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Solving the Hamilton-Jacobi-Bellman equation is important in many domainsincluding control, robotics and economics. Especially for continuous control,solving this differential equation and its extension the Hamilton-Jacobi-Isaacsequation, is important as it yields the optimal policy that achieves themaximum reward on a give task. In the case of the Hamilton-Jacobi-Isaacsequation, which includes an adversary controlling the environment andminimizing the reward, the obtained policy is also robust to perturbations ofthe dynamics. In this paper we propose continuous fitted value iteration (cFVI)and robust fitted value iteration (rFVI). These algorithms leverage thenon-linear control-affine dynamics and separable state and action reward ofmany continuous control problems to derive the optimal policy and optimaladversary in closed form. This analytic expression simplifies the differentialequations and enables us to solve for the optimal value function using valueiteration for continuous actions and states as well as the adversarial case.Notably, the resulting algorithms do not require discretization of states oractions. We apply the resulting algorithms to the Furuta pendulum and cartpole.We show that both algorithms obtain the optimal policy. The robustness Sim2Realexperiments on the physical systems show that the policies successfully achievethe task in the real-world. When changing the masses of the pendulum, weobserve that robust value iteration is more robust compared to deepreinforcement learning algorithm and the non-robust version of the algorithm.Videos of the experiments are shown at https://sites.google.com/view/rfvi

Continuous-Time Fitted Value Iteration for Robust Policies