Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

Abstract: Deep Selective State-Space Models (SSMs), characterized by input-dependent, time-varying parameters, offer significant expressive power but pose challenges for stability analysis, especially with discontinuous gating signals. In this paper, we investigate the stability and regularity properties of continuous-time selective SSMs through the lens of passivity and Input-to-State Stability (ISS). We establish that intrinsic energy dissipation guarantees exponential forgetting of past states. Crucially, we prove that the unforced system dynamics possess an underlying minimal quadratic energy function whose defining matrix exhibits robust $\text{AUC}_{\text{loc}}$ regularity, accommodating discontinuous gating. Furthermore, assuming a universal quadratic storage function ensures passivity across all inputs, we derive parametric LMI conditions and kernel constraints that limit gating mechanisms, formalizing "irreversible forgetting" of recurrent models. Finally, we provide sufficient conditions for global ISS, linking uniform local dissipativity to overall system robustness. Our findings offer a rigorous framework for understanding and designing stable and reliable deep selective SSMs.