Deep reinforcement learning based finite-horizon optimal control for a discrete-time affine nonlinear system

Approximate dynamic programming (ADP) aims to obtain an approximate numerical solution to the discrete time Hamilton-Jacobi-Bellman (HJB) equation. Heuristic dynamic programming (HDP) is a two-stage iterative scheme of ADP by separating the HJB equation into two equations, one for the value function and another for the policy function, which are referred to as the critic and the actor, respectively. Previous ADP implementations have been limited by the choice of function approximator, which requires significant prior domain knowledge or a large number of parameters to be fitted. However, recent advances in deep learning brought by the computer science community enable the use of deep neural networks (DNN) to approximate high-dimensional nonlinear functions without prior domain knowledge. Motivated by this, we examine the potential of DNNs as function approximators of the critic and the actor. In contrast to the infinite-horizon optimal control problem, the critic and the actor of the finite horizon optimal control (FHOC) problem are time-varying functions and have to satisfy a boundary condition. DNN structure and training algorithm suitable for FHOC are presented. Illustrative examples are provided to demonstrate the validity of the proposed method.