# On the Effect of Instability on Learning Continuous-Time Linear Control Systems

arXiv: https://arxiv.org/abs/2409.11327 · HTML: https://mlsanity.com/paper/2409.11327/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 17 Sep 2024.

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

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [On the Sample Complexity of the Linear Quadratic Regulator](https://mlsanity.com/paper/1710.01688/) (2017)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Near optimal finite time identification of arbitrary linear dynamical systems](https://mlsanity.com/paper/1812.01251/) (2018)
- [Non-asymptotic Identification of LTI Systems from a Single Trajectory](https://mlsanity.com/paper/1806.05722/) (2018)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Non-asymptotic Identification of Linear Dynamical Systems Using Multiple Trajectories](https://mlsanity.com/paper/2009.00739/) (2020)
- [Logarithmic regret for episodic continuous-time linear-quadratic reinforcement learning over a finite-time horizon](https://mlsanity.com/paper/2006.15316/) (2020)
- [Exploration versus exploitation in reinforcement learning: a stochastic control approach](https://mlsanity.com/paper/1812.01552/) (2018)
- [Joint Learning of Linear Time-Invariant Dynamical Systems](https://mlsanity.com/paper/2112.10955/) (2021)
- [On Applications of Bootstrap in Continuous Space Reinforcement Learning](https://mlsanity.com/paper/1903.05803/) (2019)
- [Optimal consensus control of the Cucker-Smale model](https://mlsanity.com/paper/1802.01529/) (2018)
- [Bayesian Algorithms Learn to Stabilize Unknown Continuous-Time Systems](https://mlsanity.com/paper/2112.15094/) (2021)

## Abstract (arXiv)

We study the problem of system identification for stochastic continuous-time dynamics, based on a single finite-length state trajectory. We present a method for estimating the possibly unstable open-loop matrix by employing properly randomized control inputs. Then, we establish theoretical performance guarantees showing that the estimation error decays with trajectory length, a measure of excitability, and the signal-to-noise ratio, while it grows with dimension. Numerical illustrations that showcase the rates of learning the dynamics, will be provided as well. To perform the theoretical analysis, we develop new technical tools that are of independent interest. That includes non-asymptotic stochastic bounds for highly non-stationary martingales and generalized laws of iterated logarithms, among others.