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Halanay inequality

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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

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Let be a real number and be a non-negative number. If satisfies where and are constants with , then where and .

See also

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References

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  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.