An optimal control problem without control costs
A two-dimensional diffusion process is controlled until it enters a given subset of $ \mathbb{R}^2 $. The aim is to find the control that minimizes the expected value of a cost function in which there are no control costs. The optimal control can be expressed in terms of the value function, which gi...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-01-01
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| Series: | Mathematical Biosciences and Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2023239?viewType=HTML |
| _version_ | 1811170584485167104 |
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| author | Mario Lefebvre |
| author_facet | Mario Lefebvre |
| author_sort | Mario Lefebvre |
| collection | DOAJ |
| description | A two-dimensional diffusion process is controlled until it enters a given subset of $ \mathbb{R}^2 $. The aim is to find the control that minimizes the expected value of a cost function in which there are no control costs. The optimal control can be expressed in terms of the value function, which gives the smallest value that the expected cost can take. To obtain the value function, one can make use of dynamic programming to find the differential equation it satisfies. This differential equation is a non-linear second-order partial differential equation. We find explicit solutions to this non-linear equation, subject to the appropriate boundary conditions, in important particular cases. The method of similarity solutions is used. |
| first_indexed | 2024-04-10T16:59:03Z |
| format | Article |
| id | doaj.art-e111729d87d0498ca02dbdf575c6c1e3 |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-10T16:59:03Z |
| publishDate | 2023-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-e111729d87d0498ca02dbdf575c6c1e32023-02-07T01:19:33ZengAIMS PressMathematical Biosciences and Engineering1551-00182023-01-012035159516810.3934/mbe.2023239An optimal control problem without control costsMario Lefebvre 0Department of Mathematics and Industrial Engineering, Polytechnique Montréal, C.P. 6079, Succursale Centre-ville, Montréal, H3C 3A7, CanadaA two-dimensional diffusion process is controlled until it enters a given subset of $ \mathbb{R}^2 $. The aim is to find the control that minimizes the expected value of a cost function in which there are no control costs. The optimal control can be expressed in terms of the value function, which gives the smallest value that the expected cost can take. To obtain the value function, one can make use of dynamic programming to find the differential equation it satisfies. This differential equation is a non-linear second-order partial differential equation. We find explicit solutions to this non-linear equation, subject to the appropriate boundary conditions, in important particular cases. The method of similarity solutions is used.https://www.aimspress.com/article/doi/10.3934/mbe.2023239?viewType=HTMLstochastic optimal controldiffusion processesfirst-passage timedynamic programmingpartial differential equation |
| spellingShingle | Mario Lefebvre An optimal control problem without control costs Mathematical Biosciences and Engineering stochastic optimal control diffusion processes first-passage time dynamic programming partial differential equation |
| title | An optimal control problem without control costs |
| title_full | An optimal control problem without control costs |
| title_fullStr | An optimal control problem without control costs |
| title_full_unstemmed | An optimal control problem without control costs |
| title_short | An optimal control problem without control costs |
| title_sort | optimal control problem without control costs |
| topic | stochastic optimal control diffusion processes first-passage time dynamic programming partial differential equation |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2023239?viewType=HTML |
| work_keys_str_mv | AT mariolefebvre anoptimalcontrolproblemwithoutcontrolcosts AT mariolefebvre optimalcontrolproblemwithoutcontrolcosts |