Complete Stability of Neural Networks With Nonmonotonic Piecewise Linear Activation Functions

Abstract

This brief studies the complete stability of neural networks with nonmonotonic piecewise linear activation functions. By applying the fixed-point theorem and the eigenvalue properties of the strict diagonal dominance matrix, some conditions are derived, which guarantee that such $n$ -neuron neural networks are completely stable. More precisely, the following two important results are obtained: 1) The corresponding neural networks have exactly $5^n$ equilibrium points, among which $3^n$ equilibrium points are locally exponentially stable and the others are unstable; 2) as long as the initial states are not equal to the equilibrium points of the neural networks, the corresponding solution trajectories will converge toward one of the $3^n$ locally stable equilibrium points. A numerical example is provided to illustrate the theoretical findings via computer simulations.