Finite-time boundary collision in planar linear quadratic regulator gradient flows
Abstract
Policy gradient methods for the linear quadratic regulator optimize feedback gains using a cost over an infinite horizon. When this cost is evaluated at one fixed initial state, it need not diverge near every part of the stability boundary. We study whether the Euclidean gradient flow can reach this boundary in finite optimization time. For controllable planar systems with one input and positive definite quadratic weights, an exact representation of the cost yields a necessary and sufficient condition for the existence of such a collision. The condition reduces to a scalar root calculation and includes the critical case, where a double direction root attracts an open set of stabilizing gains. For a triangular family, the criterion gives an explicit algebraic parameter region, and every trajectory either converges to the Riccati gain or collides with the stability boundary. A sharp threshold for compact cost sublevels provides a convergence certificate. Along every colliding trajectory, the stability margin and the smallest eigenvalue of the accumulated state Gramian vanish linearly, although the Gramian remains positive definite before collision. Numerical experiments examine the dependence on initialization, slow passage near a critical direction, and the effect of adding excitation in a second state direction.
Related
- (Cheap) Stochastic Policy Gradient Converges with High Probability for Linear Quadratic Regulator
- Characterizing Identifiability and Generalization for Inverse Receding-Horizon Linear-Quadratic Regulator Problems
- Continuous-Time Constrained Linear Quadratic Regulator for Switched Linear Systems
- Gradient-Only Online Convex Optimization with a Single Quadratic Leader
- Positivity-preserving scalar auxiliary variable schemes for gradient flows via a quadratic reformulation
BibTeX
@misc{liu2026finitetime,
title = {{Finite-time boundary collision in planar linear quadratic regulator gradient flows}},
author = {Kang Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00297},
eprint = {2610.00297},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00297},
}
RIS
TY - UNPB
TI - Finite-time boundary collision in planar linear quadratic regulator gradient flows
AU - Kang Liu
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00297
UR - https://arxiv.org/abs/2610.00297
AB - Policy gradient methods for the linear quadratic regulator optimize feedback gains using a cost over an infinite horizon. When this cost is evaluated at one fixed initial state, it need not diverge near every part of the stability boundary. We study whether the Euclidean gradient flow can reach this boundary in finite optimization time. For controllable planar systems with one input and positive definite quadratic weights, an exact representation of the cost yields a necessary and sufficient condition for the existence of such a collision. The condition reduces to a scalar root calculation and includes the critical case, where a double direction root attracts an open set of stabilizing gains. For a triangular family, the criterion gives an explicit algebraic parameter region, and every trajectory either converges to the Riccati gain or collides with the stability boundary. A sharp threshold for compact cost sublevels provides a convergence certificate. Along every colliding trajectory, the stability margin and the smallest eigenvalue of the accumulated state Gramian vanish linearly, although the Gramian remains positive definite before collision. Numerical experiments examine the dependence on initialization, slow passage near a critical direction, and the effect of adding excitation in a second state direction.
ER -