On global robust stability of interval Hopfield neural networks with delay (Q2477244)

Differential equations with delay \[ x_i'(t)= -c_ix_i(t)+ \sum^n_{j=1} a_{ij} f_j(x_j(t))+ \sum^n_{j=1} b_{ij}f_j(x_j(t- \tau))+ u_i,\;i= 1,\dots, n,\tag{1} \] are considered, where \(u= (u_1,u_2,\dots, u_n)\) is a constant vector. The functions \(f_j(x_j)\) satisfy the following conditions: (A1) \(|f_j(\xi)|\leq M_j\), \(j= 1,\dots, n\); (A2) \(0\leq {f_j(\xi_1)- f_j(\xi_2)\over \xi_1- \xi_2}\leq L_j\), \(j= 1,\dots, n\). Therefore, the constants \(c_i\), \(a_{ij}\) and \(b_{ij}\) may be considered as intervalized as follows: \[ \underline{c_i}\leq c_i\leq\overline{c_i},\quad \underline{a_{ij}}\leq \overline{a_{ij}}\quad\text{and}\quad \underline{b_{ij}}\leq b_{ij}\leq\overline{b_{ij}},\quad i,j= 1,\dots, n. \] The present global robust stability criterion. Theorem. Under the assumptions (A1) and (A2) the equilibrium point \(x=x^*\) of (1) is globally robust stable if there are a positive diagonal matrix \(P= \text{diag}\{p_1,p_2,\dots, p_n\}\) and a positive definite matrix \(D\), such that \[ 2rI_n+ S-\| D\|_2 I_n-\| P\|^2_2\| D^{-1}\|_2(\| B^*\|_2+\| B_*\|_2)^2\quad I_n> 0, \] where \(I_n\) denotes the \(n\times n\) identity matrix, \(r= \min_i\{{p_i\underline{c_i}\over L_i}\}\), \(S= \{s_{ij}\}\) is matrix \[ s_{ij}= \begin{cases} -2p_i\overline{a_{ij}},\quad &\text{if }i= j,\\ -\widetilde a_{ij},\quad &\text{if }i\neq j,\end{cases}\;\widetilde a_{ij}= \max\{|p_i\overline{a_{ij}}+ p_j\overline{a_{ji}}|,\,|p_i\underline{a_{ij}}+ p_j \underline{a_{ji}}|\}. \] The proof is based on a Lyapunov-Krasovskii functional. The derivative of the functional is negative define, that is zero solution of the system is stable.