  We focus on an optimal control problem, introduced by Bressan and Shen as a
model for fish harvesting. We consider the time-dependent case and we establish
existence and uniqueness of an optimal strategy, and sufficient conditions for
optimality. We also consider a related differential game that models the
situation where there are several competing fish companies and we prove
existence of Nash equilibria. From the technical viewpoint, the most relevant
point is establishing the uniqueness result. This amounts to prove precise
a-priori estimates for solutions of suitable parabolic equations with
measure-valued coefficients. All the analysis is developed in the case when the
fishing domain is one-dimensional.
