# Optimal steering for non-Markovian Gaussian processes

## TL;DR

This paper derives a closed-form optimal control law for steering a non-Markovian Gaussian process with a finite-dimensional Markov realization to a desired terminal distribution, minimizing energy over a finite horizon.

## Contribution

It provides the first closed-form solution for finite-energy steering of a non-Markovian process with a Markov realization, advancing control of partially observable stochastic systems.

## Key findings

- Closed-form optimal control law derived.
- Applicable to non-Markovian processes with Markov realizations.
- Progress towards controlling systems with partial, noisy observations.

## Abstract

At present, the problem to steer a non-Markovian process with minimum energy between specified end-point marginal distributions remains unsolved. Herein, we consider the special case for a non-Markovian process y(t) which, however, assumes a finite-dimensional stochastic realization with a Markov state process that is fully observable. In this setting, and over a finite time horizon [0,T], we determine an optimal (least) finite-energy control law that steers the stochastic system to a final distribution that is compatible with a specified distribution for the terminal output process y(T); the solution is given in closed-form. This work provides a key step towards the important problem to steer a stochastic system based on partial observations of the state (i.e., an output process) corrupted by noise, which will be the subject of forthcoming work.