Abstract
By using Ao's decomposition for stochastic dynamical systems, a new notion of potential function has been introduced by Ao and his collabora-tors recently. We show that this potential function agrees with the generalized Lyapunov function of the deterministic part of the stochastic dynamical sys-tem. We further prove the existence of Ao's potential function in dimensions 1 and 2 via the solution theory of first-order partial differential equations. Our framework reveals the equivalence between Ao's potential function and Lyapunov function, the latter being one of the most significant central notions in dynamical systems. Using this equivalence, our existence proof can also be interpreted as the proof of existence of Lyapunov function for a general dynamical system.
Recommended Citation
H. Wang et al., "THE GENERALIZED LYAPUNOV FUNCTION as AO’S POTENTIAL FUNCTION: EXISTENCE in DIMENSIONS 1 and 2," Journal of Applied Analysis and Computation, vol. 13, no. 1, pp. 359 - 375, Jan 2023.
The definitive version is available at https://doi.org/10.11948/20220149
Department(s)
Mathematics and Statistics
Keywords and Phrases
Ao's Decomposition; Potential Function; Stochastic Dynamical System; the Generalized Lyapunov Function
International Standard Serial Number (ISSN)
2158-5644; 2156-907X
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2023, All rights reserved.
Publication Date
01 Jan 2023
