Asymptotical stability for fractional‐order Hopfield neural networks with multiple time delays. (11th May 2022) - Yao, Zichen; Yang, Zhanwen; Fu, Yongqiang; Li, Jiachen
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 ) $$ {}_0^CD_t^{α }X(t)= MX(t)+ CX\left(t-τ \right) $$, which is widely studied when | Arg ( λ M ) | > π 2 $$ \mid Arg\left(λ_M\right)\mid >\frac{π }{2} $$ . However, the stability is rarely known when α π 2 < | Arg ( λ M ) | ≤ π 2 $$ \frac{α π}{2}<\mid Arg\left(λ_M\right)\mid \le \frac{π }{2} $$ . Hence, this work is mainly devoted to the stability analysis for α π 2 < | Arg ( λ M ) | ≤ π 2 $$ \frac{α π}{2}<\mid Arg\left(λ_M\right)\mid \le \frac{π }{2} $$ . 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 $$ \mid Arg\left(λ_M\right)\mid >\frac{π }{2} $$ . 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.