A Mixed Linear Quadratic Optimal Control Problem with a Controlled Time Horizon

TL;DR

This paper addresses a complex stochastic control problem involving two interconnected diffusion processes with different time horizons, proposing a solution via Riccati equations and establishing optimality conditions.

Contribution

It introduces a novel mixed linear quadratic control framework with a controlled time horizon, linking it to Riccati equations and providing a new approach for solving such problems.

Findings

Equivalent to solving a sequential RLQ and OT problem

Optimal cost expressed through coupled Riccati equations

Derived new optimality conditions for MLQ problem

Abstract

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.