Recycled and non-recycled exhaustible resource: an optimal control strategy for input allocation

The United Nations aim to perform a transition toward a sustainable environment where people can live by decoupling economic growth from resource use. Through the definition of the Agenda 2030 and the corresponding sustainable development goals, this transition asks for a lower dependence on non-renewable resources and for the use of recycled materials in a finite term perspective. In this respect, we provide an optimal control model which searches for an efficient allocation of labor between non-recycling and recycling sectors exploiting a given non-renewable resource. The optimization process is carried out over a finite time horizon in accordance with the need of rapidly achieving the targets imposed by the ecological transition. By employing the classical tools of optimal control theory, a complete theoretical analysis of the model well-posedness is developed under the assumption of linear production in both sectors. The approach is applied in order to simulate a hypothetical test case.