An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909253929795584 |
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| author | Santos, Mauricio C. Carreño, Nicolás Morales, Roberto |
| author_facet | Santos, Mauricio C. Carreño, Nicolás Morales, Roberto |
| contents | In this article, we study the existence of insensitizing controls for a nonlinear reaction-diffusion equation with dynamic boundary conditions. Here, we have a partially unknown data of the system, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. More precisely, the functional considered here depends on the norm of the state in a subset of the bulk together with the norm of the tangential gradient of the state on the boundary. This problem is equivalent to a (relaxed) null controllability problem for an optimality system of cascade type, with a zeroth-order coupling term in the bulk and a second-order coupling term on the boundary. To achieve this result, we linearize the system around the origin and analyze it by the duality approach and we prove a new Carleman estimate for the corresponding adjoint system. Then, a local null controllability result for the nonlinear system is proven by using an inverse function theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions Santos, Mauricio C. Carreño, Nicolás Morales, Roberto Optimization and Control Analysis of PDEs In this article, we study the existence of insensitizing controls for a nonlinear reaction-diffusion equation with dynamic boundary conditions. Here, we have a partially unknown data of the system, and the problem consists in finding controls such that a specific functional is insensitive for small perturbations of the initial data. More precisely, the functional considered here depends on the norm of the state in a subset of the bulk together with the norm of the tangential gradient of the state on the boundary. This problem is equivalent to a (relaxed) null controllability problem for an optimality system of cascade type, with a zeroth-order coupling term in the bulk and a second-order coupling term on the boundary. To achieve this result, we linearize the system around the origin and analyze it by the duality approach and we prove a new Carleman estimate for the corresponding adjoint system. Then, a local null controllability result for the nonlinear system is proven by using an inverse function theorem. |
| title | An Insensitizing control problem involving tangential gradient terms for a reaction-diffusion equation with dynamic boundary conditions |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2407.09882 |