Optimal control for sustainable consumption of natural resources

In this paper we study optimal policies for a central planner interested in maximizing utility in an economy driven by a renewable resource. It is shown that the optimal consumption path is sustainable only when the intrinsic growth rate of the resource is greater than the social discount rate. The model is formulated as an infinite horizon optimal control problem. We deal with the mathematical details of the problem, develop a precise notion for optimality and establish the existence of optimal control at least when the condition for sustainability is met. We apply the appropriate version of the Pontryagin maximum principle and show a numerical simulation of the optimal feedback law. In the end we present the results along with physical interpretations.