Abstract
The paper considers a general class of neural networks possessing discontinuous neuron activations and neuron interconnection matrices belonging to the class of Mโmatrices or Hโmatrices. 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 byโproduct, it is proved that the considered neural networks, although they are described by a system of differential equations with discontinuous rightโhand side, enjoy the property of uniqueness of the solution starting at a given initial condition. The results are proved by a generalized Lyapunovโlike approach and by using tools from the theory of differential equations with discontinuous rightโhand side. At the core of the approach is a basic lemma, which holds under the assumption of Mโmatrices or Hโmatrices, and enables to study the limiting behaviour of a suitably defined distance between any pair of solutions to the neural network. Copyright ยฉ 2006 John Wiley & Sons, Ltd.