# Geometry of the discrete Hamilton--Jacobi equation. Applications in optimal control

## TL;DR

This paper explores the geometric structure of the discrete Hamilton--Jacobi equation, proposing two interpretations and applying them to optimal control, with numerical comparisons of solutions.

## Contribution

It introduces two novel geometric interpretations of the discrete Hamilton--Jacobi theory and demonstrates their equivalence and application in optimal control.

## Key findings

- Two equivalent geometric interpretations are proposed.
- Numerical solutions are compared and analyzed.
- Applications in optimal control are demonstrated.

## Abstract

In this paper, we review the discrete Hamilton--Jacobi theory from a geometric point of view. In the discrete realm, the usual geometric interpretation of the Hamilton--Jacobi theory in terms of vector fields is not straightforward. Here, we propose two alternative interpretations: one is the interpretation in terms of projective flows, the second is the temptative of constructing a discrete Hamiltonian vector field renacting the usual continuous interpretation. Both interpretations are proven to be equivalent and applied in optimal control theory. The solutions achieved through both approaches are sorted out and compared by numerical computation.