An upper Riemann-Stieltjes approach to stochastic design problems

Abstract

In this paper we study a class of stochastic design problems formulated in terms of general inequality conditions on expectations. These inequalities can be used to express various mean square or almost sure stabilization conditions for stochastic systems. In contrast with existing probabilistic methods that only solve such problems with a certain probability (degree of confidence), we propose a novel method that provides a full guarantee that the constructed solution truly solves the original problem. The main idea of our method is based on overapproximating the expectations by suitably constructed upper Riemann-Stieltjes sums and imposing the inequalities on these sums instead. Next to the full guarantee on the constructed solution, the method offers three other advantages. First, it applies to arbitrary probability distributions. Second, under rather mild conditions we can derive a “converse theorem” that states that if the original problem is solvable, our method will find a solution by sufficiently refining the upper Riemann-Stieltjes sums. Finally, we will show that convexity of the function used in the expectation can be exploited to obtain convex design conditions in our approach.