2025/05/16 by Nikola Zubić, Davide Scaramuzza, Zubić, Nikola +1
Engineering · Physics and Astronomy · #Advanced Memory and Neural Computing #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems
paper · pdf · doi:10.48550/arxiv.2505.11602
openalex publication_date 2025/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.