Coclite, Giuseppe Maria; Garavello, Mauro; Spinolo, Laura V. Optimal strategies for a time-dependent harvesting problem. (English) Zbl 1405.35099 Discrete Contin. Dyn. Syst., Ser. S 11, No. 5, 865-900 (2018). Summary: We focus on an optimal control problem, introduced by Bressan and Shen in [5] as a model for fish harvesting. We consider the time-dependent case and we establish existence and uniqueness of an optimal strategy. We also study a related differential game, 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 focuses on one-dimensional fishing domains. Cited in 18 Documents MSC: 35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations 35Q93 PDEs in connection with control and optimization 49J20 Existence theories for optimal control problems involving partial differential equations 49N25 Impulsive optimal control problems 49N90 Applications of optimal control and differential games Keywords:optimal control; differential games; measured-valued solutions; fish harvesting PDFBibTeX XMLCite \textit{G. M. Coclite} et al., Discrete Contin. Dyn. Syst., Ser. S 11, No. 5, 865--900 (2018; Zbl 1405.35099) Full Text: DOI arXiv References: [1] O. Arino; J. A. Montero, Optimal control of a nonlinear elliptic population system, Proc. Edinburgh Math. Soc., 43, 225-241 (2000) · Zbl 0944.35010 [2] L. Boccardo; T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87, 149-169 (1989) · Zbl 0707.35060 [3] A. Bressan; G. M. Coclite; W. Shen, A multi-dimensional optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 51, 1186-1202 (2013) · Zbl 1268.49002 [4] A. Bressan; W. Shen, Measure-valued solutions for a differential game related to fish harvesting, SIAM J. Control Optim., 47, 3118-3137 (2008) · Zbl 1176.49039 [5] A. Bressan; W. Shen, Measure valued solutions to a harvesting game with several players, Advances in Dynamic Games, 11, 399-423 (2011) · Zbl 1218.91132 [6] J. R. Cannon, The One-Dimensional Heat Equation, With a foreword by Felix E. Browder. Encyclopedia of Mathematics and its Applications, 23. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. · Zbl 0567.35001 [7] A. Cañada; J. L. Gámez; J. A. Montero, Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type, SIAM J. Control Optim., 36, 1171-1189 (1998) · Zbl 0916.49003 [8] G. M. Coclite; M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 55, 913-935 (2017) · Zbl 1375.35584 [9] M. Delgado; J. A. Montero; A. Suárez, Optimal control for the degenerate elliptic logistic equation, Appl. Math. Optim., 45, 325-345 (2002) · Zbl 1017.49006 [10] M. Delgado; J. A. Montero; A. Suárez, Study of the optimal harvesting control and the optimality system for an elliptic problem, SIAM J. Control Optim., 42, 1559-1577 (2003) · Zbl 1048.49001 [11] S. M. Lenhart; J. A. Montero, Optimal control of harvesting in a parabolic system modeling two subpopulations, Math. Models Methods Appl. Sci., 11, 1129-1141 (2001) · Zbl 1013.92050 [12] S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Universitext. Springer-Verlag Italia, Milan, 2008. · Zbl 1146.35001 [13] J. Simon, Compact sets in the space \(\begin{document}L_p (0, T ; B)\end{document} \), Ann. Mat. Pura Appl. (4), 146, 65-96 (1987) · Zbl 0629.46031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.