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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.01521 |
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| _version_ | 1866909765331845120 |
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| author | Buttazzo, Giuseppe Casado-Díaz, Juan Maestre, Faustino |
| author_facet | Buttazzo, Giuseppe Casado-Díaz, Juan Maestre, Faustino |
| contents | We investigate optimal control problems governed by the elliptic partial differential equation $-Δu=f$ subject to Dirichlet boundary conditions on a given domain $Ω$. The control variable in this setting is the right-hand side $f$, and the objective is to minimize a cost functional that depends simultaneously on the control $f$ and on the associated state function $u$.
We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form $α\le f\leβ$ are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control $f$ assumes only the extremal values $α$ and $β$. More precisely, the control takes the form $f=\alpha1_E+\beta1_{Ω\setminus E}$, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets $E$.
Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal sources for elliptic PDEs Buttazzo, Giuseppe Casado-Díaz, Juan Maestre, Faustino Optimization and Control 49Q10 We investigate optimal control problems governed by the elliptic partial differential equation $-Δu=f$ subject to Dirichlet boundary conditions on a given domain $Ω$. The control variable in this setting is the right-hand side $f$, and the objective is to minimize a cost functional that depends simultaneously on the control $f$ and on the associated state function $u$. We establish the existence of optimal controls and analyze their qualitative properties by deriving necessary conditions for optimality. In particular, when pointwise constraints of the form $α\le f\leβ$ are imposed a priori on the control, we examine situations where a {\it bang-bang} phenomenon arises, that is where the optimal control $f$ assumes only the extremal values $α$ and $β$. More precisely, the control takes the form $f=\alpha1_E+\beta1_{Ω\setminus E}$, thereby placing the problem within the framework of shape optimization. Under suitable assumptions, we further establish certain regularity properties for the optimal sets $E$. Finally, in the last part of the paper, we present numerical simulations that illustrate our theoretical findings through a selection of representative examples. |
| title | Optimal sources for elliptic PDEs |
| topic | Optimization and Control 49Q10 |
| url | https://arxiv.org/abs/2509.01521 |