Control Realization of Schrödinger Bridges over Discrete-Time Nonlinear Systems via Maximum-Entropy Distribution Steering
Abstract
This paper shows how a Schrodinger bridge (SB) structure emerges from maximum-entropy (MaxEnt) distribution steering of discrete-time nonlinear systems. We also establish that the resulting bridge can be realized by an optimal control policy. We consider general state and input costs, non-Gaussian endpoint distributions, and input constraints. Assuming the existence of a feasible policy with finite objective value, we show that the state-path measure under a MaxEnt optimal control policy is the unique SB. The reference measure of this SB is induced by an optimal policy of the corresponding terminal-unconstrained problem. Moreover, we give an equivalent characterization of the optimal state-path measure using a killed reference and its normalized Feynman--Kac measure, thereby interpreting the state cost as killing. We then characterize the finite-cost feasibility by the existence of an endpoint coupling having finite Kullback--Leibler divergence from the reference endpoint distribution. For compactly supported marginals with bounded target density, we derive reachability-based sufficient conditions guaranteeing this feasibility. We further provide specializations to fully actuated control-affine and reachable linear systems.
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BibTeX
@misc{ito2026control,
title = {{Control Realization of Schrödinger Bridges over Discrete-Time Nonlinear Systems via Maximum-Entropy Distribution Steering}},
author = {Kaito Ito and Kenji Kashima},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17965},
eprint = {2610.17965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17965},
}
RIS
TY - UNPB
TI - Control Realization of Schrödinger Bridges over Discrete-Time Nonlinear Systems via Maximum-Entropy Distribution Steering
AU - Kaito Ito
AU - Kenji Kashima
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17965
UR - https://arxiv.org/abs/2610.17965
AB - This paper shows how a Schrodinger bridge (SB) structure emerges from maximum-entropy (MaxEnt) distribution steering of discrete-time nonlinear systems. We also establish that the resulting bridge can be realized by an optimal control policy. We consider general state and input costs, non-Gaussian endpoint distributions, and input constraints. Assuming the existence of a feasible policy with finite objective value, we show that the state-path measure under a MaxEnt optimal control policy is the unique SB. The reference measure of this SB is induced by an optimal policy of the corresponding terminal-unconstrained problem. Moreover, we give an equivalent characterization of the optimal state-path measure using a killed reference and its normalized Feynman--Kac measure, thereby interpreting the state cost as killing. We then characterize the finite-cost feasibility by the existence of an endpoint coupling having finite Kullback--Leibler divergence from the reference endpoint distribution. For compactly supported marginals with bounded target density, we derive reachability-based sufficient conditions guaranteeing this feasibility. We further provide specializations to fully actuated control-affine and reachable linear systems.
ER -