From Wikipedia, the free encyclopedia
Theorem in Mathematics
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
{\displaystyle t_{0}}
be a real number and
τ
{\displaystyle \tau }
be a non-negative number. If
v
:
t
0
−
τ
,
∞
)
→
R
+
{\displaystyle v:[t_{0}-\tau ,\infty )\rightarrow \mathbb {R} ^{+}}
satisfies
d
d
t
v
(
t
)
≤
−
α
v
(
t
)
+
β
sup
s
∈
t
−
τ
,
t
v
(
s
)
,
t
≥
t
0
{\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
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
are constants with
α
>
β
>
0
{\displaystyle \alpha >\beta >0}
, then
v
(
t
)
≤
k
e
−
η
(
t
−
t
0
)
,
t
≥
t
0
{\displaystyle v(t)\leq ke^{-\eta \left(t-t_{0}\right)},t\geq t_{0}}
where
k
>
0
{\displaystyle k>0}
and
η
>
0
{\displaystyle \eta >0}
.
See also
References
^ Halanay (1966).
Differential Equations: Stability, Oscillations, Time Lags . Academic Press. p. 378.
ISBN
978-0-08-095529-2 .
^ 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 .
^ 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 .
^ 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 .