Fixed-time synchronization (FTS) of delayed memristor-based neural networks (MNNs) with discontinuous activations is studied in this paper. Both continuous and discontinuous activations are considered for MNNs. And the mixed delays which are closer to reality are taken into the system. Besides, two kinds of control schemes are proposed, including feedback and adaptive control strategies. Based on some lemmas, mathematical inequalities and the designed controllers, a few synchronization criteria are acquired. Moreover, the upper bound of settling time (ST) which is independent of the initial values is given. Finally, the feasibility of our theory is attested by simulation examples.

Memristor is nonlinear resistance having memory function, which represents relationship between magnetic flux and charge. Based on the connections between circuits, Chua proposed the existence of it in 1971 [

Due to the chaotic characteristics of MNNs, it appeared in many fields, such as image protection [

In engineering practice, people always expect the system to realize synchronization as soon as possible. Finite time synchronization attracted people’s attention because of its fast error convergence rate and robustness [

The connection weights of MNNs are switchable, so it belongs to a switching system, which is of great significance to the research of switching systems [

From above discussion, we find that few people have studied FTS of MNNs, especially the case with mixed delays and discontinuous activation functions. Hence, inspired by above conditions, the adaptive FTS of delayed MNNs with discontinuous activations are studied in this study. And our contributions are enumerated in following aspects. (1) Complex MNNs model with mixed delay and discontinuous activations is considered. (2) A feedback control scheme and an adaptive control algorithm are proposed for continuous and discontinuous activations, respectively, and the FTS criterion is obtained. Besides, the results are extended to finite-time synchronization. (3) The ST is not affected by the initial value of system and can be adjusted by the controller and system parameters.

The drive system of MNNs is

where _{i}_{ij}_{ij}_{i}_{ij}_{ij}, _{ij} severally and which resistances are _{ij}_{ij}, _{ij}, respectively.

There exist constants _{ij}, _{ij}, _{ij}, _{ij}, _{ij}, _{ij}, such that _{j} indicates switching threshold and _{j} > 0. Due to the solutions to MNNs are in the sense of Filippov, so set-valued mappings and differential inclusions [

Recur to set-valued mappings, it acquires

If the activation functions are continuous, one can obtain

then we set

There also exist

where the

The error is

where

If activation functions are discontinuous, we set

The error is calculated as

where

for any solution

where _{1} > 0, _{2} > 0, 0 < _{1} < 1, _{2} > 1. And the ST is calculated as

The FTS of MNNs with continuous and discontinuous activations will be studied in this section.

To realize FTS, control algorithm is given as

where

and it can obtain

where

Along the error system (5), the derivative of

By means of Assumption 1 and Lemma 3, we yield

and

One can also get

Substitute

According to Throrem 1 and Lemma 2, it has

where

where

and

where

_{v}_{v}

According to

And the nether control adaptive algorithm is designed to ensure the FTS of MNNs.

and adaptive law is

where

Take derivative of

With Assumption 3, it gets

and

Besides, one can obtain

and

So it yields

According to Theorem 2, it has

where

where

and adaptive law is

and

where _{1i} to _{4i} are positive constants and

Examples are offered in this section verifing validity and superiority of above theoretical derivation.

Consider drive system of MNNs with two neurons as

The weighted matrices are

where _{j} = 1. The delays are set as

To attain FTS, the parameters are selected as

The chaotic trajectory with control is showen in

The initial values are set as

The FTS of delayed MNNs with two kinds of activations are discussed. A feedback control algorithm is given for continuous activations and an adaptive control scheme is given for discontinuous activations. Besides, the Filippov theory is used to solove the noncontinuity of MNNs and obtain the synchronization criteria. In addition, through formula derivation, we also draw the conclusion of finite-time synchronization under the same model, so our results are more comprehensive. Finally, two simulation results to prove the feasibility of theoretical derivation. Compared with integer-order MNNs, fractional-order MNNs have more complex dynamic behavior and show stronger chaos. Therefore, the synchronization of fractional-order MNNs are our research direction in the future.

The authors would like to thanks the editor office for the deep advice to improve our work.