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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 & 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

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