  This paper considers time-inconsistent problems when control and stopping
strategies are required to be made simultaneously (called stopping control
problems by us). We first formulate the timeinconsistent stopping control
problems under general multi-dimensional controlled diffusion model and propose
a formal definition of their equilibria. We show that an admissible pair
$(\hat{u},C)$ of controlstopping policy is equilibrium if and only if the
auxiliary function associated with it solves the extended HJB system, providing
a methodology to verify or exclude equilibrium solutions. We provide several
examples to illustrate applications to mathematical finance and control theory.
For a problem whose reward function endogenously depends on the current wealth,
the equilibrium is explicitly obtained. For another model with a
non-exponential discount, we prove that any constant proportion strategy can
not be equilibrium. We further show that general non-constant equilibrium
exists and is described by singular boundary value problems. This example shows
that considering our combined problems is essentially different from
investigating them separately. In the end, we also provide a two-dimensional
example with a hyperbolic discount.
