arXiv 2402.027762024-02-05math.APmath.OC

广义Korteweg-de Vries-Burgers-Huxley方程的边界控制：适定性、镇定与数值研究

Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies

Manil T. Mohan, Shri Lal Raghudev Ram Singh

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AI总结：

本文针对广义Korteweg-de Vries-Burgers-Huxley方程，利用单调性论证和多个定理建立Neumann边值问题的适定性，并通过反馈边界控制证明闭环系统的L²和H¹稳定性，数值实验验证了理论结果。

摘要：

本文研究如下广义Korteweg-de Vries-Burgers-Huxley方程的边界控制问题：$$u_t=νu_{xx}-μu_{xxx}-αu^δu_x+βu(1-u^δ)(u^δ-γ), \u2009x\in[0,1],\\,t>0,$$其中$ν,μ,α,β>0,$ $δ\in[1,\infty)$，$γ\in(0,1)$，并受Neumann边界条件约束。我们首先通过应用单调性论证、Hartman-Stampacchia定理、Minty-Browder定理和Crandall-Liggett定理，建立了Neumann边值问题的适定性。通过证明Minty-Browder定理的一个适当版本，成功处理了三阶线性项带来的额外困难。利用合适的反馈边界控制，我们证明了对于充分大的$ν>0$，闭环系统具有$\mathrm{L}^2$-和$\mathrm{H}^1$-稳定性。本文的分析结论得到了数值研究的支持和验证。

英文摘要：

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.

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