Measure-Valued Solutions for a Differential Game Related to Fish Harvesting

Abstract. In this paper we study a model for the harvesting of marine resources, described by an elliptic equation. Since the cost functionals have sublinear growth w.r.t. the pointwise intensity of fishing effort, optimal solutions are in general measure-valued. For the control problem, we prove the existence of optimal strategies. Uniqueness is established within a class of measures with small total mass. We also study the differential game, modeling the presence of several competing fishing companies, and prove the existence of a Nash equilibrium solution. This is obtained as a fixed point of a continuous transformation in a space of positive Radon measures.