Mixed Deterministic and Random Optimal Control of Linear Stochastic Systems with Quadratic Costs

TL;DR

This paper studies the optimal control of linear stochastic systems with quadratic costs, involving both deterministic and random controllers, and provides explicit solutions via Riccati equations for various scenarios.

Contribution

It introduces a novel framework combining deterministic and random controls for stochastic systems and derives explicit optimal feedback laws using coupled Riccati equations.

Findings

Optimal control exists under certain conditions.

Explicit feedback laws are derived for both controllers.

Solutions extend to singular and infinite horizon cases.

Abstract

In this paper, we consider the mixed optimal control of a linear stochastic system with a quadratic cost functional, with two controllers-one can choose only deterministic time functions, called the deterministic controller, while the other can choose adapted random processes, called the random controller. The optimal control is shown to exist under suitable assumptions. The optimal control is characterized via a system of fully coupled forward-backward stochastic differential equations (FB-SDEs) of mean-field type. We solve the FBSDEs via solutions of two (but decoupled) Riccati equations, and give the respective optimal feedback law for both determinis-tic and random controllers, using solutions of both Riccati equations. The optimal state satisfies a linear stochastic differential equation (SDE) of mean-field type. Both the singular and infinite time-horizonal cases are also addressed.