Discrete Hamilton-Jacobi Theory

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. We describe a discrete analogue of Jacobi's solution and also prove a discrete version of the geometric Hamilton-Jacobi theorem. The theory applied to discrete linear Hamiltonian systems yields the discrete Riccati equation as a special case of the discrete Hamilton-Jacobi equation. We also apply the theory to discrete optimal control problems, and recover some well-known results, such as the Bellman equation (discrete-time HJB equation) of dynamic programming and its relation to the costate variable in the Pontryagin maximum principle. This relationship between the discrete Hamilton-Jacobi equation and Bellman equation is exploited to derive a generalized form of the Bellman equation that has controls at internal stages.Comment: 26 pages, 2 figure

The Hamilton-Jacobi Difference Equation

. We study a system of difference equations which, like Hamilton's equations, preserves the standard symplectic structure on R 2m . In particular, we construct a differential-difference equation which we call the Hamilton-Jacobi difference equation, the analog of the Hamilton-Jacobi equation for our discrete system. We solve the HamiltonJacobi difference equation in a simple case. The Hamilton-Jacobi equation is of great importance in analytic dynamics [1]. Here an analogous difference equation is derived for canonical systems of difference equations. It is shown that the general solution of the Hamilton-Jacobi difference equation is equivalent to the general solution of the canonical system of difference equations to which it is related. We consider a sequence (X (0) ; Y (0) ); (X (1) ; Y (1) ); : : : ; (X (n) ; Y (n) ); : : : in R m \Theta R m , generated from the initial point (X (0) ; Y (0) ) via the difference equations X (n+1) k = X (n) k + h @H n (X ..

X (n+1)

Abstract. We study a system of difference equations which, like Hamilton’s equations, preserves the standard symplectic structure on R 2m. In particular, we construct a differential-difference equation which we call the Hamilton-Jacobi difference equation, the analog of the Hamilton-Jacobi equation for our discrete system. We solve the Hamilton-Jacobi difference equation in a simple case. The Hamilton-Jacobi equation is of great importance in analytic dynamics [1]. Here an analogous difference equation is derived for canonical systems of difference equations. It is shown that the general solution of the Hamilton-Jacobi difference equation is equivalent to the general solution of the canonical system of difference equations to which it is related. We consider a sequence (X (0) , Y (0)), (X (1) , Y (1)),...,(X (n) , Y (n)),... in R m × R m, generated from the initial point (X (0) , Y (0) ) via the difference equations (1