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

Abstract

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.

Figures (11)

• Figure 1: Time evolution of uncontrolled GKdVBH equation for $\nu = 1, \alpha=1,\beta=1,\delta=1,\gamma=0.5$ and $u_{0}(x) = \sin(\pi x)$.
• Figure 2: Time evolution of uncontrolled GKdVBH equation for $\nu = 1, \alpha=1,\beta=1,\delta=2,\gamma=0.5$ and $u_{0}(x) = \sin(\pi x)$.
• Figure 3: Time evolution of uncontrolled GKdVBH equation for $\nu = 1, \alpha=1,\beta=1,\delta=3,\gamma=0.5$ and $u_{0}(x) = \sin(\pi x)$.
• Figure 4: Time evolution of controlled GKdVBH equation for $\nu = 1, \alpha=1,\beta=1,\delta=1,\gamma=0.5,\eta = 1, \mu = 0.1$ and $u_{0}(x) = \sin(\pi x)$.
• Figure 5: Time evolution of uncontrolled GKdVBH equation for $\nu = 1, \alpha=1,\beta=1,\delta=2,\gamma=0.5,\eta = 1, \mu = 0.1$ and $u_{0}(x) = \sin(\pi x)$.

Theorems & Definitions (23)

• Remark 1.1
• proof
• Remark 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof