A Control-Theoretic View of Mamba on Stability and Robustness

Selective State Space Models (SSMs) such as Mamba have emerged as efficient alternatives to Transformers, achieving linear complexity through input-dependent parameterization. However, this selectivity transforms the system from linear time-invariant (LTI) to linear parameter-varying (LPV), where individually stable matrices can produce unbounded trajectories under switching. Existing work focuses on empirical performance, leaving global stability, robustness bounds, and practical certification unresolved. This paper develops a control-theoretic framework providing the first comprehensive stability and robustness analysis for selective SSMs. We prove BIBO stability by viewing selective scans as continuous-time LTI sampling and establish two-term robustness bounds with linear growth in sequence length. For general LPV systems, we provide common quadratic Lyapunov function conditions and develop algorithms to extract certificate constants directly from trained weights. These results bridge control theory and SSM architectures, enabling formal guarantees for safety-critical deployment.

Mamba is a fast new AI architecture for processing long sequences such as text or sensor data. Its speed comes from continuously adapting its internal computation to each input. This flexibility makes the model effective but mathematically tricky, because such adaptive systems can in principle behave erratically or become unstable. We provide the comprehensive mathematical guarantees that Mamba is well-behaved. Its internal state cannot blow up, small input perturbations cause only proportional output changes that grow slowly with sequence length rather than explosively, and these guarantees can be computed directly from a trained model's weights. The analysis bridges Mamba and classical control theory, giving practitioners tools to verify these models before deploying them in settings where reliability matters, such as medical monitoring or industrial control.