# Data-driven approximations of dynamical systems operators for control

## TL;DR

This paper reviews recent advances in data-driven methods for approximating dynamical systems operators, focusing on their applications in control, and discusses innovations involving sparsity, big data, and machine learning.

## Contribution

It provides a unified overview of recent developments in data-driven transport operator approximations for control, highlighting innovations in sparsity and machine learning.

## Key findings

- Enhanced low-dimensional approximations of transport operators.
- Integration of sparsity techniques for better control.
- Emerging methods leveraging big data and machine learning.

## Abstract

The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.