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem as well as the condition
 
that satisfies the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem as well as the condition
  
<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/u/u095/u095260/u09526028.png" /></td> </tr></table>
+
$$
 +
| g ( t, x) |  < \delta | x |
 +
$$
  
be uniformly stable, it is necessary and sufficient that the upper singular exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526029.png" /> be less than zero.
+
be uniformly stable, it is necessary and sufficient that the upper singular exponent $  \Omega  ^ {0} ( A) $
 +
be less than zero.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Persidskii,  "On stability of motion in a first approximation"  ''Mat. Sb.'' , '''40''' :  3  (1933)  pp. 284–293  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Persidskii,  "On stability of motion in a first approximation"  ''Mat. Sb.'' , '''40''' :  3  (1933)  pp. 284–293  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.P. Demidovich,  "Lectures on the mathematical theory of stability" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.L. Daletskii,  M.G. Krein,  "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:27, 6 June 2020


Lyapunov stability, uniform with respect to the initial time. A solution $ x _ {0} ( t) $, $ t \in \mathbf R ^ {+} $, of a system of differential equations

$$ \dot{x} = f ( t, x),\ \ x \in \mathbf R ^ {n} , $$

is called uniformly stable if for every $ \epsilon > 0 $ there is a $ \delta > 0 $ such that for every $ t _ {0} \in \mathbf R ^ {+} $ and every solution $ x ( t) $ of the system satisfying the inequality

$$ | x ( t _ {0} ) - x _ {0} ( t _ {0} ) | < \delta , $$

the inequality

$$ | x ( t) - x _ {0} ( t) | < \epsilon $$

holds for all $ t \geq t _ {0} $.

A Lyapunov-stable fixed point of an autonomous system of differential equations $ \dot{x} = f ( x) $, $ x \in \mathbf R ^ {n} $, is uniformly stable, but, in general, a Lyapunov-stable solution need not be uniformly stable. For example, the solution $ x ( t) = 0 $, $ t \in \mathbf R ^ {+} $, of the equation

$$ \tag{1 } \dot{x} = [ \sin \mathop{\rm ln} ( 1 + t) - \alpha ] x $$

is stable for each $ \alpha \in ( 1/ \sqrt 2 , 1) $ but is not uniformly stable for such $ \alpha $.

Suppose one is given a linear system of differential equations

$$ \tag{2 } \dot{x} = A ( t) x,\ \ x \in \mathbf R ^ {n} , $$

where $ A ( \cdot ) $ is a mapping $ \mathbf R ^ {+} \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} ) $ that is summable on each interval.

In order that the solution $ x = 0 $ of (2) be uniformly stable, it is necessary that the upper singular exponent $ \Omega ^ {0} ( A) $ of (2) be less than or equal to zero (cf. also Singular exponents). For example, in the case of equation (1), the upper singular exponent $ \Omega ^ {0} ( A) = 1 - \alpha $, and the Lyapunov characteristic exponent $ \lambda _ {1} ( A) = ( 1/ \sqrt 2 ) - \alpha $. For the existence of a $ \delta > 0 $ such that the solution $ x = 0 $ of any system

$$ \dot{x} = A ( t) x + g ( t, x),\ \ x \in \mathbf R ^ {n} , $$

that satisfies the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem as well as the condition

$$ | g ( t, x) | < \delta | x | $$

be uniformly stable, it is necessary and sufficient that the upper singular exponent $ \Omega ^ {0} ( A) $ be less than zero.

References

[1] K. Persidskii, "On stability of motion in a first approximation" Mat. Sb. , 40 : 3 (1933) pp. 284–293 (In Russian)
[2] B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[3] Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)

Comments

The upper singular exponent is also called the Bohl exponent, cf. also Singular exponents.

References

[a1] N. Rouché, "Stability theory by Liapunov's direct method" , Springer (1977)
[a2] J.K. Hale, "Ordinary differential equations" , Wiley (1969)
[a3] W.A. Coppel, "Stability and asymptotic behavior of differential equations" , D.C. Heath (1965)
How to Cite This Entry:
Uniform stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_stability&oldid=49072
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article