  This paper is concerned with stability analysis and synthesis for
discrete-time linear systems with stochastic dynamics. Equivalence is first
proved for three stability notions under some key assumptions on the randomness
behind the systems. In particular, we use the assumption that the stochastic
process determining the system dynamics is independent and identically
distributed (i.i.d.) with respect to the discrete time. Then, a Lyapunov
inequality condition is derived for stability in a necessary and sufficient
sense. Although our Lyapunov inequality will involve decision variables
contained in the expectation operation, an idea is provided to solve it as a
standard linear matrix inequality; the idea also plays an important role in
state feedback synthesis based on the Lyapunov inequality. Motivating numerical
examples are further discussed as an application of our approach.
