A gradient flow on a continuous-time Bellman error parametrized by feedback gain converges to the optimal LQR controller and stays inside the stabilizing region.
Rudin, Principles of Mathematical Analysis, 3rd ed
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Bridging Continuous-time LQR and Reinforcement Learning via Gradient Flow of the Bellman Error
A gradient flow on a continuous-time Bellman error parametrized by feedback gain converges to the optimal LQR controller and stays inside the stabilizing region.