Abstract
In this paper, we investigate local asymptotic stability ensured by the addition of Gaussian white noise into dynamical systems. There are different stability notions for stochastic systems, such as asymptotic stability in probability (ASiP) and uniform almost sure asymptotic stability (UASAS). The local ASiP property is incapable of ensuring that sample paths converge to the origin with probability one, whereas the local UASAS property is capable of it. However, in general, the local UASAS property requires tight conditions. Here, we provide our notion of local almost sure asymptotic stability (local ASAS) to relax the conditions with both almost sure convergence of sample paths to the origin and the existence of bounded (weak) invariant sets. We find that the addition of Gaussian white noise always prevents the origin from being locally UASAS as long as we consider smooth Lyapunov functions; however, it is possible to make the origin locally ASAS. The result is confirmed by a simple example of elimination of unstable equilibria by deliberately adding Gaussian white noise.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
