Halanay inequality

Halanay inequality is a comparison theorem for differential equations with delay.^[1] This inequality and its generalizations have been applied to analyze the stability of delayed differential equations, and in particular, the stability of industrial processes with dead-time^[2] and delayed neural networks.^[3][4]

Let ${\displaystyle t_{0}}$ be a real number and ${\displaystyle \tau }$ be a non-negative number. If ${\displaystyle v:[t_{0}-\tau ,\infty )\rightarrow \mathbb {R} ^{+}}$ satisfies $${\displaystyle {\frac {d}{dt}}v(t)\leq -\alpha v(t)+\beta \left[\sup _{s\in [t-\tau ,t]}v(s)\right],t\geq t_{0}}$$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are constants with ${\displaystyle \alpha >\beta >0}$, then $${\displaystyle v(t)\leq ke^{-\eta \left(t-t_{0}\right)},t\geq t_{0}}$$ where ${\displaystyle k>0}$ and ${\displaystyle \eta >0}$.

• Grönwall's inequality