Optimal exploitation of renewable resource stocks: Necessary conditions

TL;DR

This paper develops a mathematical framework for optimizing renewable resource exploitation, incorporating impulsive controls and deriving necessary conditions for optimality using Pontryagin's maximum principle.

Contribution

It formulates Pontryagin's maximum principle for control problems with impulsive and measurable controls, providing a near-complete description of optimal policies for all initial states.

Findings

Derived first-order necessary conditions of optimality.

Defined switch functions to characterize optimal trajectories.

Provided a comprehensive description of optimal policies in the phase plane.

Abstract

We study a model for the exploitation of renewable stocks developed in Clark et al. (Econometrica 47 (1979), 25-47). In this particular control problem, the control law contains a measurable and an impulsive control component. We formulate Pontryagin's maximum principle for this kind of control problems, proving first order necessary conditions of optimality. Manipulating the correspondent Lagrange multipliers we are able to define two special switch functions, that allow us to describe the optimal trajectories and control policies nearly completely for all possible initial conditions in the phase plane.