ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control
Abstract
The classical linear quadratic regulator (LQR) minimizes the expected cumulative return but fails to account for performance variability, rendering it inadequate for risk-aware applications. To address this, we introduce the variance of the cumulative return as a risk measure in LQR. We derive the first exact closed-form expression for the variance of the discounted in?finite horizon return within the discrete-time Distributional LQR framework, for i.i.d. disturbances with symmetric probability densities. For Gaussian disturbances, this expression elegantly simplifies to a form dependent only on the disturbance covariance. Leveraging these theoretical foundations, we formulate a mean-variance optimal control problem that explicitly manages the trade-off between expected return and performance variability. To address the resulting non-convex optimization problem, we propose a novel adjoint gradient descent algorithm for a penalized formulation of the original problem, and establish that all iterates remain stabilizing and converge to a stationary point of the penalized objective. The effectiveness of this framework and the inherent risk-performance trade-off? are demonstrated through numerical experiments.
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BibTeX
@misc{teng2026exactvariance,
title = {{ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control}},
author = {Ruyi Teng and Zifan Wang and Yulong Gao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39981},
eprint = {2609.39981},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39981},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control
AU - Ruyi Teng
AU - Zifan Wang
AU - Yulong Gao
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39981
UR - https://arxiv.org/abs/2609.39981
AB - The classical linear quadratic regulator (LQR) minimizes the expected cumulative return but fails to account for performance variability, rendering it inadequate for risk-aware applications. To address this, we introduce the variance of the cumulative return as a risk measure in LQR. We derive the first exact closed-form expression for the variance of the discounted in?finite horizon return within the discrete-time Distributional LQR framework, for i.i.d. disturbances with symmetric probability densities. For Gaussian disturbances, this expression elegantly simplifies to a form dependent only on the disturbance covariance. Leveraging these theoretical foundations, we formulate a mean-variance optimal control problem that explicitly manages the trade-off between expected return and performance variability. To address the resulting non-convex optimization problem, we propose a novel adjoint gradient descent algorithm for a penalized formulation of the original problem, and establish that all iterates remain stabilizing and converge to a stationary point of the penalized objective. The effectiveness of this framework and the inherent risk-performance trade-off? are demonstrated through numerical experiments.
ER -