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Learning optimal feedback control laws capable of executing optimaltrajectories is essential for many robotic applications. Such policies can belearned using reinforcement learning or planned using optimal control. Whilereinforcement learning is sample inefficient, optimal control only plans anoptimal trajectory from a specific starting configuration. In this paper wepropose deep optimal feedback control to learn an optimal feedback policyrather than a single trajectory. By exploiting the inherent structure of therobot dynamics and strictly convex action cost, we can derive principled costfunctions such that the optimal policy naturally obeys the action limits, isglobally optimal and stable on the training domain given the optimal valuefunction. The corresponding optimal value function is learned end-to-end byembedding a deep differential network in the Hamilton-Jacobi-Bellmanndifferential equation and minimizing the error of this equality whilesimultaneously decreasing the discounting from short- to far-sighted to enablethe learning. Our proposed approach enables us to learn an optimal feedbackcontrol law in continuous time, that in contrast to existing approachesgenerates an optimal trajectory from any point in state-space without the needof replanning. The resulting approach is evaluated on non-linear systems andachieves optimal feedback control, where standard optimal control methodsrequire frequent replanning.

HJB Optimal Feedback Control with Deep Differential Value Functions and Action Constraints