### Abstract

The purpose of this paper is to review and highlight some connections between the problem of nonlinear smoothing and optimal control of the Liouville equation. The latter has been an active area of recent research interest owing to work in mean-field games and optimal transportation theory. The nonlinear smoothing problem is considered here for continuous-time Markov processes. The observation process is modeled as a nonlinear function of a hidden state with an additive Gaussian measurement noise. A variational formulation is described based upon the relative entropy formula introduced by Newton and Mitter. The resulting optimal control problem is formulated on the space of probability distributions. The Hamilton’s equation of the optimal control are related to the Zakai equation of nonlinear smoothing via the log transformation. The overall procedure is shown to generalize the classical Mortensen’s minimum energy estimator for the linear Gaussian problem.

Original language | English (US) |
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Title of host publication | Studies in Systems, Decision and Control |

Publisher | Springer |

Pages | 295-311 |

Number of pages | 17 |

DOIs | |

State | Published - 2020 |

### Publication series

Name | Studies in Systems, Decision and Control |
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Volume | 304 |

ISSN (Print) | 2198-4182 |

ISSN (Electronic) | 2198-4190 |

### Keywords

- Bayesian inference
- Duality
- Markov processes
- Nonlinear filtering
- Optimal control
- Stochastic smoothing

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Control and Systems Engineering
- Automotive Engineering
- Social Sciences (miscellaneous)
- Economics, Econometrics and Finance (miscellaneous)
- Control and Optimization
- Decision Sciences (miscellaneous)

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## Cite this

*Studies in Systems, Decision and Control*(pp. 295-311). (Studies in Systems, Decision and Control; Vol. 304). Springer. https://doi.org/10.1007/978-3-030-51264-4_12