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Difference between revisions of "Lyapunov function"

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$$  
 
$$  
\dot{x}  ^ {i}  =  f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) ,\ \  
+
\dot{x}  ^ {i}  =  f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) ,\ \ i = 1 \dots n .
i = 1 \dots n .
 
 
$$
 
$$
  
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is called a Lyapunov function if it has the following properties:
 
is called a Lyapunov function if it has the following properties:
  
1) $ V ( x) > 0 $
+
1) $V(x) > 0$ for $x \neq x_{0}$;
for $ x \neq x _ {0} $;
 
  
2) $ V ( x _ {0} ) = 0 $;
+
2) $V(x_{0}) = 0$;
  
 
3)
 
3)
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$$  
 
$$  
 
0  \geq   
 
0  \geq   
\frac{d V ( x) }{dx}
+
\frac{d V(x)}{dx}
 
+
f ( x , t )  =  \sum_{i=1 }^ { n }  
f ( x , t )  =  \sum _ { i= } 1 ^ { n }  
 
 
 
 
\frac{\partial  V ( x  ^ {1} \dots x  ^ {n} ) }{\partial  x  ^ {i} }
 
\frac{\partial  V ( x  ^ {1} \dots x  ^ {n} ) }{\partial  x  ^ {i} }
 
 
f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) .
 
f ^ { i } ( x  ^ {1} \dots x  ^ {n} , t ) .
 
$$
 
$$
  
The function $ V ( x) $
+
The function $V(x)$ was introduced by A.M. Lyapunov (see [[#References|[1]]]).
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|Lyapunov stability]]). This lemma is the basis for one of the methods for investigating stability (the so-called second method of Lyapunov).
+
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====
 
====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>
+
<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>
====Comments====
+
</table>
For additional references see [[Lyapunov stability|Lyapunov stability]].
 

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=47728
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article