Theorem 3.1.

Assume that ()–() hold, then the equilibrium point of the system (2.1) can be exponentially stabilized by impulses if one of the following conditions hold.

()

() and where

Proof.

First, we consider the following positive definite Lyapunov functional:

Then we can compute that

The time derivative of along the trajectories of system (2.5) is obtained as

Next we will consider conditions () and (), respectively.

Case 1.

If holds, that is, , then by (3.3) and (3.4), we get

which implies that the equilibrium point of the system (2.1) is exponentially stable without impulses. So the conclusion of Theorem 3.1 holds obviously.

Case 2.

If holds, then there exist and such that

Then one may choose a sequence such that and define

It is obvious that since (3.6) holds.

For any , let

For any we can prove that for each solution of system (2.5) through , implies that

First, for , by (3.4), we have

Then considering (3.3) and the choice of , we get

So we obtain

By the fact that , we get

which, together with (3.6) and (3.7), yields

that is,

By following the similar inductive arguments as before, we derive that

This completes our proof of Case 2.

The proof of Theorem 3.1 is complete.

Corollary 3.2.

Assume that , hold, then the equilibrium point of system (2.1) is exponentially stable if the following condition holds:

Corollary 3.3.

Assume that conditions in Theorem 3.1 hold, then the equilibrium point of the system (2.1) can be exponentially stabilized by periodic impulses.

Proof.

In fact, we need only to choose the sequence such that and define

As a special case of system (2.1), we consider the following neural network model:

we can obtain theorem as follows.

Theorem 3.4.

Assume that hold, then the equilibrium point of the system (3.19) can be exponentially stabilized by impulses if one of the following conditions holds

.

and where

Proof.

In fact, we need only to mention a few points since the rest is the same as in the proof of Theorem 3.1. First, instead of (3.4) we can get that

Second, instead of (3.6) and (3.7) we choose constants and such that

Then one may choose a sequence such that and define

Corollary 3.5.

Assume that conditions in Theorem 3.4 hold, then the equilibrium point of the system (3.19) can be exponentially stabilized by periodic impulses.

Proof.

Here we need only to choose the sequence such that . Let