Advertisement

Stochastic Stability

  • Zeev Schuss
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 170)

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
$$\dot{x}= b(x, t)$$
(11.1)
is 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)
$$|x(t) -\xi(t)| < \varepsilon\ {\rm for}\ t \geq T,$$
whenever
$$|x(t_0) - \xi(t_0)| < \delta$$
(11.2)
for some \(t_0 \leq T\). The solution \(\xi(t)\) is said to be asymptotically stable if it is stable and, in addition,
$$\mathop{\lim}\limits_{t\to1} |x(t) - \xi(t)| = 0 $$
(11.3)
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 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Zeev Schuss
    • 1
  1. 1.Department of Applied MathematicsSchool of Mathematical Science Tel Aviv UniversityTel AvivIsrael

Personalised recommendations