  This paper investigates stochastic invariance for control systems through
probabilistic controlled invariant sets (PCISs). As a natural complement to
robust controlled invariant sets~(RCISs), we propose finite- and
infinite-horizon PCISs, and explore their relation to RICSs. We design
iterative algorithms to compute the PCIS within a given set. For systems with
discrete spaces, the computations of the finite- and infinite-horizon PCISs at
each iteration are based on linear programming and mixed integer linear
programming, respectively. The algorithms are computationally tractable and
terminate in a finite number of steps. For systems with continuous spaces, we
show how to discretize the spaces and prove the convergence of the
approximation when computing the finite-horizon PCISs. In addition, it is shown
that an infinite-horizon PCIS can be computed by the stochastic backward
reachable set from the RCIS contained in it. These PCIS algorithms are
applicable to practical control systems. Simulations are given to illustrate
the effectiveness of the theoretical results for motion planning.
