Optimal covariance control for stochastic linear systems subject to integral quadratic state constraints

This work is concerned with an optimal covariance control problem for stochastic linear systems subject to quadratic state integral constraints. In particular, our objective is to design a feedback control law that will steer the covariance matrix of the (random) terminal state vector of a stochastic linear system to a designated positive semi-definite matrix while minimizing the expected value of the control effort required for this “covariance transition” or “Schrödinger bridge” subject to integral quadratic state inequality constraints. We address this problem by imbedding it into a one-parameter family of unconstrained covariance control problems that are more tractable, both analytically and computationally, than the original, constrained covariance control problem. In this way, the original problem is essentially reduced to a finite-dimensional optimal parameter selection problem, which can be addressed by means of gradient descent-type algorithms.