  A mixed linear quadratic (MLQ, for short) optimal control problem is
considered. The controlled stochastic system consists of two diffusion
processes which are in different time horizons. There are two control actions:
a standard control action $u(\cdot)$ enters the drift and diffusion
coefficients of both state equations, and a stopping time $\tau$, a possible
later time after the first part of the state starts, at which the second part
of the state is initialized with initial condition depending on the first
state. A motivation of MLQ problem from a two-stage project management is
presented. It turns out that solving MLQ problem is equivalent to sequentially
solve a random-duration linear quadratic (RLQ, for short) problem and an
optimal time (OT, for short) problem associated with Riccati equations. In
particular, the optimal cost functional can be represented via two coupled
stochastic Riccati equations. Some optimality conditions for MLQ problem is
therefore obtained using the equivalence among MLQ, RLQ and OT problems.
