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Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controls

Jeongho Kim, Jaeuk Shin, Insoon Yang; 22(206):1−34, 2021.


In this paper, we propose Q-learning algorithms for continuous-time deterministic optimal control problems with Lipschitz continuous controls. A new class of Hamilton-Jacobi-Bellman (HJB) equations is derived from applying the dynamic programming principle to continuous-time Q-functions. Our method is based on a novel semi-discrete version of the HJB equation, which is proposed to design a Q-learning algorithm that uses data collected in discrete time without discretizing or approximating the system dynamics. We identify the conditions under which the Q-function estimated by this algorithm converges to the optimal Q-function. For practical implementation, we propose the Hamilton-Jacobi DQN, which extends the idea of deep Q-networks (DQN) to our continuous control setting. This approach does not require actor networks or numerical solutions to optimization problems for greedy actions since the HJB equation provides a simple characterization of optimal controls via ordinary differential equations. We empirically demonstrate the performance of our method through benchmark tasks and high-dimensional linear-quadratic problems.

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