Measure valued solutions for an optimal harvesting problem

The paper deals with the harvesting model introduced by one of the authors in a joint paper with A. Bressan and W. Shen. A cost term which plays an essential role in the existence proof is removed here leading to some new phenomena. Since a solution not always exists in this case, we propose a relaxed version of the problem which is always solvable, has a geometrical interpretation in the functions space and whose unique solution, under suitable conditions, solves also the original problem. Finally, the notion of strategy limit of a sequence of smooth measures is introduced as a long-term control to fish populations.