Abstract
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.
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References
FAO Fisheries Department. The State of World Fisheries and Aquaculture-2016 (SOFIA). FAO, Rome (2016)
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)
Quaas, M.F., Reusch, T.B.H., Schmidt, J.O., Tahvonen, O., Voss, R.: It is the economy, stupid! projecting the fate of fish populations using ecological-economic modeling. Glob. Change Biol. 22(1), 264–270 (2016)
Wilen, J.: Renewable resource economists and policy: What differences have we made? J. Environ. Econ. Manag. 39, 306–327 (2000)
Clark, C.W.: Mathematical Bioeconomics, 3rd edn. Wiley, New York (2010)
Brock, W., Xepapadeas, A.: Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. J. Econ. Dyn. Control 32(9), 2745–2787 (2008)
Brock, W., Xepapadeas, A.: Pattern formation, spatial externalities and regulation in coupled economic-ecological systems. J. Environ. Econ. Manag. 59(2), 149–164 (2010)
Neubert, M.G.: Marine reserves and optimal harvesting. Ecol. Lett. 6, 843–849 (2003)
Ding, W., Lenhart, S.: Optimal harvesting of a spatial explicit fishery model. Nat. Resour. Model. 22(2), 173–211 (2009)
Bressan, A., Coclite, G.M., Shen, W.: An optimal harvesting problem. SIAM J. Control Optim. 51(2), 1186–1202 (2013)
Joshi, H.R., Lenhart, S.: Solving a parabolic identification problem by optimal control methods. Houst. J. Math. 30(4), 1219–1242 (2004)
Joshi, H.R., Herrera, G.E., Lenhart, S., Neubert, M.G.: Optimal dynamic harvest of a mobile renewable resource. Nat. Resour. Model. 22(2), 322–343 (2009)
Coclite, G.M., Garavello, M.: A time-depending optimal harvesting problem with measure-valued solutions. SIAM J. Control Optim. 55(2), 913–935 (2017)
Kelly Jr., M.R., Xing, Y., Lenhart, S.: Optimal fish harvesting for a population model by a nonlinear parabolic partial differential equation. Nat. Resour. Model. 29(1), 36–70 (2016)
Stadler, G.: Elliptic optimal control problems with \(L^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44(2), 159–181 (2009)
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)
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)
Kröner, A., Vexler, B.: A priori error estimates for elliptic optimal control problems with a bilinear state equation. J. Comput. Appl. Math. 230(2), 781–802 (2009)
Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120(2), 345–386 (2012)
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)
Wachsmuth, D.: The regularity of the positive part of functions in \(L^2(I; H^1(\varOmega ))\cap H^1(I; H^1(\varOmega )^*)\) with applications to parabolic equations. Comm. Math. Univ. Carol. 57(3), 327–332 (2016)
Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39, 143–177 (1999)
Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1986)
Evans, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics, vol. 140, 2nd edn. Academic Press, London (2003)
Troeltzsch, F.: Optimale Steuerung Partieller Differentialgleichungen, 2nd edn. Vieweg-Teubner, Braunschweig (2009)
Becker, R., Meidner, D., Vexler, B.: Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22(5), 813–833 (2007)
Casas, E., Tröltzsch, F.: A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53(1), 173–206 (2012)
Casas, E., Tröltzsch, F.: Second order analysis for optimal control problems: improving results expected from abstract theory. SIAM J. Optim. 22(1), 261–279 (2012)
Sanchirico, J., Wilen, J.: A bioeconomic model of marine reserve creation. J. Environ. Econ. Manag. 42, 257–276 (2001)
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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.
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Communicated by Stefan Ulbrich.
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Braack, M., Quaas, M.F., Tews, B. et al. Optimization of Fishing Strategies in Space and Time as a Non-convex Optimal Control Problem. J Optim Theory Appl 178, 950–972 (2018). https://doi.org/10.1007/s10957-018-1304-7
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DOI: https://doi.org/10.1007/s10957-018-1304-7