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Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations. (English) Zbl 1103.34044

The authors consider a class of discontinuous neural networks described by a system of diffential equations of the form \[ \dot x=Bx+Tg(x)+I, \] where \(x\in{\mathbb R}^n\) is the vector of neuron state variables, \(B\) is a diagonal matrix with negative coefficients modeling the neuron selfinhibitions, \(T\) is the matrix of neuron interconnections, \(g(x)\) is the neuron activation function and \(I\in{\mathbb R}^n\) is the vector of neuron biasing inputs. The main hypothesis is that the components of \(g(x)\) are bounded, non-decreasing and piece-wise continuous functions. Then, a solution must be understood in the sense of Filippov. In this context, new results on global exponential convergence and global convergence in finite time are proved. The proofs make use of a generalized Lyapunov-like approach that could be of independent interest for proving convergence of other nonsmooth dynamical systems.

MSC:

34D23 Global stability of solutions to ordinary differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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