# On the Sample Complexity of Stabilizing LTI Systems on a Single Trajectory

arXiv: https://arxiv.org/abs/2202.07187 · HTML: https://mlsanity.com/paper/2202.07187/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2022.

## Built on by (most cited first, 7 of 7)

- [Learning Stabilizing Policies via an Unstable Subspace Representation](https://mlsanity.com/paper/2505.01348/) (2025)
- [Learning to Stabilize Unknown LTI Systems on a Single Trajectory under Stochastic Noise](https://mlsanity.com/paper/2406.00234/) (2024)
- [On the Hardness of Learning to Stabilize Linear Systems](https://mlsanity.com/paper/2311.11151/) (2023)
- [Canonical Bayesian Linear System Identification](https://mlsanity.com/paper/2507.11535/) (2025)
- [Model-Free Output Feedback Stabilization via Policy Gradient Methods](https://mlsanity.com/paper/2601.19284/) (2026)
- [Almost Surely √T Regret for Adaptive LQR](https://mlsanity.com/paper/2301.05537/) (2023)
- [Symplectic Inductive Bias for Data-Driven Target Reachability in Hamiltonian Systems](https://mlsanity.com/paper/2604.17213/) (2026)

## Abstract (arXiv)

Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (LTI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in n, the state dimension, which incurs a state norm that blows up exponentially in n. We propose a novel algorithm based on spectral decomposition that only needs to learn "a small part" of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our algorithm stabilizes an LTI system on a single trajectory with tildeO(k) samples, where k is the instability index of the system. This represents the first sub-linear sample complexity result for the stabilization of LTI systems under the regime when k = o(n).