# Polynomial Profits in Renewable Resources Management

## TL;DR

This paper develops a mathematical framework using renewal equations on graphs to optimize biological resource exploitation, proving the existence of optimal controls and characterizing their structure, including conditions for bang-bang solutions.

## Contribution

It introduces a novel polynomial cost function approach in renewable resource management and provides conditions for optimal control structures, including bang-bang controls.

## Key findings

- Optimal control exists for the resource exploitation model.
- Under certain conditions, the optimal control is bang-bang.
- The polynomial dependence simplifies the computation of optimal controls.

## Abstract

A system of renewal equations on a graph provides a framework to describe the exploitation of a biological resource. In this context, we formulate an optimal control problem, prove the existence of an optimal control and ensure that the target cost function is polynomial in the control. In specific situations, further information about the form of this dependence is obtained. As a consequence, in some cases the optimal control is proved to be necessarily bang--bang, in other cases the computations necessary to find the optimal control are significantly reduced.