Robustness and Perturbation Analysis of a Class of Nonlinear Systems with Applications to Neural Networks

We study robustness properties of a large class of nonlinear systems, by addressing the following question: given a nonlinear system with specified asymptotically stable equilibria, under what conditions will a perturbed model of the system possess asymptotically stable equilibria which are close (in distance) to the asymptotically stable equilibria of the unperturbed system? In arriving at our results, we establish robustness stability results for the perturbed systems considered and we determine conditions which ensure the existence of asymptotically stable equilibria of the perturbed system which are near the asymptotically stable equilibria of the original unperturbed system. These results involve quantitative estimates of the distance between the corresponding equilibrium points of the unperturbed and perturbed systems. We apply the above results in the qualitative analysis of a large class of artificial neural networks.