Regularized Transport Between Singular Covariance Matrices

We consider the problem of steering a linear stochastic system between two endpoint degenerate Gaussian distributions in finite time. This accounts for those situations which some but not all state entries are uncertain at initial, $t=0$ , and final time, notation="LaTeX">$t=T$ . entails nontrivial technical challenges, as singularity terminal covariance causes control to grow unbounded time notation="LaTeX">$T$ Consequently, entropic interpolation ( xmlns:xlink="http://www.w3.org/1999/xlink">Schrödinger bridge ) is provided by diffusion process, xmlns:xlink="http://www.w3.org/1999/xlink">finite energy thereby placing this case outside most current theory. In article, we show that feasible can be derived limiting earlier results nondegenerate cases, it expressed closed form. Moreover, such belongs same xmlns:xlink="http://www.w3.org/1999/xlink">reciprocal class uncontrolled evolution. By doing so, also highlight symmetry problem, contrasting dual formulations forward reverse directions, where each, grows approaches (in direction, respectively).