Global Finite-Time Stability for Stochastic Impulsive Systems via Comparison Approach

GKL‐finite‐time stability via comparison principles for stochastic impulsive systems

“…Remark Theorem 4.2 is an extension of FTS reported in previous works [21, 23]. But, in previous studies [21, 23], it requires $$$ \alpha >0 $$$ and $$$ d<0 $$$. Here, in Theorem 4.2, we relaxed this condition.…”

“…Remark The FTS notions in Definitions 2.1‐ 2.2 are the extensions of the FTS notions for deterministic impulsive systems and SIS (see previous works [21, 23]).…”

“…Remark Example 5.2 Case‐(i) was studied in Gao et al [23], where it required $$$ \theta >0 $$$. In Example 5.2 Case‐(ii) with $$$ \theta <0 $$$, the jump is a destabilizing factor for FTS.…”

“…In Example 5.2 Case‐(ii) with $$$ \theta <0 $$$, the jump is a destabilizing factor for FTS. The result in Gao et al [23] fails to judge the FTS.…”

“…In Ai and Zong [22], the notion of FTS was extended to FTS in probability and finite‐time input‐to‐state stability via the It $$$ \hat{\mathrm{o}} $$$'s operator. In Gao et al [23], the FTS driven only by jump was investigated and the requirement on both stabilizing flow and jump was partly relaxed. However, most existing FTS work for impulsive systems is based on the conditions of LF $$$ V $$$ as $$$ {D}^{+}V\le -\alpha {V}^{\eta } $$$, $$$ V\left({t}^{+}\right)\le {e}^dV(t) $$$, for some $$$ \alpha >0,d\le 0,\eta \in \left(0,1\right) $$$.…”