In these lectures, we will first present an overview on the foundations of stochastic control systems, and their optimization under various performance criteria. We will present a tutorial on methods utilized to arrive at optimal solutions for each of the criteria. 

Then, we will discuss approximations and robustness properties: In stochastic control, typically, an ideal model is assumed or an estimate model is learned, and the control design is based on this model, raising the problem of performance loss due to the mismatch between the assumed model and the actual model. In addition, even when a complete model is available, often computational methods dictate the use of approximate models. 

We will initially consider robustness to quantized approximations for stochastic control problems with standard Borel spaces and present conditions under which finite models obtained through quantization of the state and action sets can be used to construct approximately optimal policies. We will establish that weak continuity of the transition kernel in the state and action is sufficient for the convergence of finite approximations. Under further conditions, we will obtain explicit rates of convergence to the optimal cost of the original problem as the quantization rate increases. We will also extend our analysis to partially observed models and non-linear filtering processes as well as decentralized/distributed stochastic control where one can construct a sequence of finite models whose solutions constructively converge to the optimal cost. 

We will then investigate robustness to more general modeling errors and study the mismatch loss of optimal control policies designed for incorrect models applied to the true system, as the incorrect model approaches the true model under various criteria. 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. For partially observed models, by imposing further assumptions on the measurement channel, we show that the optimal cost can be made continuous under weak convergence of transition kernels as well. 

These entail 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. 

Throughout the lectures, many examples will be discussed.