# Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati equations

arXiv: https://arxiv.org/abs/2011.11430 · HTML: https://mlsanity.com/paper/2011.11430/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 1 of 1)

- [A General Control-Theoretic Approach for Reinforcement Learning: Theory and Algorithms](https://mlsanity.com/paper/2406.14753/) (2024)

## Abstract (arXiv)

Sylvester, Lyapunov, and algebraic Riccati equations are the bread and butter of control theorists. They are used to compute infinite-horizon Gramians, solve optimal control problems in continuous or discrete time, and design observers. While popular numerical computing frameworks (e.g., scipy) provide efficient solvers for these equations, these solvers are still largely missing from most automatic differentiation libraries. Here, we derive the forward and reverse-mode derivatives of the solutions to all three types of equations, and showcase their application on an inverse control problem.