This paper introduces a way to build controllers for nonlinear systems that are stable by design. The authors use a structured state‑space model (SSM) layer they call the L2‑Recurrent Unit (L2RU). The key feature is a “free parametrization” that forces any instance of the L2RU to have a prescribed L2 gain — a number that measures how much outputs can grow in response to inputs. Because the L2 bound is built into the layer, the controller can be trained without extra constraints to keep the closed loop stable.

What the researchers did is give a new mathematical parametrization of linear time‑invariant (LTI) blocks inside SSMs so that every choice of the layer parameters yields the same guaranteed L2 bound. They wrap these linear recurrent units with the usual nonlinear functions used in SSMs to make the full L2RU architecture. The paper shows how this construction generalizes prior work and can be implemented efficiently, for example with parallel scan algorithms that process long input sequences quickly.

How this leads to guaranteed stability is explained using standard control ideas. If a plant (the system to be controlled) has an estimated L2 gain γ̂ and the controller’s L2 gain is γ, then the small‑gain theorem says the feedback loop is stable when γ̂·γ < 1. By fixing γ for the L2RU, the designer can pick γ small enough to satisfy that inequality, which makes closed‑loop stability hold regardless of how the controller parameters are optimized. The authors also point out that the L2RU fits into a “performance‑boosting” framework that parametrizes all controllers that preserve stability around a pre‑stabilized plant, provided an explicit L2 bound is available.

Why this matters: neural network controllers are powerful but can easily destabilize the system. Existing methods to enforce stability typically require constrained optimization, projections, or expensive post‑hoc checks. The L2RU decouples stability from the optimization problem: you can tune the controller to minimize any nonlinear cost without adding stability constraints. The SSM structure also keeps computation efficient, contrasting with continuous‑time neural ODE methods and some recurrent architectures that are costly or hard to run in parallel. The paper demonstrates the idea on a formation‑control task for wheeled mobile robots, reporting that the L2RU controller ensures collision and obstacle avoidance while keeping stability and performance.