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Regularized transport between singular covariance matrices

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Document pages: 8 pages

Abstract: We consider the problem of steering a linear stochastic system between twoend-point degenerate Gaussian distributions in finite time. This accounts forthose situations in which some but not all of the state entries are uncertainat the initial, t = 0, and final time, t = T . This problem entails non-trivialtechnical challenges as the singularity of terminal state-covariance causes thecontrol to grow unbounded at the final time T. Consequently, the entropicinterpolation (Schroedinger Bridge) is provided by a diffusion process which isnot finite-energy, thereby placing this case outside of most of the currenttheory. In this paper, we show that a feasible interpolation can be derived asa limiting case of earlier results for non-degenerate cases, and that it can beexpressed in closed form. Moreover, we show that such interpolation belongs tothe same reciprocal class of the uncontrolled evolution. By doing so we alsohighlight a time-symmetry of the problem, contrasting dual formulations in theforward and reverse time-directions, where in each the control grows unboundedas time approaches the end-point (in the forward and reverse time-direction,respectively).

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