  This paper introduces the formalism required to analyze a certain class of
stochastic control problems that involve a super diffusion as the underlying
controlled system. To establish the existence of these processes, we show that
they are weak scaling limits of controlled branching processes. By proving
their uniqueness in law, we can establish a dynamic programming principle for
our stochastic control problem. This lays the groundwork for a PDE
characterization of the associated value function, which is based on the
concept of derivations in the space of finite positive measures. We also
establish a verification theorem. To illustrate this approach, we focus on an
exponential-type value function and show how a regular solution to a
finite-dimensional HJB equation can be used to construct a smooth solution to
the HJB equation in the space of finite measures.
