A mixed linear quadratic optimal control problem with a controlled time horizon

A mixed linear quadratic (MLQ) 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(·) enters the drift and diffusion coefficients of both state equations, and a stopping time τ, 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 an MLQ problem is equivalent to sequentially solve a random-duration linear quadratic (RLQ) problem and an optimal time (OT) 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. In case of seeking the optimal time in the family of deterministic times (even through somewhat restrictive, such seeking is still reasonable from practical standpoint), we give a more explicit characterization of optimal actions.