[ PDF | arXiv:0911.2258 [math.OC] ]
(with A.M. Bloch and T. Ohsawa)
We develop a discrete analogue of the Hamilton-Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton-Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time, and is shown to recover the Hamilton-Jacobi equation in the continuous-time limit. 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 also prove a discrete analogue of the geometric Hamilton-Jacobi theorem of Abraham and Marsden. These results are readily applied to discrete optimal control setting, and some well-known results in discrete optimal control theory, such as the Bellman equation (discrete-time Hamilton-Jacobi-Bellman equation) of dynamic programming, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case of the discrete Hamilton-Jacobi equation.