Bridging Continuous-time LQR and Reinforcement Learning via Gradient Flow of the Bellman Error

In this paper, we present a novel method for computing the optimal feedback gain of the infinite-horizon Linear Quadratic Regulator (LQR) problem via an ordinary differential equation. We introduce a novel continuous-time Bellman error, derived from the Hamilton-Jacobi-Bellman (HJB) equation, which quantifies the suboptimality of stabilizing policies and is parametrized in terms of the feedback gain. We analyze its properties, including its effective domain, smoothness, and coerciveness, and show the existence of a unique stationary point within the stability region. Furthermore, we derive a closed-form gradient expression of the Bellman error that induces a gradient flow. This converges to the optimal feedback and generates a unique trajectory that exclusively comprises stabilizing feedback policies. Additionally, this work advances interesting connections between LQR theory and Reinforcement Learning (RL) by redefining suboptimality of the Algebraic Riccati Equation (ARE) as a Bellman error, adapting a state-independent formulation, and leveraging Lyapunov equations to overcome the infinite-horizon challenge. We validate our method in a simulation and compare it to the state of the art.