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]

Statement

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 ${\displaystyle \eta >0}$ .

See also

• Grönwall's inequality

References

1. Halanay (1966). Differential Equations: Stability, Oscillations, Time Lags. Academic Press. p. 378. ISBN 978-0-08-095529-2.
2. Bresch-Pietri, D.; Chauvin, J.; Petit, N. (2012). "Invoking Halanay inequality to conclude on closed-loop stability of a process with input-varying delay1". IFAC Proceedings Volumes. 45 (14): 266–271. doi:10.3182/20120622-3-US-4021.00011.
3. Chen, Tianping (2001). "Global exponential stability of delayed Hopfield neural networks". Neural Networks. 14 (8): 977–980. doi:10.1016/S0893-6080(01)00059-4. PMID 11681757.
4. Li, Hongfei; Li, Chuandong; Zhang, Wei; Xu, Jing (2018). "Global Dissipativity of Inertial Neural Networks with Proportional Delay via New Generalized Halanay Inequalities". Neural Processing Letters. 48 (3): 1543–1561. doi:10.1007/s11063-018-9788-6. ISSN 1370-4621. S2CID 34828185.