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Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

Nikola Zubić, Davide Scaramuzza

TL;DR

This paper analyzes the stability and regularity of continuous-time selective State-Space Models (SSMs) with input-dependent gating by applying energy-based methods from passivity and ISS theory. It shows that intrinsic energy dissipation guarantees exponential forgetting of past states, and that the unforced dynamics admit a minimal quadratic energy function with a matrix Q_0(t) in the locally absolutely upper semicontinuous class, robust to discontinuous gating. When a universal quadratic storage function exists, Q(t) must be AUC_loc with non-increasing rank, leading to parametric LMIs and a universal kernel constraint that gates the gating mechanism to avoid observable energy-less directions, formalizing irreversible forgetting. The work further establishes global ISS under a uniform dissipativity condition, tying a common quadratic Lyapunov function to robust stability against bounded inputs. Collectively, these results provide a principled control-theoretic framework for designing and analyzing stable, reliable deep selective SSMs in the presence of discontinuous gating and time-varying inputs.

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.

Regularity and Stability Properties of Selective SSMs with Discontinuous Gating

TL;DR

This paper analyzes the stability and regularity of continuous-time selective State-Space Models (SSMs) with input-dependent gating by applying energy-based methods from passivity and ISS theory. It shows that intrinsic energy dissipation guarantees exponential forgetting of past states, and that the unforced dynamics admit a minimal quadratic energy function with a matrix Q_0(t) in the locally absolutely upper semicontinuous class, robust to discontinuous gating. When a universal quadratic storage function exists, Q(t) must be AUC_loc with non-increasing rank, leading to parametric LMIs and a universal kernel constraint that gates the gating mechanism to avoid observable energy-less directions, formalizing irreversible forgetting. The work further establishes global ISS under a uniform dissipativity condition, tying a common quadratic Lyapunov function to robust stability against bounded inputs. Collectively, these results provide a principled control-theoretic framework for designing and analyzing stable, reliable deep selective SSMs in the presence of discontinuous gating and time-varying inputs.

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 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.
Paper Structure (31 sections, 7 theorems, 58 equations)

This paper contains 31 sections, 7 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Selective SSMs and Passivity Framework
  3. System Definition and Notation
  4. Well-Posedness and Function Space Assumptions
  5. Passivity and Dissipativity
  6. Quadratic Storage Functions and AUC Regularity
  7. Fundamental Stability and Structural Properties from Passivity
  8. Exponential Decay from Strict Dissipativity
  9. Inherent Quadratic Structure and Regularity in Unforced Dynamics
  10. Constraints Imposed by Universal Quadratic Passivity on Gating Mechanisms
  11. Regularity and Rank Monotonicity of Universal Quadratic Storage
  12. Parametric LMI and Kernel Constraints on Gating
  13. Irreversible Forgetting and Energy Consistency in Kernel Subspace
  14. Robust Stability under Input-Driven Dynamics
  15. Global Input-to-State Stability from Uniform Dissipativity
  16. ...and 16 more sections

Key Result

Theorem 3.1

Consider the continuous‐time selective state‐space model defined in Eq. eq:mamba_ssm_prelim. Suppose there exists a storage functional $V: [0, \infty) \times \mathbb{C}^N \to \mathbb{R}_{\ge 0}$ satisfying: Then, for the unforced system (i.e., when $x(t) \equiv 0$ for $t \ge 0$), the state exhibits exponential decay: there exist constants $C \ge 1$ and $\gamma > 0$ such that for any initial state

Theorems & Definitions (17)

  • Definition 2.1: Storage Function
  • Definition 2.2: Passivity
  • Definition 2.3: Strict Dissipativity ($\beta$-Strict Passivity)
  • Theorem 3.1: Exponential Decay from Strict Dissipativity
  • proof : Proof of Theorem \ref{['thm:decay_from_V']}
  • Lemma 3.2: Strict Passivity of the Minimal Available Storage Function
  • proof : Proof of Lemma \ref{['lemma:strict_passivity_restated_sec3']}
  • Theorem 3.3: Existence and Regularity of Quadratic Storage for Unforced Dynamics Amidst Gating Switches
  • proof : Proof of Theorem \ref{['thm:existence_auc_quadratic_unforced_detailed_sec3']}
  • Theorem 4.1: Regularity and Rank of Universal Quadratic Storage
  • ...and 7 more