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.
An iterative technique for the computation of the steady state gains for the discrete optimal regulator
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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.