Английская Википедия:Halanay inequality

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Шаблон:Short descriptionHalanay inequality is a comparison theorem for differential equations with delay. 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 and delayed neural networks.

Statement

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

See also

• Grönwall's inequality

References

Шаблон:Math-stub