Interpreting the dual Riccati equation through the LQ reproducing kernel

TL;DR

This paper offers a new interpretation of the dual Riccati equation in LQ control by linking it to regression in a reproducing kernel Hilbert space, revealing connections between control theory and machine learning.

Contribution

It introduces a novel perspective viewing LQ control as a regression problem in RKHS, providing insights into the evolution of the dual Riccati equation.

Findings

LQ control can be interpreted as regression in RKHS.

The dual Riccati equation describes the evolution of the LQ reproducing kernel.

New connections between control theory and kernel methods are established.

Abstract

In this study, we provide an interpretation of the dual differential Riccati equation of Linear-Quadratic (LQ) optimal control problems. Adopting a novel viewpoint, we show that LQ optimal control can be seen as a regression problem over the space of controlled trajectories, and that the latter has a very natural structure as a reproducing kernel Hilbert space (RKHS). The dual Riccati equation then describes the evolution of the values of the LQ reproducing kernel when the initial time changes. This unveils new connections between control theory and kernel methods, a field widely used in machine learning.