Minimization of output uncertainty is an important control target for stochastic systems subject to bounded random inputs. Firstly, causes that prevent realization of the minimum Shannon entropy (SE) control are examined based on the analysis of the SE definition in the continuous random variable (CRV) and on this basis, a new measure of uncertainty, which is called rational entropy (RE), is proposed, and the key properties of the RE are proved. Next, results are extended to the output stochastic distribution control (SDC) systems whose output probability density functions (PDFs) are approximated by a linear B-spline basis functions model. Then, two types of minimum RE controller with the mean constraint are given and several controller performance assessment (CPA) benchmarks for output SDC systems are presented. Finally, simulations are included to discuss the feasibility and effectiveness of the proposed measure of uncertainty and performance assessment methods.