Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

arXiv:2505.11602v2 Announce Type: replace-cross
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 ($uequiv 0$), 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.