Effects of impulse delays on L-stability of a class of nonlinear time-delay systems

Exponential stability of nonlinear switched systems with hybrid delayed impulses

“…Actually, $$$ a+b\le 1 $$$ is desirable by adjusting the value of rate efficient $$$ \beta $$$. The “pure” delayed impulses 15,20,21,24,32 and standard delayed impulses 33,34 can be considered the special case of hybrid impulses. When $$$ a=0,\kern0.3em b=1 $$$, the Lyapunov‐based condition (iii) reduces to condition $$$ V\left(t,g(x)\right)\le {e}^{-\beta }V\left({\left(t-{\tau}_n\right)}^{-},x\right),\kern0.3em t={t}_n $$$ consider in Reference 24.…”

“…Actually, a + b ≤ 1 is desirable by adjusting the value of rate efficient 𝛽. The "pure" delayed impulses 15,20,21,24,32 and standard delayed impulses 33,34 can be considered the special case of hybrid impulses. When a = 0, b = 1, the Lyapunov-based condition (iii) reduces to condition V(t, g(x)) ≤ e −𝛽 V((t − 𝜏 n ) − , x), t = t n consider in Reference 24.…”