Global dynamics of Hopfield neural networks involving variable delays. (English) Zbl 0990.34036

The following system of delayed functional-differential equations used as a model describing the dynamics of Hopfield neural networks is considered \[ \dot u(t)=-Bu(t)+Ag(u(t-\tau(t)))+J,\quad t\geq 0,\qquad u(t)=\phi(t),\quad -\tau\leq t\leq 0, \] with \(u(t)=\text{col}\{u_i(t)\} \in \mathbb{R}^n\), \(B=\text{diag}\{ b_i \}\), \(A=(a_{ij})_{n\times n}\), \(\tau(t)=(\tau_{ij}(t))\), \(g(u)=\text{col}(g_i(u_i))\) with \(g(0)=0\) continuous, \(J=\text{col}\{J_i\}\), \(\varphi=\text{col}\{\varphi_i\}\). Conditions for the uniform boundedness of the solutions are given. Existence and uniqueness of an equilibrium point under general conditions are established. Further, sufficient criteria for the global asymptotic stability are derived using a technique based on properties of nonnegative matrices and matrix inequalities. In particular, sufficient criteria for the global asymptotic stability independent of the delay are obtained.


34C11 Growth and boundedness of solutions to ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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