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A function defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611401.png" /> be a fixed point of the 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/l/l061/l061140/l0611402.png" /></td> </tr></table>
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(that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611403.png" />), where the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611404.png" /> is continuous and continuously differentiable with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611405.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611406.png" /> is a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l0611408.png" />). In coordinates this system is written in the form
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A function defined as follows. Let  $  x _ {0} $
 +
be a fixed point of the 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/l/l061/l061140/l0611409.png" /></td> </tr></table>
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$$
 +
\dot{x}  = f ( x , t )
 +
$$
  
A differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l06114010.png" /> is called a Lyapunov function if it has the following properties:
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(that is,  $  f ( x _ {0} , t ) \equiv 0 $),
 +
where the mapping  $  f ( x , t ) :  U \times \mathbf R  ^ {+} \rightarrow \mathbf R  ^ {n} $
 +
is continuous and continuously differentiable with respect to  $  x $(
 +
here  $  U $
 +
is a neighbourhood of  $  x _ {0} $
 +
in  $  \mathbf R  ^ {n} $).  
 +
In coordinates this system is written in the form
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l06114011.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l06114012.png" />;
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$$
 +
\dot{x}  ^ {i}  =  f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) ,\ \ i = 1 \dots n .
 +
$$
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l06114013.png" />;
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A differentiable function  $  V ( x) : U \rightarrow \mathbf R $
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is called a Lyapunov function if it has the following properties:
  
3)
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1) $V(x) > 0$ for $x \neq x_{0}$;
  
<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/l/l061/l061140/l06114014.png" /></td> </tr></table>
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2) $V(x_{0}) = 0$;
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061140/l06114015.png" /> was introduced by A.M. Lyapunov (see [[#References|[1]]]).
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3)
 
 
Lyapunov's lemma holds: If a Lyapunov function exists, then the fixed point is Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]). This lemma is the basis for one of the methods for investigating stability (the so-called second method of Lyapunov).
 
  
====References====
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$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.M. Lyapunov,   "Stability of motion" , Acad. Press  (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.A. Barbashin,  "Lyapunov functions" , Moscow (1970)  (In Russian)</TD></TR></table>
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0 \geq    
 +
\frac{d V(x)}{dx}
 +
f ( x , t ) = \sum_{i=1 }^ { n }
 +
\frac{\partial V ( x  ^ {1} \dots x ^ {n} ) }{\partial x  ^ {i} }
 +
f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) .
 +
$$
  
 +
The function $V(x)$ was introduced by A.M. Lyapunov (see [[#References|[1]]]).
  
 +
Lyapunov's lemma holds: If a Lyapunov function exists, then the fixed point is Lyapunov stable (cf. [[Lyapunov stability]]). This lemma is the basis for one of the methods for investigating stability (the so-called second method of Lyapunov).
  
 
====Comments====
 
====Comments====
For additional references see [[Lyapunov stability|Lyapunov stability]].
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For additional references see [[Lyapunov stability]].
 +
 
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "Stability of motion" , Acad. Press  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.A. Barbashin,  "Lyapunov functions" , Moscow  (1970)  (In Russian)</TD></TR>
 +
</table>

Latest revision as of 15:32, 1 May 2023


A function defined as follows. Let $ x _ {0} $ be a fixed point of the system of differential equations

$$ \dot{x} = f ( x , t ) $$

(that is, $ f ( x _ {0} , t ) \equiv 0 $), where the mapping $ f ( x , t ) : U \times \mathbf R ^ {+} \rightarrow \mathbf R ^ {n} $ is continuous and continuously differentiable with respect to $ x $( here $ U $ is a neighbourhood of $ x _ {0} $ in $ \mathbf R ^ {n} $). In coordinates this system is written in the form

$$ \dot{x} ^ {i} = f ^ { i } ( x ^ {1} \dots x ^ {n} , t ) ,\ \ i = 1 \dots n . $$

A differentiable function $ V ( x) : U \rightarrow \mathbf R $ is called a Lyapunov function if it has the following properties:

1) $V(x) > 0$ for $x \neq x_{0}$;

2) $V(x_{0}) = 0$;

3)

$$ 0 \geq \frac{d V(x)}{dx} f ( x , t ) = \sum_{i=1 }^ { n } \frac{\partial V ( x ^ {1} \dots x ^ {n} ) }{\partial x ^ {i} } f ^ { i } ( x ^ {1} \dots x ^ {n} , t ) . $$

The function $V(x)$ was introduced by A.M. Lyapunov (see [1]).

Lyapunov's lemma holds: If a Lyapunov function exists, then the fixed point is Lyapunov stable (cf. Lyapunov stability). This lemma is the basis for one of the methods for investigating stability (the so-called second method of Lyapunov).

Comments

For additional references see Lyapunov stability.

References

[1] A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2] E.A. Barbashin, "Lyapunov functions" , Moscow (1970) (In Russian)
How to Cite This Entry:
Lyapunov function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_function&oldid=11336
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article