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.

References

[1]	K. Persidskii, "On stability of motion in a first approximation" Mat. Sb. , 40 : 3 (1933) pp. 284–293 (In Russian)
[2]	B.P. Demidovich, "Lectures on the mathematical theory of stability" , Moscow (1967) (In Russian)
[3]	Yu.L. Daletskii, M.G. Krein, "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc. (1974) (Translated from Russian)

Comments

The upper singular exponent is also called the Bohl exponent, cf. also Singular exponents.

References

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