Theorem 3.1.
Assume that assumptions (H1), (H2), and (H3) hold. Then, the zero solution of system (2.5) is
-stable if there exist some constants
, two
matrices
, two diagonal positive definite
matrices
, a nonnegative continuous differential function
defined on
, and a constant
such that, for
and the following LMIs hold:
where
.
Proof.
Consider the Lyapunov-Krasovskii functional:
The time derivative of
along the trajectories of system (2.5) can be derived as
It follows from the assumption (3.1) that
We use the assumption (H2) and Cauchy's inequality
and get
Note that, for any
diagonal matrix
it follows that
Substituting (3.5), (3.6) and (3.7), to (3.4), we get, for
,
where
So, by assumption (3.2) and (3.8), we have
In addition, we note that
which, together with assumption (3.2) and Lemma 2.2, implies that
Thus, it yields
Hence, we can deduce that
By (3.10) and (3.14), we know that
is monotonically nonincreasing for
, which implies that
It follows from the definition of
that
where
.
It implies that
This completes the proof of Theorem 3.1.
Remark 3.2.
Theorem 3.1 provides a
-stability criterion for an impulsive differential system (2.5). It should be noted that the conditions in the theorem are dependent on the upper bound of the derivative of time-varying delay and the delay kernels
, and independent of the range of time-varying delay. Thus, it can be applied to impulsive neural networks with unbounded time-varying and continuously distributed delays.
Remark 3.3.
In [23, 24], the authors have studied
-stability for neural networks with unbounded time-varying delays and continuously distributed delays via different approaches. However, the impulsive effect is not taken into account. Hence, our developed result in this paper complements and improves those reported in [23, 24]. In particular, if we take
,
,
, then the following result can be obtained.
Corollary 3.4.
Assume that assumptions (H1), (H2) and (H3) hold. Then, the zero solution of system (2.5) is
-stable if there exist some constants
,
,
,
, two
matrices
,
, two diagonal positive definite
matrices
,
, a nonnegative continuous differential function
defined on
, and a constant
such that, for
and the following LMIs hold:
where
,
.
If we take
(
denotes a constant), then the following global bounded result can be obtained.
Corollary 3.5.
Assume that assumptions (H1), (H2), and (H3) hold. Then, the all solutions of system (2.5) have global boundedness if there exist two
matrices
,
, two diagonal positive definite
matrices
,
, such that, the following LMIs hold:
where
.
Remark 3.6.
Notice that
,
,
,
, and using the similar proof of Theorem 3.1, we can obtain the result easily.