Almost sure exponential stabilization of linear stochastic systems driven by a general α-stable process via discrete-time state feedback
Abstract
Given an unstable linear scalar differential equation dx(t)=θx(t)dt, θ>0, we show that it can be almost surely exponentially stabilized by a discrete-time feedback control driven by a general α-stable process. More precisely, consider the stochastically controlled system dx(t)=θx(t)dt+bx(⌊ t/τ⌋τ)dZ(t), where Z(t) is a general α-stable process with α∈(1,2), and ⌊ t/τ⌋ denotes the integer part of t/τ. Unlike the extensively studied driven Brownian-driven case, the α-stable process has no finite second moment, so high-order moment methods are no longer applicable. By using fractional moments, characteristic functions, and moment inequalities for α-stable stochastic integrals, we prove that the stochastically controlled system is almost surely exponentially stable provided that the inequality |bσ|α(1−βsgn(b)tan2πα/2)>(θ+bμ) (−1)/Γ(α)cosαπ/2 holds, and that τ>0 is sufficiently small.
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BibTeX
@misc{zhang2026almost,
title = {{Almost sure exponential stabilization of linear stochastic systems driven by a general $α$-stable process via discrete-time state feedback}},
author = {Xuekang Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04340},
eprint = {2611.04340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04340},
}
RIS
TY - UNPB
TI - Almost sure exponential stabilization of linear stochastic systems driven by a general α-stable process via discrete-time state feedback
AU - Xuekang Zhang
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04340
UR - https://arxiv.org/abs/2611.04340
AB - Given an unstable linear scalar differential equation dx(t)=θx(t)dt, θ>0, we show that it can be almost surely exponentially stabilized by a discrete-time feedback control driven by a general α-stable process. More precisely, consider the stochastically controlled system dx(t)=θx(t)dt+bx(⌊ t/τ⌋τ)dZ(t), where Z(t) is a general α-stable process with α∈(1,2), and ⌊ t/τ⌋ denotes the integer part of t/τ. Unlike the extensively studied driven Brownian-driven case, the α-stable process has no finite second moment, so high-order moment methods are no longer applicable. By using fractional moments, characteristic functions, and moment inequalities for α-stable stochastic integrals, we prove that the stochastically controlled system is almost surely exponentially stable provided that the inequality |bσ|^α(1−βsgn(b)tan^2πα/2)>(θ+bμ) (−1)/Γ(α)cosαπ/2 holds, and that τ>0 is sufficiently small.
ER -