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Math. Comput. Appl. 2018, 23(2), 22; https://doi.org/10.3390/mca23020022

Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model

1
Faculty of Sciences, University of Oulu, FI-90014 Oulu, Finland
2
Natural Resources Institute Finland (Luke) Oulu, FI-90014 Oulu, Finland
3
Institute of Computational Mathematics & Information Technology, Kazan Federal University, 420008 Kazan, Russia
*
Author to whom correspondence should be addressed.
Received: 11 April 2018 / Revised: 23 April 2018 / Accepted: 24 April 2018 / Published: 26 April 2018
(This article belongs to the Special Issue Optimization in Control Applications)
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Abstract

We solve numerically a forest management optimization problem governed by a nonlinear partial differential equation (PDE), which is a size-structured population model. The formulated problem is supplemented with a natural constraint for a solution to be non-negative. PDE is approximated by an explicit or implicit in time finite difference scheme, whereas the cost function is taken from the very beginning in the finite-dimensional form used in practice. We prove the stability of the constructed nonlinear finite difference schemes on the set of non-negative vectors and the solvability of the formulated discrete optimal control problems. The gradient information is derived by constructing discrete adjoint state equations. The projected gradient method is used for finding the extremal points. The results of numerical testing for several real problems show good agreement with the known results and confirm the theoretical statements. View Full-Text
Keywords: size-structured population model; nonlinear partial differential equation; finite difference approximation; optimization; gradient method size-structured population model; nonlinear partial differential equation; finite difference approximation; optimization; gradient method
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Pyy, J.; Ahtikoski, A.; Lapin, A.; Laitinen, E. Solution of Optimal Harvesting Problem by Finite Difference Approximations of Size-Structured Population Model. Math. Comput. Appl. 2018, 23, 22.

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