Optimal harvesting for a logistic model with grazing
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929210067517440 |
|---|---|
| author | Mallick, Mohan A, Ardra Sasi, Sarath |
| author_facet | Mallick, Mohan A, Ardra Sasi, Sarath |
| contents | We consider semi-linear elliptic equations of the following form: \begin{equation*} \left\{ \begin{aligned}
-Δu &= λ[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:λf_h(u), \quad && x \in Ω,
\frac{\partial u}{\partial η}&+qu = 0, \quad && x\in\partialΩ, \end{aligned} \right. \end{equation*}
where, $h\in U=\{h\in L^2(Ω): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $λ.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)=\int_Ωh(x)u_h(x)~\rm{d}x-\int_Ω(B_1+B_2 h(x))h(x)~\rm{d}x$ over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal harvesting for a logistic model with grazing Mallick, Mohan A, Ardra Sasi, Sarath Analysis of PDEs Optimization and Control We consider semi-linear elliptic equations of the following form: \begin{equation*} \left\{ \begin{aligned} -Δu &= λ[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:λf_h(u), \quad && x \in Ω, \frac{\partial u}{\partial η}&+qu = 0, \quad && x\in\partialΩ, \end{aligned} \right. \end{equation*} where, $h\in U=\{h\in L^2(Ω): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $λ.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)=\int_Ωh(x)u_h(x)~\rm{d}x-\int_Ω(B_1+B_2 h(x))h(x)~\rm{d}x$ over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system. |
| title | Optimal harvesting for a logistic model with grazing |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2401.07264 |