Discrete Hamilton-Jacobi Theory Export

@article{citeulike:6111507, abstract = {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.}, archivePrefix = {arXiv}, author = {Ohsawa, Tomoki and Bloch, Anthony M. and Leok, Melvin}, citeulike-article-id = {6111507}, citeulike-linkout-0 = {http://arxiv.org/abs/0911.2258}, citeulike-linkout-1 = {http://arxiv.org/pdf/0911.2258}, day = {11}, eprint = {0911.2258}, keywords = {discrete, hje}, month = {Nov}, posted-at = {2009-11-14 08:16:45}, priority = {2}, title = {Discrete Hamilton-Jacobi Theory}, url = {http://arxiv.org/abs/0911.2258}, year = {2009} }

TY - JOUR ID - citeulike:6111507 L3 - citeulike-article-id:6111507 N2 - 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. TI - Discrete Hamilton-Jacobi Theory KW - discrete KW - hje AU - Ohsawa, Tomoki AU - Bloch, Anthony AU - Leok, Melvin PY - 2009/Nov/11/ UR - http://arxiv.org/abs/0911.2258 ER -