Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914053555748864 |
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| author | Límaco, Juan Barreira, João Carlos Silva, Suerlan Yapu, Luis P. |
| author_facet | Límaco, Juan Barreira, João Carlos Silva, Suerlan Yapu, Luis P. |
| contents | In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called \textit{leader} and two controls called \textit{followers}. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, Límaco and Viana \cite{DemarqueLimacoViana_deg_sys2020} and the local controllability of the non-linear system is obtained using Liusternik's inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19505 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient Límaco, Juan Barreira, João Carlos Silva, Suerlan Yapu, Luis P. Optimization and Control Analysis of PDEs In this paper we use a Stackelberg-Nash strategy to show the local null controllability of a parabolic equation where the diffusion coefficient is the product of a degenerate function in space and a nonlocal term. We consider one control called \textit{leader} and two controls called \textit{followers}. To each leader we associate a Nash equilibrium corresponding to a bi-objective optimal control problem; then, we find a leader that solves the null controllability problem. The linearized degenerated system is treated adapting Carleman estimates for degenerated systems from Demarque, Límaco and Viana \cite{DemarqueLimacoViana_deg_sys2020} and the local controllability of the non-linear system is obtained using Liusternik's inverse function theorem. The nonlocal coefficient originates a multiplicative coupling in the optimality system that gives rise to interesting calculations in the applications of the inverse function theorem. |
| title | Hierarchical null controllability of a degenerate parabolic equation with nonlocal coefficient |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2509.19505 |