In this study, we adopt age of information as a measure of the staleness of information, and take initial steps towards analyzing the control performance of stochastic systems with stale information. Our goals are to cast light on a fundamental limit on the information staleness that is required for a certain level of the control performance and to specify the corresponding stalest information pattern. In the asymptotic regime, such a limit asserts a critical information staleness that is required for stabilization. We achieve these goals by formulating the problem as a stochastic optimization problem and characterizing the associated optimal solutions. These solutions are in fact a control policy, which specifies the control inputs of the plant, and a queuing policy, which specifies the staleness of information at the controller.