Methods for the numerical integration of Hamiltonian systems

Abstract

To compute an infinite horizon optimal controller for a linear periodic system via an invariant subspace method, the computation of the period map associated with the Hamilton-Jacobi-Bellman equations is required. In this paper we discuss methods for the numerical integration of such Hamiltonian systems. Two numerical integration techniques are introduced. A method is developed whereby symplectic invariants associated with the Hamilton-Jacobi-Bellman equations are preserved. Also, a shifting scheme is introduced that in effect swaps the roles of the stable and unstable invariant subspaces by using the semigroup property of state transition matrices. A shift is introduced into the resultant initial value problem ensuring that the eigenvalues of a differential equation reside in the region of absolute stability for an appropriate numerical integration routine. These techniques are then compared to standard numerical integration routines to ascertain their efficiency and accuracy.