LQ Covariance Control with Deadline Constraint and Frobenius Terminal Loss

This work advances the state-of-the-art for optimal control of the state covariance for a continuous-time stochastic linear time-varying control system. Specifically, we formulate and solve the fixed horizon linear quadratic covariance steering problem in continuous time, in which a time-varying linear state-feedback controller is optimized to steer the system state covariance close to a desired terminal value. The resulting optimality conditions yield a coupled matrix ODE two-point boundary value problem, which we solve via a novel matricial recursive algorithm and prove its convergence. The algorithm's analysis leverages linear fractional transforms parameterized by the state transition matrix of the associated Hamiltonian. The computational tractability of the resulting control algorithm has several applications across diverse domains, such as robotic tracking and autonomous rendezvous and docking missions for satellites. To illustrate the results, the proposed algorithm is tested and validated for a close-proximity rendezvous scenario by modeling the relative motion of a service spacecraft with a target satellite in low Earth orbit using the Clohessy–Wiltshire equations with stochastic disturbances.