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

In this article, 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 derive the optimal control, which entails solving two 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.