On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost

We revisit the covariance steering problem for discrete-time Gaussian linear systems with a squared Wasserstein distance terminal cost and analyze the properties of its solution in terms of existence and uniqueness. Specifically, we derive the first and second order conditions for optimality and provide analytic expressions for the gradient and the Hessian of the performance index by utilizing specialized tools from matrix calculus. Subsequently, we prove that the optimization problem always admits a global minimizer, and finally, we provide a sufficient condition for the performance index to be a strictly convex function. In particular, we show that when the terminal state covariance is lower bounded, with respect to the Lowner partial order, by the covariance matrix of the desired terminal normal distribution, then the objective function is strictly convex.