Almost Sure Asymptotic Stabilization Problems for Deterministic Affine Systems by Adding One-dimensional Wiener Processes

Abstract

This paper shows that there is no one-dimensional Wiener process making the origins of deterministic affine systems become locally asymptotically stable with probability one. We clarify that Khasminskii's stochastic Lyapunov functions (SLFs) are not equivalent to deterministic Lyapunov functions (DLFs), and claim that Bardi and Cesaroni's almost Lyapunov functions (ALFs) are the same as DLFs with probability one. We also summarize randomization problems briefly, and explain why deterministic systems become Stratonovich-type stochastic systems by randomization with one-dimensional Wiener processes. Then, we prove that the origins of the randomized systems are not locally almost surely asymptotically stable if the original systems are not locally asymptotically stable. Further, we compare asymptotic stability with probability one ensured by global SLFs with almost sure asymptotic stability ensured by local/global ALFs via linear stochastic systems and its computer simulations.