Halanay inequality explained

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

t _{0} be a real number and \tau be a non-negative number. If v: [t _{0} - \tau, inf ty) → R ^{+} satisfies $$\backslash frac\; v(t)\; \backslash leq-\backslash alpha\; v(t)+\backslash beta\backslash left[\backslash sup\; \_\{s\; \backslash in[t-\backslash tau,\; t]\}\; v(s)\backslash right],\; t\; \backslash geq\; t\_$$where \alpha and \beta are constants with \alpha>\beta> 0 , then $$v(t)\; \backslash leq\; k\; e^,\; t\; \backslash geq\; t\_$$where k> 0 and η > 0 .

See also

• Grönwall's inequality

Notes and References

1. Book: Halanay . Differential Equations: Stability, Oscillations, Time Lags . 1966 . Academic Press . 978-0-08-095529-2 . 378 . en.
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 . en . 45 . 14 . 266–271 . 10.3182/20120622-3-US-4021.00011.
3. Chen . Tianping . 2001 . Global exponential stability of delayed Hopfield neural networks . Neural Networks . en . 14 . 8 . 977–980 . 10.1016/S0893-6080(01)00059-4. 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 . en . 48 . 3 . 1543–1561 . 10.1007/s11063-018-9788-6 . 34828185 . 1370-4621.