Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control

We are concerned with the discretization of optimal control problems when a Runge-Kutta scheme is selected for the related Hamiltonian system. It is known that Lagrangian's first order conditions on the discrete model, require a symplectic partitioned Runge-Kutta scheme for state-costate equations. In the present paper this result is extended to growth models, widely used in Economics studies, where the system is described by a current Hamiltonian. We prove that a correct numerical treatment of the state-current costate system needs Lawson exponential schemes for the costate approximation. In the numerical tests a shooting strategy is employed in order to verify the accuracy, up to the fourth order, of the innovative procedure we propose.