Gradient Descent Approaches to Neural-Net-Based Solutions of the Hamilton-Jacobi-Bellman Equation

Abstract

In this paper we investigate new approaches to dynamic-programming-based optimal control of continuous time-and-space systems. We use neural networks to approximate the solution to the Hamilton-Jacobi-Bellman (HJB) equation which is, in the deterministic case studied here, a first-order, non-linear, partial differential equation. We derive the gradient descent rule for integrating this equation inside the domain, given the conditions on the boundary. We apply this approach to the "Car-on-the-hill" which is a two-dimensional highly non-linear control problem. We discuss the results obtained and point out a low quality of approximation of the value function and of the derived control. We attribute this bad approximation to the fact that the HJB equation has many generalized solutions (i.e. differentiable almost everywhere) other than the value function, and our gradient descent method converges to one among these functions, thus possibly failing to find the correct value function. We illustrate this limitation on a simple one-dimensional control problem.