# Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

arXiv: https://arxiv.org/abs/2505.11602 · HTML: https://mlsanity.com/paper/2505.11602/

## Key facts

- Cited by 2 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 14 of its 38 references that are in the map.
- First posted on arXiv 16 May 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, 14 of 14)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [HGRN2: Gated Linear RNNs with State Expansion](https://mlsanity.com/paper/2404.07904/) (2024)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Limits of Deep Learning: Sequence Modeling through the Lens of Complexity Theory](https://mlsanity.com/paper/2405.16674/) (2024)
- [S7: Selective and Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2410.03464/) (2024)
- [SST: Multi-Scale Hybrid Mamba-Transformer Experts for Time Series Forecasting](https://mlsanity.com/paper/2404.14757/) (2024)
- [Mamba State-Space Models Are Lyapunov-Stable Learners](https://mlsanity.com/paper/2406.00209/) (2024)
- [Dissipative energy functionals of passive linear time-varying systems](https://mlsanity.com/paper/2412.16347/) (2024)

## Built on by (most cited first, 2 of 2)

- [Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations](https://mlsanity.com/paper/2603.18331/) (2026)
- [On the Expressive Power and Limitations of Multi-Layer SSMs](https://mlsanity.com/paper/2604.14501/) (2026)

## Abstract (arXiv)

Selective State-Space Models (SSMs) such as Mamba have become central to long-sequence modeling. Still, their stability is poorly understood: their state-space coefficients are modulated online by a token-dependent gating signal, making the recurrence neither linear time-invariant nor classically nonlinear. We study continuous-time selective SSMs through passivity, dissipativity, and Input-to-State Stability (ISS), explicitly separating the selection signal x(·) from the driving input u(·). We obtain four results: exponential forgetting under strict dissipativity; a canonical AUCloc quadratic storage for the frozen-selection subsystem that accommodates discontinuous gating; a parametric LMI together with universal kernel constraints and "irreversible forgetting" under universal quadratic storage; and sufficient conditions for global ISS uniformly over admissible selection schedules. We then bridge to practice by deriving a sampled block LMI for the Mamba selective-scan core, which is used as a differentiable training-time regularizer. Across seven standard time-series datasets and four prediction horizons, the regularizer reduces sampled Mamba-core LMI violations by roughly 92% in 28/28 pairs at a clean-MSE cost of less than 0.018%. It improves internal Mamba passivity and state-norm diagnostics under injected perturbations. Our results turn classical control-theoretic tools into verifiable structural and training criteria for selective SSMs, while honestly scoping which guarantees transfer to a deep selective-scan architecture.