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 ($β>0$) 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.

翻译：深度选择性状态空间模型（SSMs）通过选择信号在线调制其状态空间参数，虽具备强大的表达能力，却给稳定性分析带来挑战，尤其在不连续门控条件下。本文从无源性与输入-状态稳定性（ISS）的视角研究连续时间选择性SSMs，明确区分选择调度信号$x(\cdot)$与驱动（端口）输入$u(\cdot)$。首先，我们证明状态严格耗散性（$β>0$）结合存储泛函的二次界可推出齐次轨迹（$u\equiv 0$）的指数衰减，从而产生指数遗忘效应。其次，通过冻结选择信号（$x(t)\equiv 0$），我们得到一个无源线性时变输入-输出子系统，并证明其最小可用存储必为二次型：$V_{a,0}(t,h)=\tfrac{1}{2}h^H Q_0(t)h$，其中$Q_0 \in \mathrm{AUC}_{\mathrm{loc}}$，该形式可容纳门控引发的不连续性。再次，在强假设条件下——即存在单一二次存储函数能统一证明所有允许选择调度下的无源性，我们推导出参数化线性矩阵不等式及门控的普适核约束，从而形式化了一种“不可逆遗忘”结构。最后，我们给出关于端口输入$u(\cdot)$的全局ISS的充分条件，该条件对所有允许选择调度一致成立，并通过针对性仿真研究验证了主要理论预测。