Volume 11, Issue 6
Analysis of Finite Difference Approximations of an Optimal Control Problem in Economics

Alexander Lapin, Shuhua Zhang, Sergey Lapin & Na Yan

Adv. Appl. Math. Mech., 11 (2019), pp. 1358-1375.

Published online: 2019-09

Preview Full PDF 212 3593
Export citation
  • Abstract

We consider an optimal control problem which serves as a mathematical model for several problems in economics and management. The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part. The additional constraint is non-negativity of a solution of state equation. We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time. All these approximations keep the positivity of the solutions to mesh state problem, either unconditionally or under some additional constraints to mesh steps. This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation. Based on theoretical estimates and numerical results, we draw conclusions about the quality of the proposed mesh schemes.

  • Keywords

Mean field game, optimal control problem, parabolic diffusion-advection equation, finite difference methods.

  • AMS Subject Headings

65M06, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{AAMM-11-1358, author = {Lapin , Alexander and Shuhua Zhang , and Sergey Lapin , and Na Yan , }, title = {Analysis of Finite Difference Approximations of an Optimal Control Problem in Economics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {11}, number = {6}, pages = {1358--1375}, abstract = {

We consider an optimal control problem which serves as a mathematical model for several problems in economics and management. The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part. The additional constraint is non-negativity of a solution of state equation. We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time. All these approximations keep the positivity of the solutions to mesh state problem, either unconditionally or under some additional constraints to mesh steps. This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation. Based on theoretical estimates and numerical results, we draw conclusions about the quality of the proposed mesh schemes.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0186}, url = {http://global-sci.org/intro/article_detail/aamm/13307.html} }
TY - JOUR T1 - Analysis of Finite Difference Approximations of an Optimal Control Problem in Economics AU - Lapin , Alexander AU - Shuhua Zhang , AU - Sergey Lapin , AU - Na Yan , JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1358 EP - 1375 PY - 2019 DA - 2019/09 SN - 11 DO - http://doi.org/10.4208/aamm.OA-2018-0186 UR - https://global-sci.org/intro/article_detail/aamm/13307.html KW - Mean field game, optimal control problem, parabolic diffusion-advection equation, finite difference methods. AB -

We consider an optimal control problem which serves as a mathematical model for several problems in economics and management. The problem is the minimization of a continuous constrained functional governed by a linear parabolic diffusion-advection equation controlled in a coefficient in advection part. The additional constraint is non-negativity of a solution of state equation. We construct and analyze several mesh schemes approximating the formulated problem using finite difference methods in space and in time. All these approximations keep the positivity of the solutions to mesh state problem, either unconditionally or under some additional constraints to mesh steps. This allows us to remove corresponding constraint from the formulation of the discrete problem to simplify its implementation. Based on theoretical estimates and numerical results, we draw conclusions about the quality of the proposed mesh schemes.

Alexander Lapin, Shuhua Zhang, Sergey Lapin & Na Yan. (2019). Analysis of Finite Difference Approximations of an Optimal Control Problem in Economics. Advances in Applied Mathematics and Mechanics. 11 (6). 1358-1375. doi:10.4208/aamm.OA-2018-0186
Copy to clipboard
The citation has been copied to your clipboard