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M-matrices and global convergence of discontinuous neural networks. (English) Zbl 1131.92004

Summary: The paper considers a general class of neural networks possessing discontinuous neuron activations and neuron interconnection matrices belonging to the class of M-matrices or H-matrices. A number of results are established on global exponential convergence of the state and output solutions towards a unique equilibrium point. Moreover, by exploiting the presence of sliding modes, conditions are given under which convergence in finite time is guaranteed. In all cases, the exponential convergence rate, or the finite convergence time, can be quantitatively estimated on the basis of the parameters defining the neural network.

As a by-product, it is proved that the considered neural networks, although they are described by a system of differential equations with discontinuous right-hand side, enjoy the property of uniqueness of the solutions starting at a given initial condition. The results are proved by a generalized Lyapunov-like approach and by using tools from the theory of differential equations with discontinuous right-hand side. At the core of the approach is a basic lemma, which holds under the assumption of M-matrices or H-matrices, and enables to study the limiting behaviour of a suitably defined distance between any pair of solutions to the neural network.

92B20General theory of neural networks (mathematical biology)
68T05Learning and adaptive systems
15A99Miscellaneous topics in linear algebra
94C99Circuits, networks
34D20Stability of ODE