Dual control denotes a class of control problems where the parameters governing the system are imperfectly known. The challenge is to find the optimal balance between probing, i.e. exciting the system to understand it more, and caution, i.e. selecting conservative controls based on current knowledge to achieve the control objective. Dynamic programming techniques can achieve this optimal trade-off. However, while dynamic programming performs well with discrete state and time, it is not well-suited to problems with continuous time-frames or continuous or unbounded state spaces. Another limitation is that multidimensional states often cause the dynamic programming approaches to be intractable. In this paper, we investigate whether continuous-time optimal control tools could help circumvent these caveats whilst still achieving the probing-caution balance. We introduce a stylized problem where the state is governed by one of two differential equations. It is initially unknown which differential equation governs the system, so we must simultaneously determine the true differential equation and control the system to the desired state. We show how this problem can be transformed to apply optimal control tools, and compare the performance of this approach to a dynamic programming approach. Our results suggest that the optimal control algorithm rivals dynamic programming on small problems, achieving the right balance between aggressive and smoothly varying controls. In contrast to dynamic programming, the optimal control approach remains tractable when several states are to be controlled simultaneously.
A Splitting Method For Overcoming the Curse of Dimensionality in Hamilton-Jacobi Equations Arising from Nonlinear Optimal Control and Differential Games with Applications to Trajectory Generation
Combining Trajectory Optimization, Supervised Machine Learning, and Model Structure for Mitigating the Curse of Dimensionality in the Control of Bipedal Robots
