Abstract
In this paper, the issue of exponential synchronization of fractional-order multilayer coupled neural networks with reaction-diffusion terms is investigated by using periodically intermittent control. It deserves to mention that spatial diffusions, multilayer interactions and fractional dynamics are introduced to coupled neural networks at the same time. A novel fractional-order differential inequality is established on the basis of Caputo partial fractional operator. Moreover, to realize exponential synchronization of the underlying neural networks, some sufficient conditions are presented with the help of Lyapunov method and graph theory. Theoretical results show that the exponential convergence rate is dependent on the control gain and the order of fractional derivative. Finally, an illustrative numerical example is provided to further verify the feasibility and effectiveness of our results.















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Acknowledgements
The authors really appreciate the valuable comments of the editors and reviewers. This work was supported by Shandong Province Natural Science Foundation (Nos. ZR2018MA005, ZR2018MA020, ZR2017MA008); the Key Project of Science and Technology of Weihai (No. 2014DXGJMS08) and the Innovation Technology Funding Project in Harbin Institute of Technology (No. HIT.NSRIF.201703).
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Appendices
Appendix A
Proof
Since matrix \({\hat{M}}\) is a symmetric and positive definite matrix, there exists an orthogonal matrix \(P\in {\mathbb {R}}^{n\times n}\) and a diagonal matrix \(\Lambda =\mathrm {diag}\{\lambda _{1},\lambda _{2},\ldots , \lambda _{n}\}\in {\mathbb {R}}^{n\times n}\) such that \(P\Lambda P^{\mathrm {T}}={\hat{M}}\), in which \(\lambda _{i}>0\). Then, we can see that
Let \(\Phi (x,t)=P^{\mathrm {T}}\Psi (x,t)\), where \(\Phi (x,t)=\left( \Phi _{1}(x,t),\Phi _{2}(x,t),\ldots ,\Phi _{n}(x,t)\right) ^{\mathrm {T}}\). According to (19), one has
Noticing that \(\Lambda\) is diagonal, we derive\(\Phi ^{\mathrm {T}}(x,t)\Lambda \Phi (x,t)=\sum _{i=1}^{n}\lambda _{i}\Phi _{i}^{2}(x,t).\) That is to say for \(\forall \alpha \in (0,1],~t\ge t_{0}\)
Applying Lemma 3, due to \(\lambda _{i}>0\), in line with (20), it is obtained that
Since \(\lambda _{i}\) is the diagonal element of matrix \(\Lambda\), it is clear that
Consequently, combined with (21), for \(\forall \alpha \in (0,1],~t\ge t_{0}\), we state that
Considering \(\Phi (x,t)=P^{\mathrm {T}}\Psi (x,t)\), for \(\forall \alpha \in (0,1],~t\ge t_{0}\), we obtain
Rearranging and using \(P\Lambda P^{\mathrm {T}}={\hat{M}}\) into (22), for \(\forall \alpha \in (0,1],~t\ge t_{0}\), it is concluded that
This completes the proof. \(\square\)
Appendix B
Proof
When \(t\in T_m^c\), in view of (5), there is a nonnegative function A(t) such that
Taking Laplace transform of (23), we have
where \({\mathbb {V}}(s)\) and \({\mathbb {A}}(s)\) denote the Laplace transform of V(t) and A(t), respectively. Then, taking inverse Laplace transform of (24), we get
where \(*\) represents the convolution operator. Since \((t-T_{m})^{\alpha }\) and \(E_{\alpha ,\alpha }(-\zeta _{1}\left( t-T_{m}\right) ^{\alpha })\) are nonnegative functions, it yields that
When \(t\in T_m^r\), based on (5), there exists a nonnegative function B(t) such that
Similarly to the case of \(t\in T_m^c\), for (25), one obtain
Hence, owing to the nonnegativity of \((t-(T_{m}+\theta ))^{\alpha }\) and \(E_{\alpha ,\alpha }(\zeta _{2}\left( t-(T_{m}+\theta )\right) ^{\alpha })\), it follows that
This proof is now completed. \(\square\)
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Xu, Y., Sun, F. & Li, W. Exponential synchronization of fractional-order multilayer coupled neural networks with reaction-diffusion terms via intermittent control. Neural Comput & Applic 33, 16019–16032 (2021). https://doi.org/10.1007/s00521-021-06214-0
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DOI: https://doi.org/10.1007/s00521-021-06214-0