A biological resource is a population characterized by birth, aging and death, grown in order to produce a profit. The evolution of this system is described by a structured population model, modified to take into account the selection for reproduction or for the market. This selection is the control that has to be optimized in order to maximize the profit. First we prove the well posedness of the descriptive model. Then, the profit is shown to be Gâteaux differentiable with respect to the controls. Finally, we ensure that the maximal profit can be reached by means of Bang–Bang controls.

Accepted:

DOI: 10.1051/cocv/2016027

Keywords: Conservation Laws, optimal control problems, management of biological resources, structured population dynamics

^{1}; Garavello, Mauro

^{2}

@article{COCV_2017__23_3_1073_0, author = {Colombo, Rinaldo M. and Garavello, Mauro}, title = {Control of biological resources on graphs}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1073--1097}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016027}, mrnumber = {3660460}, zbl = {1372.35306}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016027/} }

TY - JOUR AU - Colombo, Rinaldo M. AU - Garavello, Mauro TI - Control of biological resources on graphs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1073 EP - 1097 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016027/ DO - 10.1051/cocv/2016027 LA - en ID - COCV_2017__23_3_1073_0 ER -

%0 Journal Article %A Colombo, Rinaldo M. %A Garavello, Mauro %T Control of biological resources on graphs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1073-1097 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016027/ %R 10.1051/cocv/2016027 %G en %F COCV_2017__23_3_1073_0

Colombo, Rinaldo M.; Garavello, Mauro. Control of biological resources on graphs. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 1073-1097. doi : 10.1051/cocv/2016027. http://archive.numdam.org/articles/10.1051/cocv/2016027/

A nonautonomous juvenile-adult model: well-posedness and long-time behavior via a comparison principle. SIAM J. Appl. Math. 69 (2009) 1644–1661. | DOI | MR | Zbl

and ,Sensitivity analysis for a structured juvenile–adult model. Comput. Math. Appl. 64 (2012) 190–200. | MR | Zbl

, and ,S. Anita, Vol. 11 of Analysis and control of age-dependent population dynamics. Springer Science & Business Media (2000). | MR | Zbl

S. Anita, V. Capasso and V. Arnautu, An introduction to optimal control problems in life sciences and economics. Springer (2011). | MR | Zbl

On the “bang-bang” control problem. Quart. Appl. Math. 14 (1956) 11–18. | DOI | MR | Zbl

, and ,Constant versus periodic fishing: age structured optimal control approach. Math. Model. Nat. Phenom. 9 (2014) 20–37. | DOI | MR | Zbl

and ,A. Bressan and B. Piccoli, Introduction to the mathematical theory of control. Vol. 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2007). | MR | Zbl

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York (2011). | MR | Zbl

Optimal geometric control applied to the protein misfolding cyclic amplification process. Acta Appl. Math. 135 (2015) 145–173. | DOI | MR | Zbl

, , , , , and ,Stability and optimization in structured population models on graphs. Math. Biosci. Eng. 12 (2015) 311–335. | DOI | MR | Zbl

and ,Optimization of an amplification protocol for misfolded proteins by using relaxed control. J. Math. Biol. 70 (2015) 289–327. | DOI | MR | Zbl

, and ,G. Dal Maso, An introduction to $\Gamma $-convergence. Vol. 8 of Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston, Inc., Boston, MA (1993). | MR | Zbl

M. De Lara and L. Doyen, Sustainable Management of Natural Resources. Springer-Verlag, Berlin, Heidelberg (2008).

On some questions in the theory of optimal regulation: existence of a solution of the problem of optimal regulation in the class of bounded measurable functions. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959) 25–32. | MR | Zbl

,Optimal control in renewable resources modeling. Bull. Braz. Math. Soc., New Ser. 47 (2016) 347–357. | DOI | MR | Zbl

,Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance. Numer. Methods Partial Differ. Eq. 30 (2014) 1797–1820. | DOI | MR | Zbl

, , and ,N. Hritonenko and Y. Yatsenko, Mathematical modeling in economics, ecology and the environment. Vol. 88 of Springer Optimization and Its Applications. Second edition of the 1999 original. Springer, New York (2013). | MR | Zbl

R.J. LeVeque, Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). | MR | Zbl

J.D. Murray, Mathematical biology. I. Vol. 17 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York (2002). 3rd edition. An introduction. | MR | Zbl

B. Perthame, Transport equations in biology. Frontiers in Mathematics. Birkhauser Verlag, Basel (2007). | MR | Zbl

L.S. Pontrjagin, V.G. Boltjanskij, R.V. Gamkrelidze and E.F. Misčenko, Mathematische Theorie optimaler Prozesse. R. Oldenbourg, Munich-Vienna (1964). | MR | Zbl

*Cited by Sources: *