Regularized Transport Between Singular Covariance Matrices

Abstract

We consider the problem of steering a linear stochastic system between two endpoint <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">degenerate</i> Gaussian distributions in finite time. This accounts for those situations in which some but not all of the state entries are uncertain at the initial, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t=0$</tex-math></inline-formula> , and final time, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t=T$</tex-math></inline-formula> . This problem entails nontrivial technical challenges, as the singularity of terminal state covariance causes the control to grow unbounded at the final time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> . Consequently, the entropic interpolation ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Schrödinger bridge</i> ) is provided by a diffusion process, which is not <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite energy</i> , thereby placing this case outside of most of the current theory. In this article, we show that a feasible interpolation can be derived as a limiting case of earlier results for nondegenerate cases, and that it can be expressed in closed form. Moreover, we show that such interpolation belongs to the same <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reciprocal class</i> of the uncontrolled evolution. By doing so, we also highlight a time symmetry of the problem, contrasting dual formulations in the forward and reverse time directions, where in each, the control grows unbounded as time approaches the endpoint (in the forward and reverse time direction, respectively).