View source for Uniform stability

[[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

<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>

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

<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>

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>

holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095260/u09526010.png" />.

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

<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>

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" />.

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>

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.

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

<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>

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>

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.

====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>



====Comments====
The upper singular exponent is also called the Bohl exponent, cf. also [[Singular exponents|Singular exponents]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Rouché,   "Stability theory by Liapunov's direct method" , Springer  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.K. Hale,   "Ordinary differential equations" , Wiley  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.A. Coppel,   "Stability and asymptotic behavior of differential equations" , D.C. Heath  (1965)</TD></TR></table>