Numerical treatment of a class of optimal control problems arising in economics

The approximate solution of the problem of controlling an initial value problem for a linear system of autonomous ordinary differential equations is considered. The corresponding homogeneous solution to the differential equation is assumed to be non-expansive and the inhomogeneity is a linear function of the control variable that is constant along a priori given sub-intervals. The optimal control minimises a convex functional that depends, possibly in a nonlinear way, on the solution of the differential equation. Infinite time horizons are allowed. In view of the piecewise constant control, the corresponding Lagrangian can be split into the sum of Lagrangians acting on sub-intervals. The two algorithms suggested are based upon an iterative process that takes advantage of this splitting as well as of the explicit solution to the differential constraints. Convergence results are provided under suitable assumptions on the problem’s data. Finally, numerical tests for a model of global warming demonstrate the performance of the algorithms.