Optimal exploitation of renewable resource stocks: necessary conditions

We study a model for the exploitation of renewable stocks developed by Clark et al. 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.