On the continuous dependence between a matrix product and its factors for finite systems of stochastic and substochastic matrices (Q1912455)

In a number of cases the functioning of a control system is determined by a system of matrices belonging to a certain class. In this situation the parameters of interest for the control system are functions of the elements of matrices that are products from the given system of matrices. In practice a case can arise in which the true values of the elements of the matrices of the original system are unknown, but can be computed with any preassigned precision. In this situation the following problem naturally arises: What conditions must a system of matrices satisfy in order that for any \(\varepsilon> 0\) we can compute the product of matrices of the original system with a guaranteed precision \(\varepsilon\)? In the present paper, this problem is solved with respect to two types of systems of matrices: a finite system of stochastic matrices, and a finite system of substochastic matrices whose sum is a stochastic matrix. It is shown that the necessary and sufficient conditions found are effectively verifiable. The results obtained can be applied to analyze stability or synthesize different stable systems of probabilistic automata, which have wide applicability in analog simulation.