# Infinite-horizon Risk-constrained Linear Quadratic Regulator with Average Cost

arXiv: https://arxiv.org/abs/2103.15363 · HTML: https://mlsanity.com/paper/2103.15363/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2021.

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

- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Augmented RBMLE-UCB Approach for Adaptive Control of Linear Quadratic Systems](https://mlsanity.com/paper/2201.10542/) (2022)
- [Policy Gradient Methods for the Cost-Constrained LQR: Strong Duality and Global Convergence](https://mlsanity.com/paper/2406.03734/) (2024)
- [Policy Gradient-based Model Free Optimal LQG Control with a Probabilistic Risk Constraint](https://mlsanity.com/paper/2403.16767/) (2024)
- [Deterministic Policy Gradient Primal-Dual Methods for Continuous-Space Constrained MDPs](https://mlsanity.com/paper/2408.10015/) (2024)
- [Actor-Critic Learning for Risk-Constrained Linear Quadratic Regulation](https://mlsanity.com/paper/2510.22267/) (2025)
- [Model-free LQG Control with Chance Constraints](https://mlsanity.com/paper/2605.31310/) (2026)

## Abstract (arXiv)

The behaviour of a stochastic dynamical system may be largely influenced by those low-probability, yet extreme events. To address such occurrences, this paper proposes an infinite-horizon risk-constrained Linear Quadratic Regulator (LQR) framework with time-average cost. In addition to the standard LQR objective, the average one-stage predictive variance of the state penalty is constrained to lie within a user-specified level. By leveraging the duality, its optimal solution is first shown to be stationary and affine in the state, i.e., u(x,λ^*) = -K(λ^*)x + l(λ^*), where λ^* is an optimal multiplier, used to address the risk constraint. Then, we establish the stability of the resulting closed-loop system. Furthermore, we propose a primal-dual method with sublinear convergence rate to find an optimal policy u(x,λ^*). Finally, a numerical example is provided to demonstrate the effectiveness of the proposed framework and the primal-dual method.