An algorithm for quasi-linear control problems in the economics of renewable resources: The steady state and end state for the infinite and long-term horizon

This paper presents the problem of finding the optimal harvesting strategy, maximizing the expected present value of total revenues. The problem is formulated as an optimal control problem. Combining the techniques of Pontryagin’s Maximum Principle and the shooting method, an algorithm has been developed that is not affected by the values of the parameter. The algorithm is able to solve conventional problems as well as cases in which the optimal solution is shown to be bang–bang with singular arcs. In addition, we present a result that characterizes the optimal steady-state in infinite-horizon, autonomous models (except in the discount factor) and does not require the solution of the dynamic optimization problem. We also present a result that, under certain additional conditions, allows us to know a priori the final state solution when the optimization interval is finite. Finally, several numerical examples are presented to illustrate the different possibilities of the method.