  The stability of stochastic Model Predictive Control (MPC) subject to
additive disturbances is often demonstrated in the literature by constructing
Lyapunov-like inequalities that ensure closed-loop performance bounds and
boundedness of the state, but tight ultimate bounds for the state and
non-conservative performance bounds are typically not determined. In this work
we use an input-to-state stability property to find conditions that imply
convergence with probability 1 of a disturbed nonlinear system to a minimal
robust positively invariant set. We discuss implications for the convergence of
the state and control laws of stochastic MPC formulations, and we prove
convergence results for several existing stochastic MPC formulations for linear
and nonlinear systems.
