Optimal design of networks of positive linear systems under stochastic uncertainty

In this paper, we study networks of positive linear systems subject to time-invariant and random uncertainties. We present linear matrix inequalities for checking the stability of the whole network of the systems around the origin with a prescribed probability and a decay rate. Based on this condition, we then give an efficient method, based on geometric programming, to find the optimal parameters of the probability distribution describing the uncertainty. We illustrate our results by studying the stability of a viral spreading process in the presence of random uncertainties.