Halanay inequality

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Short description: Theorem in Mathematics

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]

Statement

Let [math]\displaystyle{ t_{0} }[/math] be a real number and [math]\displaystyle{ \tau }[/math] be a non-negative number. If [math]\displaystyle{ v: [t_{0}-\tau, \infty) \rightarrow \mathbb{R}^{+} }[/math] satisfies [math]\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} }[/math] where [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are constants with [math]\displaystyle{ \alpha\gt \beta\gt 0 }[/math], then [math]\displaystyle{ v(t) \leq k e^{-\eta\left(t-t_{0}\right)}, t \geq t_{0} }[/math] where [math]\displaystyle{ k\gt 0 }[/math] and [math]\displaystyle{ \eta\gt 0 }[/math].

See also

References