# American Institute of Mathematical Sciences

October  2018, 11(5): 865-900. doi: 10.3934/dcdss.2018053

## Optimal strategies for a time-dependent harvesting problem

 1 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via E. Orabona 4, 70125 Bari, Italy 2 Department of Mathematics and its Applications, University of Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy 3 IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy

Received  February 2017 Revised  March 2017 Published  June 2018

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.

Citation: Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 865-900. doi: 10.3934/dcdss.2018053
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000. Google Scholar [2] O. Arino and J. A. Montero, Optimal control of a nonlinear elliptic population system, Proc. Edinburgh Math. Soc., 43 (2000), 225-241. doi: 10.1017/S0013091500020897. Google Scholar [3] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. Google Scholar [4] A. Bressan, G. M. Coclite and W. Shen, A multi-dimensional optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202. doi: 10.1137/110853510. Google Scholar [5] A. Bressan and W. Shen, Measure-valued solutions for a differential game related to fish harvesting, SIAM J. Control Optim., 47 (2008), 3118-3137. doi: 10.1137/07071007X. Google Scholar [6] A. Bressan and W. Shen, Measure valued solutions to a harvesting game with several players, Advances in Dynamic Games, 11 (2011), 399-423. doi: 10.1007/978-0-8176-8089-3_20. Google Scholar [7] 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. doi: 10.1017/CBO9781139086967. Google Scholar [8] A. Cañada, J. L. Gámez and J. A. Montero, Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type, SIAM J. Control Optim., 36 (1998), 1171-1189. doi: 10.1137/S0363012995293323. Google Scholar [9] G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 55 (2017), 913-935. doi: 10.1137/16M1061886. Google Scholar [10] M. Delgado, J. A. Montero and A. Suárez, Optimal control for the degenerate elliptic logistic equation, Appl. Math. Optim., 45 (2002), 325-345. doi: 10.1007/s00245-001-0039-1. Google Scholar [11] M. Delgado, J. A. Montero and A. Suárez, Study of the optimal harvesting control and the optimality system for an elliptic problem, SIAM J. Control Optim., 42 (2003), 1559-1577. doi: 10.1137/S0363012902410903. Google Scholar [12] S. M. Lenhart and J. A. Montero, Optimal control of harvesting in a parabolic system modeling two subpopulations, Math. Models Methods Appl. Sci., 11 (2001), 1129-1141. doi: 10.1142/S0218202501000982. Google Scholar [13] S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Universitext. Springer-Verlag Italia, Milan, 2008. Google Scholar [14] J. Simon, Compact sets in the space $L_p (0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, New York, 2000. Google Scholar [2] O. Arino and J. A. Montero, Optimal control of a nonlinear elliptic population system, Proc. Edinburgh Math. Soc., 43 (2000), 225-241. doi: 10.1017/S0013091500020897. Google Scholar [3] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. doi: 10.1016/0022-1236(89)90005-0. Google Scholar [4] A. Bressan, G. M. Coclite and W. Shen, A multi-dimensional optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 51 (2013), 1186-1202. doi: 10.1137/110853510. Google Scholar [5] A. Bressan and W. Shen, Measure-valued solutions for a differential game related to fish harvesting, SIAM J. Control Optim., 47 (2008), 3118-3137. doi: 10.1137/07071007X. Google Scholar [6] A. Bressan and W. Shen, Measure valued solutions to a harvesting game with several players, Advances in Dynamic Games, 11 (2011), 399-423. doi: 10.1007/978-0-8176-8089-3_20. Google Scholar [7] 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. doi: 10.1017/CBO9781139086967. Google Scholar [8] A. Cañada, J. L. Gámez and J. A. Montero, Study of an optimal control problem for diffusive nonlinear elliptic equations of logistic type, SIAM J. Control Optim., 36 (1998), 1171-1189. doi: 10.1137/S0363012995293323. Google Scholar [9] G. M. Coclite and M. Garavello, A time dependent optimal harvesting problem with measure valued solutions, SIAM J. Control Optim., 55 (2017), 913-935. doi: 10.1137/16M1061886. Google Scholar [10] M. Delgado, J. A. Montero and A. Suárez, Optimal control for the degenerate elliptic logistic equation, Appl. Math. Optim., 45 (2002), 325-345. doi: 10.1007/s00245-001-0039-1. Google Scholar [11] M. Delgado, J. A. Montero and A. Suárez, Study of the optimal harvesting control and the optimality system for an elliptic problem, SIAM J. Control Optim., 42 (2003), 1559-1577. doi: 10.1137/S0363012902410903. Google Scholar [12] S. M. Lenhart and J. A. Montero, Optimal control of harvesting in a parabolic system modeling two subpopulations, Math. Models Methods Appl. Sci., 11 (2001), 1129-1141. doi: 10.1142/S0218202501000982. Google Scholar [13] S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Universitext. Springer-Verlag Italia, Milan, 2008. Google Scholar [14] J. Simon, Compact sets in the space $L_p (0, T ; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar
 [1] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [2] Peng Zhong, Suzanne Lenhart. Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations. Evolution Equations & Control Theory, 2013, 2 (4) : 749-769. doi: 10.3934/eect.2013.2.749 [3] Hiroaki Morimoto. Optimal harvesting and planting control in stochastic logistic population models. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2545-2559. doi: 10.3934/dcdsb.2012.17.2545 [4] Peng Zhong, Suzanne Lenhart. Optimal control of integrodifference equations with growth-harvesting-dispersal order. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2281-2298. doi: 10.3934/dcdsb.2012.17.2281 [5] Jingzhen Liu, Ka-Fai Cedric Yiu. Optimal stochastic differential games with VaR constraints. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889 [6] Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455 [7] Yunfeng Jia, Jianhua Wu, Hong-Kun Xu. On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1927-1941. doi: 10.3934/cpaa.2013.12.1927 [8] Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028 [9] Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics & Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012 [10] Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026 [11] Martin Bohner, Sabrina Streipert. Optimal harvesting policy for the Beverton--Holt model. Mathematical Biosciences & Engineering, 2016, 13 (4) : 673-695. doi: 10.3934/mbe.2016014 [12] Meng Liu, Chuanzhi Bai. Optimal harvesting of a stochastic delay competitive model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1493-1508. doi: 10.3934/dcdsb.2017071 [13] M. Delgado-Téllez, Alberto Ibort. On the geometry and topology of singular optimal control problems and their solutions. Conference Publications, 2003, 2003 (Special) : 223-233. doi: 10.3934/proc.2003.2003.223 [14] Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129 [15] Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 [16] Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 [17] Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595 [18] John A. Morgan. Interception in differential pursuit/evasion games. Journal of Dynamics & Games, 2016, 3 (4) : 335-354. doi: 10.3934/jdg.2016018 [19] Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017 [20] Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059

2018 Impact Factor: 0.545