About the Bang-Bang Principle for Controlled Affine Dynamics with Brownian Noise

We revisit the Bang-Bang principle, established for deterministic dynamics with affine control, in a stochastic setting where the dynamics are subject to Brownian motion. We show that such a principle does not generally apply in this context. However, we demonstrate that it does apply under certain conditions with deterministic controls. With stochastic controls, we obtain, under certain conditions, that Mayer's problems with an expectation criterion and a compact control set are equivalent to the same problems with controls taking values in the closed convex hull of the control set. This result is illustrated by a linearized dynamics of the SIR epidemiological model.