Geometry of the discrete Hamilton–Jacobi equation: applications in optimal control

Abstract

In this paper, we review the discrete Hamilton—Jacobi equation from a geometric point of view. In similarity with the continuous geometric Hamilton—Jacobi theory, we propose two different discrete geometric interpretations for the equation. The first approach is based on the construction of a discrete Hamilton—Jacobi equation using discrete projective flows. For it, we develop some former results on discrete Hamiltonian systems and provide a discrete equation explicitly, which matches some previous results depicted in the literature. The interest of our method is that it retrieves some already known results, but starting from a new outlook. The second approach is formulated in terms of discrete vector fields, whose definition is not straightfoward. For this, we revisit the discrete theory of mechanics by relying on the construction of discrete vector fields taken from optimal control backgrounds. From here, we reconstruct a discrete Hamilton—Jacobi equation in a novel way, and which has not been devised in the literature before. As a last result, both interpretations are proven to be equivalent theoretically, but the numerical results differ slightly. The discrete vector field approach seems fairly more accurate concerning numerical values in the specific example that we show, that is an optimal control problem for a nonlinear system.