Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929609135620096 |
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| author | Christof, Constantin |
| author_facet | Christof, Constantin |
| contents | We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fréchet differentiable even though the differential operator contains a nondifferentiable term. We exploit this effect to establish first-order necessary optimality conditions for minimizers of the considered control problems. The resulting KKT-conditions take the form of coupled PDE-systems that are posed in non-Muckenhoupt weighted Sobolev spaces and raise interesting questions regarding the regularity of optimal controls, the derivation of second-order optimality conditions, and the analysis of finite element discretizations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_03110 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities Christof, Constantin Optimization and Control 49J20, 49K20, 49K40 We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fréchet differentiable even though the differential operator contains a nondifferentiable term. We exploit this effect to establish first-order necessary optimality conditions for minimizers of the considered control problems. The resulting KKT-conditions take the form of coupled PDE-systems that are posed in non-Muckenhoupt weighted Sobolev spaces and raise interesting questions regarding the regularity of optimal controls, the derivation of second-order optimality conditions, and the analysis of finite element discretizations. |
| title | Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities |
| topic | Optimization and Control 49J20, 49K20, 49K40 |
| url | https://arxiv.org/abs/2406.03110 |