American Institute of Mathematical Sciences

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

Optimal strategies for a time-dependent harvesting problem

Giuseppe Maria Coclite 1, , Mauro Garavello 2, and  Laura V. Spinolo 3,

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.

Keywords: Optimal control, differential games, measured-valued solutions, fish harvesting.
Mathematics Subject Classification: Primary: 35K61, 35Q93, 49J20; Secondary: 49N25, 49N90.
Citation: Giuseppe Maria Coclite, Mauro Garavello, Laura V. Spinolo. Optimal strategies for a time-dependent harvesting problem. Discrete and 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. 
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Universitext. Springer-Verlag Italia, Milan, 2008.

[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.

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. 
[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Universitext. Springer-Verlag Italia, Milan, 2008.

[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.

[1]

Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2013, 18 (7) : 1889-1907. doi: 10.3934/dcdsb.2013.18.1889
[6]

H. W. J. Lee, Kar Hung Wong, Y. C. E. Lee. Optimal control of fish-feeding in a three-dimensional calm freshwater pond considering environmental concern. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021213
[7]

R. P. Gupta, Shristi Tiwari, Shivam Saxena. The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2791-2815. doi: 10.3934/dcdsb.2021160
[8]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[9]

Alexei Korolev, Gennady Ougolnitsky. Optimal resource allocation in the difference and differential Stackelberg games on marketing networks. Journal of Dynamics and Games, 2020, 7 (2) : 141-162. doi: 10.3934/jdg.2020009
[10]

Sebastian Aniţa, Ana-Maria Moşsneagu. Optimal harvesting for age-structured population dynamics with size-dependent control. Mathematical Control and Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043
[11]

Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4869-4903. doi: 10.3934/dcdsb.2020131
[12]

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 and Applied Analysis, 2013, 12 (5) : 1927-1941. doi: 10.3934/cpaa.2013.12.1927
[13]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations and Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[14]

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
[15]

Meng Liu, Chuanzhi Bai. Optimal harvesting of a stochastic delay competitive model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1493-1508. doi: 10.3934/dcdsb.2017071
[16]

Alejandra Fonseca-Morales, Onésimo Hernández-Lerma. A note on differential games with Pareto-optimal NASH equilibria: Deterministic and stochastic models†. Journal of Dynamics and Games, 2017, 4 (3) : 195-203. doi: 10.3934/jdg.2017012
[17]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control and Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026
[18]

Monica Motta, Caterina Sartori. Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 595-613. doi: 10.3934/dcds.2011.29.595
[19]

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
[20]

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

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (215)
  • HTML views (122)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy