Structure and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics
Abstract
Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper studies the finite-horizon continuous-time inverse LQR problem from closed-loop state--input trajectories, where both the system matrices and the quadratic cost are unknown. The finite horizon induces a time-varying optimal gain, and this endogenous excitation serves as the structural mechanism that makes joint recovery possible. We quantify this mechanism through three computable conditioning indices, which measure state richness, gain-variation richness, and injectivity of a structured cost operator. Using these indices, we establish joint identifiability conditions for the inverse problem considered here. Crucially, these conditions guarantee recovery of the ground-truth system matrices (A,B) and the true cost weighting matrices, rather than merely a behaviorally equivalent surrogate. We also develop a conditioning-aware sampled-data reconstruction method that reconstructs the gain K(·) and the closed-loop dynamics matrix Ac(·) from noisy measurements, recovers (A,B) in closed form, and identifies the quadratic weights through a convex semidefinite program. We further establish the non-asymptotic perturbation bounds and the consistency of the full reconstruction method under sub-Gaussian observation noise, with explicit dependence on the same conditioning indices. Numerical experiments support the theory and illustrate the diagnostic value of the conditioning indices.
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BibTeX
@misc{yu2026structure,
title = {{Structure and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics}},
author = {Meiling Yu and Yuan-Hua Ni and Lei Jiang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.11932},
eprint = {2608.11932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11932},
}
RIS
TY - UNPB
TI - Structure and Conditioning in Finite-Horizon Continuous-Time Inverse LQR with Unknown Dynamics
AU - Meiling Yu
AU - Yuan-Hua Ni
AU - Lei Jiang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.11932
UR - https://arxiv.org/abs/2608.11932
AB - Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper studies the finite-horizon continuous-time inverse LQR problem from closed-loop state--input trajectories, where both the system matrices and the quadratic cost are unknown. The finite horizon induces a time-varying optimal gain, and this endogenous excitation serves as the structural mechanism that makes joint recovery possible. We quantify this mechanism through three computable conditioning indices, which measure state richness, gain-variation richness, and injectivity of a structured cost operator. Using these indices, we establish joint identifiability conditions for the inverse problem considered here. Crucially, these conditions guarantee recovery of the ground-truth system matrices (A,B) and the true cost weighting matrices, rather than merely a behaviorally equivalent surrogate. We also develop a conditioning-aware sampled-data reconstruction method that reconstructs the gain K(·) and the closed-loop dynamics matrix A_c(·) from noisy measurements, recovers (A,B) in closed form, and identifies the quadratic weights through a convex semidefinite program. We further establish the non-asymptotic perturbation bounds and the consistency of the full reconstruction method under sub-Gaussian observation noise, with explicit dependence on the same conditioning indices. Numerical experiments support the theory and illustrate the diagnostic value of the conditioning indices.
ER -