# Uniform 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

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

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.

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