Journal of Dynamical and Control Systems

, Volume 21, Issue 3, pp 475–494 | Cite as

Optimal Cyclic Exploitation of Renewable Resources

  • Anton O. BelyakovEmail author
  • Alexey A. Davydov
  • Vladimir M. Veliov


The paper contributes to the topic of optimal utilization of spatially distributed renewable resources. Namely, a problem of “sustainable” optimal cyclic exploitation of a renewable resource with logistic law of recovery is investigated. The resource is distributed on a circle and is collected by a single harvester moving along the circle. The recovering and harvesting rates are position dependent, and the latter depends also on the velocity of the harvester, which is considered as a control. The existence of an optimal solution is proved, as well as necessary optimality conditions for the velocity of the harvester. On this base, a numerical approach is proposed, and some qualitative properties of the optimal solutions are established. The results are illustrated by numerical examples, which reveal some economically meaningful features of the optimal harvesting.


Spacial heterogeneity Optimal harvesting Averaged functional 

Mathematics Subject Classification (2010)

49J15 34H05 93C10 



This research financed by a joint project of the Austrian Science Foundation (FWF) and Russian Foundation for Basic Research (RFBR) under grant No I 476-N13 and 10-01-91004 AИΦ_a, respectively .


  1. 1.
    Alekseev V M, Tikhomirov V M, Fomin S V. Optimal Control. New York: Consultants Bureau; 1987.CrossRefzbMATHGoogle Scholar
  2. 2.
    Brock W, Xepapadeas A. Diffusion-induced instability and pattern formation in infinite horizon recursive optimal control. J Econ Dyn Control. 2008;32:2745–2787.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brock W, Xepapadeas A. Pattern formation, spatial externalities and regulation in coupled economic–ecological systems. J Environ Econ Manag. 2010;59:149–164.CrossRefzbMATHGoogle Scholar
  4. 4.
    Behringer S, Upmann T. Optimal harvesting of a spatial renewable resource. J Econ Dyn Control. 2014;42:105–120.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen Ph J, Foale S J. Sustaining small-scale fisheries with periodically harvested marine reserves. Mar Policy 2013;37:278–287.CrossRefGoogle Scholar
  6. 6.
    Colonius F, Kliemann W. Infinite time optimal control and periodicity. Appl Math Optim. 1989;20(1):113–130.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Ekeland I, Temam R. Convex analysis and variational problems. Classics in Mathematics Society for Industrial and Applied Mathematics. Philadelphia; 1999.Google Scholar
  8. 8.
    Koopman B O. The theory of search. Part III. The optimal distribution of searching efforts. Oper Res. 1957;5(5):613–626.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Smith M D, Sanchirico J N, Wilen J E. The economics of spatial–dynamic processes: Applications to renewable resources. J Environ Econ Manag. 2009;57:104–121.CrossRefzbMATHGoogle Scholar
  10. 10.
    Repetto R, Gillis M. Public policies and misuse of forest resources: Cambridge University Press; 1990.Google Scholar
  11. 11.
    Rudin W. Functional analysis. NewYork: McGraw-Hill Inc; 1991.zbMATHGoogle Scholar
  12. 12.
    Stone L D. Theory of optimal search. New York: Academic Press; 1975.zbMATHGoogle Scholar
  13. 13.
    Undersander D J, Albert B, Porter P, Crossley A. Pastures for profit: a hands on guide to rotational grazing. University of Wisconsin Cooperative Extension Publication A3529, Madison. WL 1994:26.Google Scholar
  14. 14.
    Warga J. Optimal control of differential and functional equations: Academic Press; 1972.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Anton O. Belyakov
    • 1
    • 2
    Email author
  • Alexey A. Davydov
    • 3
    • 4
  • Vladimir M. Veliov
    • 1
  1. 1.Institute of Mathematical Methods in EconomicsVienna University of Technology ArgentinierstrViennaAustria
  2. 2.Institute of MechanicsLomonosov Moscow State UniversityMoscowRussia
  3. 3.Functional Analysis and Its Applications DepartmentVladimir State University named after Alexander and Nikolay StoletovsVladimirRussia
  4. 4.International Institute for Applied Systems AnalysisLaxenburgAustria

Personalised recommendations