Stochastic optimal control problems naturally arise in contexts such as optimal investment, optimal consumption, and economic growth. Moreover, many fundamental models in robust finance - such as G-Brownian motion, G-diffusions, or G-semimartingales - can be translated to frameworks of stochastic control. A central aspect of these problems is the connection between value functions and Hamilton-Jacobi-Bellman (HJB) equations. For controlled diffusions with sufficiently regular coefficients, this link is typically established either through the comparison method, relying on a comparison principle for discontinuous viscosity solutions, or via the verification approach, which requires the existence of classical or Sobolev solutions. In this talk, we consider a general class of controlled diffusions for which these traditional methods break down. We present a new approach that connects stochastic control problems and HJB equations by combining probabilistic and analytic techniques. Furthermore, we discuss uniqueness results, leading to stochastic representations of HJB equations in terms of control problems, and provide stability results for associated value functions.