Hierarchical exact controllability for a parabolic equation with Hardy potential
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912580543447040 |
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| author | Lin, Haiyang You, Bo |
| author_facet | Lin, Haiyang You, Bo |
| contents | The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hierarchical exact controllability for a parabolic equation with Hardy potential Lin, Haiyang You, Bo Optimization and Control Analysis of PDEs The main objective of this paper is to study the hierarchical exact controllability for a parabolic equation with Hardy potential by Stackelberg-Nash strategy. In linear case, we employ Lax-Milgram theorem to prove the existence of an associated Nash equilibrium pair corresponding to a bi-objective optimal control problem for each leader, which is responsible for an exact controllability property. Then the observability inequality of a coupled parabolic system is established by using global Carleman inequalities, which results in the existence of a leader that drives the controlled system exactly to any prescribed trajectory. In semilinear case, we first prove the well-posedness of the coupled parabolic system to obtain the existence of Nash quasi-equilibrium pair and show that Nash quasi-equilibrium is equivalent to Nash equilibrium. Based on these results, we establish the existence of a leader that drives the controlled system exactly to a prescribed (but arbitrary) trajectory by Leray-Schauder fixed point theorem. |
| title | Hierarchical exact controllability for a parabolic equation with Hardy potential |
| topic | Optimization and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2509.08471 |