  In control theory, typically a nominal model is assumed based on which an
optimal control is designed and then applied to an actual (true) system. This
gives rise to the problem of performance loss due to the mismatch between the
true model and the assumed model. A robustness problem in this context is to
show that the error due to the mismatch between a true model and an assumed
model decreases to zero as the assumed model approaches the true model. We
study this problem when the state dynamics of the system are governed by
controlled diffusion processes. In particular, we will discuss continuity and
robustness properties of finite horizon and infinite-horizon
$\alpha$-discounted/ergodic optimal control problems for a general class of
non-degenerate controlled diffusion processes, as well as for optimal control
up to an exit time. Under a general set of assumptions and a convergence
criterion on the models, we first establish that the optimal value of the
approximate model converges to the optimal value of the true model. We then
establish that the error due to mismatch that occurs by application of a
control policy, designed for an incorrectly estimated model, to a true model
decreases to zero as the incorrect model approaches the true model. We will see
that, compared to related results in the discrete-time setup, the
continuous-time theory will let us utilize the strong regularity properties of
solutions to optimality (HJB) equations, via the theory of uniformly elliptic
PDEs, to arrive at strong continuity and robustness properties.
