Optimally Controlled Moving Sets with Geographical Constraints
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929706333372416 |
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| author | Bressan, Alberto Marchini, Elsa M. Staicu, Vasile |
| author_facet | Bressan, Alberto Marchini, Elsa M. Staicu, Vasile |
| contents | The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset \R^2$ bounded by geographical barriers. If no control is applied, the contaminated set $Ω(t)\subset V$ expands with unit speed in all directions. By implementing a control, a region of area $M$ can be cleared up per unit time.
Given an initial set $Ω(0)=Ω_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapstoΩ(t)$ which eradicates the contamination in finite time, so that $Ω(T)=\emptyset$ for some $T>0$. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval $[0,T]$. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions $t\mapsto Ω(t)$ are explicitly constructed in a number of cases. \end{abstract} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05968 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimally Controlled Moving Sets with Geographical Constraints Bressan, Alberto Marchini, Elsa M. Staicu, Vasile Optimization and Control 49Q10 The paper is concerned with a family of geometric evolution problems, modeling the spatial control of an invasive population within a region $V\subset \R^2$ bounded by geographical barriers. If no control is applied, the contaminated set $Ω(t)\subset V$ expands with unit speed in all directions. By implementing a control, a region of area $M$ can be cleared up per unit time. Given an initial set $Ω(0)=Ω_0\subseteq V$, three main problems are studied: (1) Existence of an admissible strategy $t\mapstoΩ(t)$ which eradicates the contamination in finite time, so that $Ω(T)=\emptyset$ for some $T>0$. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval $[0,T]$. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions $t\mapsto Ω(t)$ are explicitly constructed in a number of cases. \end{abstract} |
| title | Optimally Controlled Moving Sets with Geographical Constraints |
| topic | Optimization and Control 49Q10 |
| url | https://arxiv.org/abs/2502.05968 |