  In stochastic control applications, typically only an ideal model (controlled
transition kernel) is assumed and the control design is based on the given
model, raising the problem of performance loss due to the mismatch between the
assumed model and the actual model. Toward this end, we study continuity
properties of discrete-time stochastic control problems with respect to system
models (i.e., controlled transition kernels) and robustness of optimal control
policies designed for incorrect models applied to the true system. We study
both fully observed and partially observed setups under an infinite horizon
discounted expected cost criterion. We show that continuity and robustness
cannot be established under weak and setwise convergences of transition kernels
in general, but that the expected induced cost is robust under total variation.
By imposing further assumptions on the measurement models and on the kernel
itself (such as continuous convergence), we show that the optimal cost can be
made continuous under weak convergence of transition kernels as well. Using
these continuity properties, we establish convergence results and error bounds
due to mismatch that occurs by the application of a control policy which is
designed for an incorrectly estimated system model to a true model, thus
establishing positive and negative results on robustness.Compared to the
existing literature, we obtain strictly refined robustness results that are
applicable even when the incorrect models can be investigated under weak
convergence and setwise convergence criteria (with respect to a true model), in
addition to the total variation criteria. These entail positive implications on
empirical learning in (data-driven) stochastic control since often system
models are learned through empirical training data where typically weak
convergence criterion applies but stronger convergence criteria do not.
