Optimal covariance control for discrete-time stochastic linear systems subject to constraints

This work deals with an optimal covariance control problem for stochastic discrete-time linear systems subject to mean sum constraints involving quadratic functions of the state and the control input sequences under the assumption of full state information. We show that the stochastic optimal control problem is equivalent to a deterministic nonlinear program, which, under a judicious choice of the decision variable, can be brought to a form in which its performance index is a convex, quadratic function subject to both equality and inequality quadratic constraints. The key challenge here stems from the fact that the equality constraints that result from the terminal constraints on the state covariance may not be necessarily convex. We show, however, that by employing a simple relaxation technique, the nonlinear program is associated with a convex program, which can be addressed by means of robust and efficient algorithms. Despite the fact that the solution to the relaxed convex program will not necessarily give closed-loop trajectories whose endpoints follow exactly the goal Gaussian distribution, a representative sample of such trajectories are expected to have endpoints that will be more concentrated near the origin than if there were drawn from the goal Gaussian distribution. Finally, numerical simulations that illustrate some key ideas of the paper are presented.