
Mathematics 2009
Discrete HamiltonJacobi TheoryDOI: 10.1137/090776822 Abstract: We develop a discrete analogue of HamiltonJacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete HamiltonJacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric HamiltonJacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete HamiltonJacobi equation. We also apply the theory to discrete optimal control problems, and recover some wellknown results, such as the Bellman equation (discretetime HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete HamiltonJacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.
