LQRax

LQRax is JAX-enabled continuous-time Riccati equation solver for LQR problems.

• It accelerates numerical simulation through JAX's scan mechanism;
• It allows rapid prototyping of iterative LQR (iLQR) for nonlinear control by supporting auto-differentiation on the loss function and dynamics;
• It also enables batch-based large-scale optimal control using JAX's vmap mechanism.

This repo is currently under active development.

Install

Follow the instructions to install JAX before installing this package.

To install: pip install lqrax

Usage

There are two modules: LQR and iLQR,

The LQR module solves the following time-varying LQR problem:

$$ \arg\min_{u(t)} \int_0^T \Big[ (x(t)-x_{ref}(t))^\top Q (x(t)-x_{ref}(t)) + u(t)^\top R u(t) \Big] dt $$ $$ \text{s.t. } \dot{x}(t) = A(t) x(t) + B(t) u(t), \quad x(0) = x_0 $$

The iLQR module solves a different time-varying LQR problem:

$$ \arg\min_{v(t)} \int_0^T \Big[ z(t)^\top Q z(t) + v(t)^\top R v(t) + z(t)^\top a(t) + v(t)^\top b(t) \Big] dt $$ $$ \text{s.t. } \dot{z}(t) = A(t) z(t) + B(t) v(t), \quad z(0) = 0. $$

This formulation is often used as the sub-problem for iterative linear quadratic regulator (iLQR) to calculate the steepest descent direction on the control for a general nonlinear control problem:

$$ \arg\min_{u(t)} \int_0^T l(x(t), u(t)) dt, \text{ s.t. } \dot{x}(t) = f(x(t), u(t)), $$

where the $z(t)$ and $v(t)$ are perturbations on the system's state $x(t)$ and control $u(t)$, and $A(t)$ and $B(t)$ are the linearized system dynamics $f(x(t), u(t))$ on the current system trajectory with respect to the state and control.

Examples

Example	Code	Example	Code
LQR for double integrator	[Jupyter Notebook] [Google Colab]	Nonlinear control with iLQR	[Jupyter Notebook] [Google Colab]
Multi-agent iLQGames	[Jupyter Notebook] [Google Colab]	Ergodic control	[Jupyter Notebook] [Google Colab]

Please also checkout Linear Quadratic Flow Matching that uses this package (the LQR module).

Copyright and License

The implementations contained herein are copyright (C) 2024 - 2025 by Max Muchen Sun, and are distributed under the terms of the GNU General Public License (GPL) version 3 (or later). Please see the LICENSE for more information.

If you use the package in your research, please cite this repository. You can see the citation information at the right side panel under "About". The BibTeX file is attached below:

@software{sun_lqrax_2025,
    author = {["Sun"], Max Muchen},
    license = {GPL-3.0},
    month = march,
    title = {{LQRax: JAX-enabled continuous-time LQR solver}},
    url = {https://github.com/MaxMSun/lqrax},
    version = {0.0.6},
    year = {2025}
}

Contact: msun@u.northwestern.edu