Controllability Analysis for Vision State Space Models: A Structural Interpretability Framework

State space models (SSMs) have emerged as a compelling alternative to transformers for visual recognition, offering linear computational complexity while maintaining competitive accuracy. However, the lack of interpretability tools designed specifically for the recurrent dynamics of these models remains a significant gap: existing saliency methods originate from convolutional or attention-based architectures and do not account for the sequential state propagation that governs information flow in SSMs. We introduce controllability analysis, a framework grounded in control theory that quantifies the structural influence of each input position on the internal state dynamics of vision SSMs. We derive two complementary indices: a Jacobian-based measure that tracks sensitivity propagation through the full sequence via an efficient $O(LN)$ backward recursion, and a Gramian-based measure that admits a closed-form, fully parallelizable solution for diagonal state matrices. Unlike gradient-based attribution methods that produce class-specific explanations, controllability indices measure intrinsic model properties that are invariant to the target class. Experiments on seven image classification benchmarks spanning diverse visual domains (microscopy, optical coherence tomography (OCT), dermatoscopy, radiography, natural images, apparel, and remote sensing) demonstrate that controllability analysis produces perfectly class-agnostic explanations (cross-class correlation of 1.0 compared to -0.45 to 0.18 for Grad-CAM across datasets) and reveals layerwise information flow patterns that vary across imaging domains. On the dataset where the model is best trained (BloodMNIST, 98% test accuracy), the proposed Jacobian index also outperforms Grad-CAM on standard insertion/deletion faithfulness (Jacobian $0.63 {\,}\pm {\,}0.02$ versus Grad-CAM $0.47 {\,}\pm {\,}0.03$ , mean ± standard deviation across three training seeds, with nonoverlapping per-seed 95% bootstrap CIs); on the lower accuracy datasets Grad-CAM remains stronger on this class-specific metric, a gap we discuss as a diagnostic of incomplete alignment between the model's internal dynamics and the classification task rather than a failure of the structural framework. A complementary internal-attention IoU metric, measuring alignment between each method's saliency and the model's own L₂-magnitude attention pattern, shows controllability outperforming Grad-CAM on six of the seven datasets. Directional decomposition of the controllability profiles further shows that influence progressively sharpens with depth, transitioning from diffuse responses in early layers to sparse, high-magnitude peaks in deep layers that align with semantically meaningful structures. The proposed framework provides a principled foundation for understanding, debugging, and improving vision SSMs across application domains.