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

In this article, we study the finite-horizon optimal covariance steering problem for a continuous-time linear stochastic system subject to both additive and multiplicative noise. The noise can be continuous or it may contain jumps. The additive noise does not depend on the state or the control, whereas the multiplicative noise has a magnitude proportional to the current state. The cost is assumed to be quadratic in both the state and the control. The optimal control for steering the covariance from some initial to some final value is provided. Furthermore, the existence and uniqueness of the optimal control is shown. In the process, we also provide a result of independent interest regarding the maximal interval of existence of the solution of a matrix Riccati differential equation.