  Solving the Hamilton-Jacobi-Bellman equation is important in many domains
including control, robotics and economics. Especially for continuous control,
solving this differential equation and its extension the Hamilton-Jacobi-Isaacs
equation, is important as it yields the optimal policy that achieves the
maximum reward on a give task. In the case of the Hamilton-Jacobi-Isaacs
equation, which includes an adversary controlling the environment and
minimizing the reward, the obtained policy is also robust to perturbations of
the dynamics. In this paper we propose continuous fitted value iteration (cFVI)
and robust fitted value iteration (rFVI). These algorithms leverage the
non-linear control-affine dynamics and separable state and action reward of
many continuous control problems to derive the optimal policy and optimal
adversary in closed form. This analytic expression simplifies the differential
equations and enables us to solve for the optimal value function using value
iteration for continuous actions and states as well as the adversarial case.
Notably, the resulting algorithms do not require discretization of states or
actions. We apply the resulting algorithms to the Furuta pendulum and cartpole.
We show that both algorithms obtain the optimal policy. The robustness Sim2Real
experiments on the physical systems show that the policies successfully achieve
the task in the real-world. When changing the masses of the pendulum, we
observe that robust value iteration is more robust compared to deep
reinforcement learning algorithm and the non-robust version of the algorithm.
Videos of the experiments are shown at https://sites.google.com/view/rfvi
