Stabilization of companion form systems by mean zero noise

Kao, John; Wihstutz, Volker

doi:10.1080/17442509408833910

Cited by 10 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From the vast literature we just mention Ariaratnam and Xie [7], Arnold, Eizenberg and Wihstutz [17], Kao and Wihstutz [196], Pardoux and Wihstutz [270,271], Pinsky [278], and Pinsky and Wihstutz [279]. 9), etc.…”

Section: Furstenberg-khasminskii Formulasmentioning

confidence: 99%

Random Dynamical Systems

Arnold¹

1998

Springer Monographs in Mathematics

2,434

3,522

View full text Add to dashboard Cite

Section: Furstenberg-khasminskii Formulasmentioning

confidence: 99%

Random Dynamical Systems

Arnold¹

1998

Springer Monographs in Mathematics

2,434

3,522

View full text Add to dashboard Cite

“…Stabilization has been proved in the limit ω → ∞ using Liapunov exponent techniques (see, for example, [16]). The dependence of the stability on ω can be deduced by separating u asymptotically into fast and slow components [2]:…”

mentioning

confidence: 99%

Stochastic stabilization of cosmological photons

Dettmann

Keating

Prado

2004

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The stability of photon trajectories in models of the Universe that have constant spatial curvature is determined by the sign of the curvature: they are exponentially unstable if the curvature is negative and stable if it is positive or zero. We demonstrate that random fluctuations in the curvature provide an additional stabilizing mechanism. This mechanism is analogous to the one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean curvature is negative it is capable of stabilizing the photon trajectories; when the mean curvature is zero or positive it determines the characteristic frequency with which neighbouring trajectories oscillate about each other. In constant negative curvature models of the Universe that have compact topology, exponential instability implies chaos (e.g. mixing) in the photon dynamics. We discuss some consequences of stochastic stabilization in this context.

show abstract

“…Another approach goes back to Kao and Wihstutz [9,10], who investigate stabilization of linear systems in companion form by noise. Roughly speaking, they consider the scalar equation y (n) + a n−1 y (n−1) + · · · + a 1 y + a 0 y = 0 with real coefficients a j , which are perturbed by mean zero noise.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of linear systems by rotation

Crauel

Damm

Ilchmann

2007

Journal of Differential Equations

View full text Add to dashboard Cite

We introduce the concept of "stabilization by rotation" for deterministic linear systems with negative trace. This concept encompasses the well-known concept of "vibrational stabilization" introduced by Meerkov in the 1970s and is a deterministic version of 'stabilization by noise' for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called "gain function" (possibly a constant) which is sufficiently large. To overcome the problem of what is "sufficiently large", we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.

show abstract

Stabilization of companion form systems by mean zero noise

Cited by 10 publications

References 13 publications

Random Dynamical Systems

Random Dynamical Systems

Stochastic stabilization of cosmological photons

Stabilization of linear systems by rotation

Contact Info

Product

Resources

About