# Uniform stability

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Lyapunov stability, uniform with respect to the initial time. A solution , , of a system of differential equations is called uniformly stable if for every there is a such that for every and every solution of the system satisfying the inequality the inequality holds for all .

A Lyapunov-stable fixed point of an autonomous system of differential equations , , is uniformly stable, but, in general, a Lyapunov-stable solution need not be uniformly stable. For example, the solution , , of the equation (1)

is stable for each but is not uniformly stable for such .

Suppose one is given a linear system of differential equations (2)

where is a mapping that is summable on each interval.

In order that the solution of (2) be uniformly stable, it is necessary that the upper singular exponent 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 , and the Lyapunov characteristic exponent . For the existence of a such that the solution of any system that satisfies the conditions of the existence and uniqueness theorem for the solution of the Cauchy problem as well as the condition be uniformly stable, it is necessary and sufficient that the upper singular exponent be less than zero.

How to Cite This Entry:
Uniform stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_stability&oldid=17527
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article