# An Optimal Control Derivation of Nonlinear Smoothing Equations

## TL;DR

This paper explores the connection between nonlinear smoothing for continuous-time Markov processes and optimal control of the Liouville equation, using a variational approach and relating it to classical estimation methods.

## Contribution

It introduces a variational formulation linking nonlinear smoothing with optimal control of the Liouville equation, extending classical estimation techniques to nonlinear settings.

## Key findings

- Derives Hamilton's equations related to nonlinear smoothing.
- Connects the optimal control problem to the Zakai equation.
- Generalizes Mortensen's minimum energy estimator to nonlinear cases.

## 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.