Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

TL;DR

This paper analyzes the stability and regularity of selective deep state-space models with discontinuous gating, providing conditions for stability, passivity, and exponential forgetting, supported by theoretical proofs and simulations.

Contribution

It introduces a stability analysis framework for selective SSMs with discontinuous gating, including passivity conditions, storage function characterizations, and ISS criteria.

Findings

Exponential decay of trajectories under state-strict dissipativity.

Passive LTV subsystem with quadratic storage bounds.

Conditions for global input-to-state stability.

Abstract

Deep selective State-Space Models (SSMs), whose state-space parameters are modulated online by a selection signal, offer significant expressive power but pose challenges for stability analysis, especially under discontinuous gating. We study continuous-time selective SSMs through the lenses of passivity and Input-to-State Stability (ISS), explicitly distinguishing the selection schedule $x(\cdot)$ from the driving (port) input $u(\cdot)$. First, we show that state-strict dissipativity ($\beta>0$) together with quadratic bounds on a storage functional implies exponential decay of homogeneous trajectories ($u\equiv 0$), yielding exponential forgetting. Second, by freezing the selection ($x(t)\equiv 0$) we obtain a passive LTV input-output subsystem and prove that its minimal available storage is necessarily quadratic, $V_{a,0}(t,h)=\tfrac{1}{2}h^H Q_0(t)h,$ with $Q_0 \in \mathrm{AUC}_{\mathrm{loc}}$, accommodating discontinuities induced by gating. Third, under the strong hypothesis that a single quadratic storage certifies passivity uniformly over all admissible selection schedules, we derive a parametric LMI and universal kernel constraints on gating, formalizing an "irreversible forgetting" structure. Finally, we give sufficient conditions for global ISS with respect to the port input $u(\cdot)$, uniformly over admissible selection schedules, and we validate the main predictions in targeted simulation studies.