Theorem 3.1.

Assume that assumptions (H_{1}), (H_{2}), and (H_{3}) 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 (H_{2}) 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 (H_{1}), (H_{2}) and (H_{3}) 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 (H_{1}), (H_{2}), and (H_{3}) 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.