# L2RU: a Structured State Space Model with prescribed L2-bound

arXiv: https://arxiv.org/abs/2503.23818 · HTML: https://mlsanity.com/paper/2503.23818/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Core SSM architectures for language, one of 10 directions in State-space sequence models (S4, Mamba).
- Builds on 20 of its 27 references that are in the map.
- First posted on arXiv 31 Mar 2025.

## Placed in

- State-space sequence models (S4, Mamba) / [Core SSM architectures for language](https://mlsanity.com/ssm/core-ssm-architectures-for-language/)

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [HiPPO: Recurrent Memory with Optimal Polynomial Projections](https://mlsanity.com/paper/2008.07669/) (2020)
- [On the Parameterization and Initialization of Diagonal State Space Models](https://mlsanity.com/paper/2206.11893/) (2022)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [State Space Models as Foundation Models: A Control Theoretic Overview](https://mlsanity.com/paper/2403.16899/) (2024)
- [Deep Convolutional Networks in System Identification](https://mlsanity.com/paper/1909.01730/) (2019)
- [On Recurrent Neural Networks for learning-based control: recent results and ideas for future developments](https://mlsanity.com/paper/2111.13557/) (2021)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Distributed neural network control with dependability guarantees: a compositional port-Hamiltonian approach](https://mlsanity.com/paper/2112.09046/) (2021)
- [LMI Properties and Applications in Systems, Stability, and Control Theory](https://mlsanity.com/paper/1903.08599/) (2019)
- [Learning to Boost the Performance of Stable Nonlinear Systems](https://mlsanity.com/paper/2405.00871/) (2024)
- [Unconstrained learning of networked nonlinear systems via free parametrization of stable interconnected operators](https://mlsanity.com/paper/2311.13967/) (2023)
- [Structured state-space models are deep Wiener models](https://mlsanity.com/paper/2312.06211/) (2023)
- [Optimal distributed control with stability guarantees by training a network of neural closed-loop maps](https://mlsanity.com/paper/2404.02820/) (2024)

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

- [Controller Design for Structured State-space Models via Contraction Theory](https://mlsanity.com/paper/2604.07069/) (2026)

## Abstract (arXiv)

Structured state-space models (SSMs) have recently emerged as a powerful architecture at the intersection of machine learning and control, featuring layers composed of discrete-time linear time-invariant (LTI) systems followed by pointwise nonlinearities. These models combine the expressiveness of deep neural networks with the interpretability and inductive bias of dynamical systems, offering strong performance on long-sequence tasks with favorable computational complexity. However, their adoption in applications such as system identification and optimal control remains limited by the difficulty of enforcing stability and robustness in a principled and tractable manner. We introduce L2RU, a class of SSMs endowed with a prescribed L2-gain bound, guaranteeing input--output stability and robustness for all parameter values. The L2RU architecture is derived from free parametrizations of LTI systems satisfying an L2 constraint, enabling unconstrained optimization via standard gradient-based methods while preserving rigorous stability guarantees. Specifically, we develop two complementary parametrizations: a non-conservative formulation that provides a complete characterization of square LTI systems with a given L2-bound, and a conservative formulation that extends the approach to general (possibly non-square) systems while improving computational efficiency through a structured representation of the system matrices. Both parametrizations admit efficient initialization schemes that facilitate training long-memory models. We demonstrate the effectiveness of the proposed framework on a nonlinear system identification benchmark, where L2RU achieves improved performance and training stability compared to existing SSM architectures, highlighting its potential as a principled and robust building block for learning and control.