The LQR-Schr"odinger Bridge

We consider the Schr"odinger bridge problem in discrete time, where the pathwise cost is replaced by a sum of quadratic functions, taking the form of a linear quadratic regulator (LQR) cost. This cost comprises potential terms that act as attractors and kinetic terms that control the diffusion of the process. When the two boundary marginals are Gaussian, we show that the LQR-Schr"odinger bridge problem can be solved in closed form. We follow the dynamic programming principle, interpreting the Kantorovich potentials as cost-to-go functions. Under the LQR-Gaussian assumption, these potentials can be propagated exactly in a backward and forward passes, leading to a system of dual Riccati equations, well known in estimation and control. This system converges rapidly in practice. We then show that the optimal process is Markovian and compute its transition kernel in closed form as well as the Gaussian marginals. Through numerical experiments, we demonstrate that this approach can be used to construct complex, non-homogeneous Gaussian processes with acceleration and loops, given well-chosen attractive potentials. Moreover, this approach allows extending the Bures transport between Gaussian distributions to more complex geometries with negative curvature.