Asymptotical stability for fractional‐order Hopfield neural networks with multiple time delays. (11th May 2022)

• Record Type: Journal Article
• Authors: Yao, Zichen; Yang, Zhanwen; Fu, Yongqiang; Li, Jiachen
• Is Part Of: Mathematical methods in the applied sciences. Volume 45:Number 16(2022)
• Page Start: 10052
• Page End: 10069
• Publication Date: 2022-05-11
• Subjects: asymptotical stability -- Caputo's fractional derivative -- Hopfield neural networks -- nonlinear equations -- time delays; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8355
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : This paper is concerned with the asymptotical stability of fractional‐order Hopfield neural networks with multiple delays. The problem is actually a generalization of stability for linear fractional‐order delayed differential equations: 0 C D t α X ( t ) = M X ( t ) + C X ( t − τ ) $$ {}_0^C{\mathrm{D}}_t^{\alpha }X(t)= MX(t)+ CX\left(t-\tau \right) $$, which is widely studied when | Arg ( λ M ) | > π 2 $$ \mid \mathrm{Arg}\left({\lambda}_M\right)\mid >\frac{\pi }{2} $$ . However, the stability is rarely known when α π 2 < | Arg ( λ M ) | ≤ π 2 $$ \frac{\alpha \pi}{2}<\mid \mathrm{Arg}\left({\lambda}_M\right)\mid \le \frac{\pi }{2} $$ . Hence, this work is mainly devoted to the stability analysis for α π 2 < | Arg ( λ M ) | ≤ π 2 $$ \frac{\alpha \pi}{2}<\mid \mathrm{Arg}\left({\lambda}_M\right)\mid \le \frac{\pi }{2} $$ . By virtue of the Laplace transform method and a decoupling technique for the characteristic equation, we propose a necessary and sufficient condition to ensure the stability, which improves the existing stability results for | Arg ( λ M ) | > π 2 $$ \mid \mathrm{Arg}\left({\lambda}_M\right)\mid >\frac{\pi }{2} $$ . Afterward, by a linearization technique, a necessary and sufficient stability condition is also presented for fractional‐order Hopfield neural networks with multiple delays. The conditions are established by delay‐independent coefficient‐type criteria. Finally, several numerical simulations are given to show the effectiveness of our results.