[Paper Review] Well-posedness and exponential decay estimates for a Korteweg-de Vries-Burgers equation with time-delay

Vilmos Komornik, Cristina Pignotti|arXiv (Cornell University)|Jun 6, 2019

Stability and Controllability of Differential Equations22 references19 citations

TL;DR

This paper establishes well-posedness and exponential decay for a Korteweg–de Vries–Burgers equation with time-delayed feedback using a Lyapunov functional approach and semigroup theory. It derives explicit conditions on the damping coefficients $λ_0$ and $λ$—including $\|\beta + \beta_0\|_p < \left(\frac{\alpha_0 - \alpha}{c_p}\right)^{1 - \frac{1}{2p}}$—that guarantee exponential stability, even when the undelayed damping is indefinite, extending beyond prior smallness assumptions.

ABSTRACT

We consider the KdV-Burgers equation and its linear version in presence of a delay feedback. We prove well-posedness of the models and exponential decay estimates under appropriate conditions on the damping coefficients. Our arguments rely on a Lyapunov functional approach combined with a step by step procedure and semigroup theory.

Motivation & Objective

• To analyze the well-posedness and long-term behavior of the Korteweg–de Vries–Burgers equation with time-delayed feedback.
• To overcome limitations of prior methods that required smallness assumptions on the delay coefficient $\lambda$.
• To establish exponential decay estimates for both linear and nonlinear versions of the time-delayed KdV–Burgers equation.
• To generalize stability results to cases where the undelayed damping coefficient $\lambda_0$ is indefinite, not necessarily bounded below by a positive constant.
• To provide explicit, quantitative conditions on $\lambda_0$ and $\lambda$ ensuring exponential stability despite the destabilizing potential of time delay.

Proposed method

• A Lyapunov functional approach is used to derive energy estimates for the system, combining time-delayed and undelayed damping terms.
• A step-by-step procedure is applied to prove well-posedness in the space $C([- au, +∞); L^2(\mathbb{R}))$ for initial data in $C([- au,0]; L^2(\mathbb{R}))$.
• Semigroup theory is employed to establish the generation of a strongly continuous semigroup for the linearized model without delay.
• The energy functional $\mathcal{E}(t)$ is defined as the sum of $\|u(t)\|_{L^2}^2$ and $\|u_x(t)\|_{L^2}^2$, which is used to track decay.
• Hölder and Young inequalities are applied to control the nonlinear and delayed terms in the energy derivative, particularly $\|\beta + \beta_0\|_p$ and $\|u\|_{L^{2q}}^2$.
• A critical condition $\|\beta + \beta_0\|_p < \left(\frac{\alpha_0 - \alpha}{c_p}\right)^{1 - \frac{1}{2p}}$ is derived to ensure exponential decay, where $c_p$ is a Sobolev embedding constant.

Experimental results

Research questions

• RQ1 Under what conditions on the damping coefficients $\lambda_0$ and $\lambda$ does the time-delayed KdV–Burgers equation admit a unique global solution?
• RQ2 Can exponential decay of the $L^2$-norm be guaranteed for the linearized model with time delay, even when the undelayed damping is indefinite?
• RQ3 How can the destabilizing effect of time delay be compensated to ensure stability, and what are the explicit bounds on the feedback coefficients?
• RQ4 Can the stability results be extended from the linear to the nonlinear KdV–Burgers equation with time-delayed feedback?
• RQ5 What is the precise quantitative decay rate $\tilde{\gamma}$ in terms of the coefficients $\lambda_0$, $\lambda$, and the $L^p$-norm of the delay influence?

Key findings

• The problem (1.2) is well-posed in $C([- au, +∞); L^2(\mathbb{R}))$ for initial data in $C([- au,0]; L^2(\mathbb{R}))$ when $\lambda_0, \lambda \in L^\infty(\mathbb{R})$, via a step-by-step semigroup argument.
• Exponential decay of the energy $\mathcal{E}(t)$ is established under the condition $\|\beta + \beta_0\|_p < \left(\frac{\alpha_0 - \alpha}{c_p}\right)^{1 - \frac{1}{2p}}$, where $\alpha_0 > \alpha \geq 0$, ensuring $\tilde{\gamma} > 0$.
• The decay rate is given explicitly by $\tilde{\gamma} = \min\left\{2\left(\alpha_0 - \alpha - \frac{2p-1}{2p}\left(\frac{2}{p}\right)^{\frac{1}{2p-1}}\|\beta + \beta_0\|_p^{\frac{2p}{2p-1}}\right), 1\right\}$, with $p > 1$.
• The energy estimate $\mathcal{E}(t) \leq C(u_0)e^{-\tilde{\gamma}t}$ holds for all $t \geq 0$, with $C(u_0)$ depending on the initial data.
• The results extend to the nonlinear model (1.1), where the same exponential decay estimate holds under the same conditions on $\lambda_0$ and $\lambda$, via a mild solution framework.
• The method avoids smallness assumptions on $\|\lambda\|_\infty$, distinguishing it from prior works that required such constraints.