Halanay Inequalities

Abstract

In 1966 Halanay [71] studied the stability of the delay differential equation

$$\displaystyle{ x^{{\prime}}(t) = -px(t) + qx(t-\tau ),\ \ \tau> 0, }$$

and proved that if

$$\displaystyle{ f^{{\prime}}(t) \leq -\alpha f(t) +\beta \sup _{ s\in \left [t-\tau,t\right ]}f(s)\ \ for\ \ t \geq t_{0} }$$

and α > β > 0, then there exist γ > 0 and K > 0 such that

$$\displaystyle{ f(t) \leq Ke^{-\gamma (t-t_{0})}\ \ for\ \ t \geq t_{ 0}. }$$