Optimization of Fishing Strategies in Space and Time as a Non-convex Optimal Control Problem
- 16 Downloads
The behavior of a fishing fleet and its impact onto the biomass of fish can be described by a nonlinear parabolic diffusion–reaction equation. Looking for an optimal fishing strategy leads to a non-convex optimal control problem with a bilinear control action. In this work, we present such an optimal control formulation, prove its well-posedness and derive first- and second-order optimality conditions. These results provide a basis for tailored finite element discretization as well as for Newton type optimization algorithms. First numerical test problems show typical features as so-called No-Take-Zones and maximal fishing quota (total allowable catches) as parts of an optimal fishing strategy.
KeywordsFishing strategies Optimal control Non-convex optimization
Mathematics Subject Classification35K20 35K45 35K57 49K20 49K40 65M60
This work was supported by the German Science Foundation (DFG) through the Excellence Cluster Future Ocean by project number CP 1336. This support is gratefully acknowledged.
- 1.FAO Fisheries Department. The State of World Fisheries and Aquaculture-2016 (SOFIA). FAO, Rome (2016)Google Scholar
- 2.Pauly, D., Zeller, D.: Catch reconstructions reveal that global marine fisheries catches are higher than reported and declining. Nature Communications 7, article number 10244 (2016)Google Scholar
- 5.Clark, C.W.: Mathematical Bioeconomics, 3rd edn. Wiley, New York (2010)Google Scholar
- 16.Ito, K., Kunisch, K.: Lagrange Multiplier Approach to Variational Problems and Applications, pp. xvi+341. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2008)Google Scholar
- 17.Ulbrich, M.: On a nonsmooth Newton method for nonlinear complementarity problems in function space with applications to optimal control. In: Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.) Complementarity: Applications, Algorithms and Extensions, pp. 341–360. Kluwer Academic Publishers, Dordrecht (2001)CrossRefGoogle Scholar
- 20.Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, pp. xx+313. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1980 original (2000)Google Scholar
- 24.Evans, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, London (2003)Google Scholar