Optimal Covariance Steering for Continuous-Time Linear Stochastic Systems With Additive Noise

TL;DR

This paper develops a method to optimally steer the covariance of continuous-time linear stochastic systems with additive noise, including jump noise, by solving coupled matrix ODEs to minimize control energy.

Contribution

It introduces a novel covariance steering approach for systems with complex noise, establishing controllability and solving coupled matrix ODEs for optimal control.

Findings

Proves controllability of covariance in linear stochastic systems.

Derives unique solutions for coupled matrix ODEs governing optimal control.

Provides a framework for covariance steering with jump noise.

Abstract

In this paper, we study the problem of how to optimally steer the state covariance of a general continuous-time linear stochastic system over a finite time interval subject to additive noise. Optimality here means reaching a target state covariance with minimal control energy. The additive noise may include a combination of white Gaussian noise and abrupt "jump noise" that is discontinuous in time. We first establish the controllability of the state covariance for linear time-varying stochastic systems. We then turn to the derivation of the optimal control, which entails solving two dynamically coupled matrix ordinary differential equations (ODEs) with split boundary conditions. We show the existence and uniqueness of the solution to these coupled matrix ODEs, and thus those of the optimal control.