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 [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
- ↑ Halanay (1966) (in en). Differential Equations: Stability, Oscillations, Time Lags. Academic Press. pp. 378. ISBN 978-0-08-095529-2. https://books.google.com/books?id=YThILEV5ZPIC.
- ↑ Bresch-Pietri, D.; Chauvin, J.; Petit, N. (2012). "Invoking Halanay inequality to conclude on closed-loop stability of a process with input-varying delay1" (in en). IFAC Proceedings Volumes 45 (14): 266–271. doi:10.3182/20120622-3-US-4021.00011. https://linkinghub.elsevier.com/retrieve/pii/S1474667015347340.
- ↑ Chen, Tianping (2001). "Global exponential stability of delayed Hopfield neural networks" (in en). Neural Networks 14 (8): 977–980. doi:10.1016/S0893-6080(01)00059-4. PMID 11681757. https://linkinghub.elsevier.com/retrieve/pii/S0893608001000594.
- ↑ Li, Hongfei; Li, Chuandong; Zhang, Wei; Xu, Jing (2018). "Global Dissipativity of Inertial Neural Networks with Proportional Delay via New Generalized Halanay Inequalities" (in en). Neural Processing Letters 48 (3): 1543–1561. doi:10.1007/s11063-018-9788-6. ISSN 1370-4621. http://link.springer.com/10.1007/s11063-018-9788-6.
Original source: https://en.wikipedia.org/wiki/Halanay inequality.
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