Theory and Applications of Stochastic Processes pp 399-441 | Cite as

# Stochastic Stability

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## Abstract

The notion of stability in deterministic and stochastic systems is not the same. The solution \(\xi(t)\) of a deterministic system of differential equations
is
whenever
for some \(t_0 \leq T\). The solution \(\xi(t)\)
is said to be for any solution \(x(t)\) satisfying (11.2). If eq. (11.3) holds for all solutions of eq. (11.1), then \(\xi(t)\) is said to be

$$\dot{x}= b(x, t)$$

(11.1)

*stable*if for any positive number \(\varepsilon\) there exist two numbers, \(\delta > 0\) and \(T\), such that for any solution \(x(t)\) of (11.1)$$|x(t) -\xi(t)| < \varepsilon\ {\rm for}\ t \geq T,$$

$$|x(t_0) - \xi(t_0)| < \delta$$

(11.2)

*asymptotically stable*if it is stable and, in addition,$$\mathop{\lim}\limits_{t\to1} |x(t) - \xi(t)| = 0 $$

(11.3)

*globally stable*.## Keywords

Equilibrium Point Stability Criterion Inverted Pendulum Colored Noise Stochastic Stability
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media, LLC 2010