# State-space models through the lens of ensemble control

arXiv: https://arxiv.org/abs/2603.13587 · HTML: https://mlsanity.com/paper/2603.13587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 23 references that are in the map.
- First posted on arXiv 13 Mar 2026.

## Builds on (most cited first, 14 of 14)

- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [RoboMamba: Efficient Vision-Language-Action Model for Robotic Reasoning and Manipulation](https://mlsanity.com/paper/2406.04339/) (2024)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Theoretical Foundations of Deep Selective State-Space Models](https://mlsanity.com/paper/2402.19047/) (2024)
- [Point Mamba: A Novel Point Cloud Backbone Based on State Space Model with Octree-Based Ordering Strategy](https://mlsanity.com/paper/2403.06467/) (2024)
- [A Mean-Field Optimal Control Formulation of Deep Learning](https://mlsanity.com/paper/1807.01083/) (2018)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [An Optimal Control Approach to Deep Learning and Applications to Discrete-Weight Neural Networks](https://mlsanity.com/paper/1803.01299/) (2018)
- [Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiency](https://mlsanity.com/paper/2505.13499/) (2025)
- [Optimal Control of ensembles of dynamical systems](https://mlsanity.com/paper/2203.16485/) (2022)
- [Self-Healing Robust Neural Networks via Closed-Loop Control](https://mlsanity.com/paper/2206.12963/) (2022)
- [WSSM: Geographic-enhanced hierarchical state-space model for global station weather forecast](https://mlsanity.com/paper/2501.11238/) (2025)
- [A theoretical investigation of Brockett's ensemble optimal control problems](https://mlsanity.com/paper/1809.09904/) (2018)

## Abstract (arXiv)

State-space models (SSMs) are effective architectures for sequential modeling, but a rigorous theoretical understanding of their training dynamics is still lacking. We formulate continuous-time SSM parameter-path optimization as an ensemble optimal control problem: each input sequence generates a corresponding state trajectory in the ensemble, while the trainable parameter path forms a common control shared by all trajectories. Within this formulation, model evaluation is represented by the forward state equation and backward sensitivity propagation by the corresponding adjoint equation. We show that the Hamiltonian gradient with respect to the control variable represents the negative first-variation density of the reduced ensemble objective. Using this representation, we analyze a Bregman mirror-descent scheme for the continuous-time ensemble objective; in Euclidean geometry this reduces to functional projected gradient descent. State-adjoint and Hamiltonian-gradient stability yield two-sided relative curvature of the reduced objective and relative convexity under sufficient regularization. We further show that in the relatively convex regime the optimal-control problem admits a minimizer, which is unique under relative strong convexity. For SSMs, the explicit stability-generated defect kappastabSSM gives an O(1/k) objective rate at τ=kappastabSSM and geometric convergence to the unique minimizer for τ>kappastabSSM, under the stated stepsize condition.