Discrete Hamilton-Jacobi theory and discrete optimal control

We develop a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi´s solution to the Hamilton-Jacobi equation. We prove discrete analogues of Jacobi´s solution to the Hamilton-Jacobi equation and of the geometric Hamilton-Jacobi theorem. These results are readily applied to the discrete optimal control setting, and some well-known results in discrete optimal control theory, such as the Bellman equation, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case.