Runge-Kutta Discretizations of Infinite Horizon Optimal Control Problems with Steady-State Invariance

Direct numerical approximation of a continuous-time infinite horizon control problem, requires to recast the model as a discrete-time, finite-horizon control model. The quality of the optimization results can be heavily degraded if the discretization process does not take into account features of the original model to be preserved. Restricting their attention to optimal growh problems with a steady state, Mercenier and Michel in [1] and [2], studied the conditions to be imposed for ensuring that discrete first-order approximation models have the same steady states as the infinite-horizon continuous-times counterpart. Here we show that Mercenier and Michel scheme is a first order partitioned Runge-Kutta method applied to the state-costate differential system which arises from the Pontryagin maximum principle. The main consequence is that it is possible to consider high order schemes which generalize that algorithm by preserving the steady-growth invariance of the solutions with respect to the discretization process. Numerical examples show the efficiency and accuracy of the proposed methods when applied to the classical Ramsey growth model.