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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.02776 |
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| _version_ | 1866913223115014144 |
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| author | Mohan, Manil T. Singh, Shri Lal Raghudev Ram |
| author_facet | Mohan, Manil T. Singh, Shri Lal Raghudev Ram |
| contents | A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=νu_{xx}-μu_{xxx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), \ x\in[0,1], \ t>0,$$ where $ν,μ,α,β>0,$ $δ\in[1,\infty)$, $γ\in(0,1)$ subject to Neumann boundary conditions is considered in this work. We first establish the well-posedness of the Neumann boundary value problem by an application of monotonicity arguments, the Hartman-Stampacchia theorem, the Minty-Browder theorem, and the Crandall-Liggett theorem. The additional difficulties caused by the third order linear term is successfully handled by proving a proper version of the Minty-Browder theorem. By using suitable feedback boundary controls, we demonstrate $\mathrm{L}^2$- and $\mathrm{H}^1$-stability properties of the closed-loop system for sufficiently large $ν>0$. The analytical conclusions from this work are supported and validated by numerical investigations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies Mohan, Manil T. Singh, Shri Lal Raghudev Ram Analysis of PDEs Optimization and Control A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=νu_{xx}-μu_{xxx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), \ x\in[0,1], \ t>0,$$ where $ν,μ,α,β>0,$ $δ\in[1,\infty)$, $γ\in(0,1)$ subject to Neumann boundary conditions is considered in this work. We first establish the well-posedness of the Neumann boundary value problem by an application of monotonicity arguments, the Hartman-Stampacchia theorem, the Minty-Browder theorem, and the Crandall-Liggett theorem. The additional difficulties caused by the third order linear term is successfully handled by proving a proper version of the Minty-Browder theorem. By using suitable feedback boundary controls, we demonstrate $\mathrm{L}^2$- and $\mathrm{H}^1$-stability properties of the closed-loop system for sufficiently large $ν>0$. The analytical conclusions from this work are supported and validated by numerical investigations. |
| title | Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2402.02776 |