Almost sure exponential stabilization of linear stochastic systems driven by a general $α$-stable process via discrete-time state feedback

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/ττ)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driven Brownian-driven case, the $α$-stable process has no finite second moment, so high-order moment methods are no longer applicable.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) -1Γ(α)cosαπ/2 holds, and that $τ>0$ is sufficiently small.