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In this paper, we establish a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time. Numerical simulations show that the upper bound estimate for the settling time in this paper is tighter than those given in the existing fixed-time stability theorems. By designing a simple feedback controller, the fixed-time synchronization of neural networks with discrete delay is investigated based on the fixed-time stability theorem established in this paper. A numerical example is included to validate the effectiveness of the obtained theoretical results.

Neural networks can be described by differential equations [

Different from infinite-time synchronization (including asymptotic synchronization and exponential synchronization), finite-time synchronization [

The drawback of finite-time synchronization is that the settling time varies with the initial conditions of the drive-response systems. Since different initial conditions lead to different settling times, the corresponding settling times will be confusing when we consider many different initial conditions. Moreover, if the initial conditions of the studied dynamical systems are unknown beforehand, it is even impossible to estimate the settling times. Therefore, it will be desirable that finite-time synchronization can be achieved in a fixed time interval, irrespective of the initial conditions of the studied systems.

In 2012, a new concept named “fixed-time stability” was introduced by Polyakov [

As typical nonlinear systems, neural networks usually present some strange dynamic behaviors, so the fixed-time synchronization of neural networks can also be applied in many fields, such as secure communication and image encryption. Based on the fixed-time stability theorems in [

Inspired by the aforementioned discussion, in this paper, we establish a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time. Numerical simulations show that the upper bound estimate for the settling time in this paper is tighter than those given in [

Consider the following neural network with discrete delay:

In this paper, system (

The synchronization errors between systems (

Let

Throughout this paper, the following assumptions will be needed:

Based on assumptions

If

The origin of system (

The origin of system (

Suppose that

Then, the origin of system (

Suppose that

Then, the origin of system (

Suppose that

Then, the origin of system (

Suppose

for any

First, we derive a new fixed-time stability theorem, which provides a novel fixed-time stability criterion and a novel upper bound estimate formula for the settling time.

Suppose that

Then, the origin of system (

It is obvious that

Based on Lemma

Let

Therefore,

Since

This means

So we can obtain that

Two cases will be considered separately:

If

If

Then, we have

Since

Next, we investigate the fixed-time synchronization of systems (

Suppose assumptions

We choose the following Lyapunov function:

The derivative of

Since

Let

It is obvious that

Suppose assumptions

Similarly, we can prove that

Based on Lemma

Suppose assumptions

Similarly, we can prove that

Based on Lemma

Now, we consider the following controller:

Suppose assumptions

Similarly, we can prove that

Based on Lemma

To study the finite-time synchronization of systems (

Suppose assumptions

Similarly, we can prove that

Based on Lemma

Based on the fixed-time stability theorems in [

In the controllers designed in this paper, the switching item

In this section, a numerical example is provided to validate the obtained theoretical results.

Consider the following neural network:

Let

The corresponding response system is

Choose

Based on Theorem

The synchronization errors between systems (

The synchronization errors between system (

Based on Corollary

Based on Corollary

In Example

In this paper, the fixed-time synchronization of neural networks with discrete delay is investigated by utilizing the newly developed fixed-time stability theorem, which can give the settling time a tighter upper bound estimate compared with the existing fixed-time stability theorems. The settling time of fixed-time synchronization/stability is bounded by a fixed constant, irrespective of the initial conditions of the considered systems. The obtained fixed-time synchronization criteria can be verified easily, and numerical simulations are provided to demonstrate the validity of the theoretical results. In the future, we will study the application of neural networks in associative memory.

The data used to support the findings of this study are available from the corresponding author upon request.

Julian Shen and Wei Wei are co-first authors.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant nos. 61771071 and 11771196).