On global asymptotic stability of a class of nonlinear systems arising in neural network theory
The author considers nonlinear dynamical systems of the form (1) \(\dot x= -{\mathcal D}(x)+ {\mathcal T}(x)h(x)\), where \(x= (x_1,\dots, x_n)\in \mathbb{R}^n\), \({\mathcal D}(x)= ({\mathcal D}_1(x_1),\dots, {\mathcal D}_n(x_n))\) and \(h(x)= (h_1(x_1),\dots, h_n(x_n))\) are vector fields of \(\mathbb{R}^n\) and \({\mathcal T}(x)\) is an \(n\times n\) matrix. \(\mathcal D\), \(h\) and \(\mathcal T\) are assumed to be as regular as required to guarantee existence, uniqueness and continuous dependence of solutions. Moreover, \({\mathcal D}(0)= h(0)= 0\). This class of systems includes for instance the so-called additive neural network model widely employed for real time signal processing. Using a Lyapunov approach, and under appropriate assumptions, the author proves a number of theorems on global asymptotic stability of (1), which turn out to be more general than previous results on the same subject. Finally, the author discusses applications to neural network models.
scientific article; zbMATH DE number 665731