Delay is Necessary for a Potential to Achieve Exponential Stabilization of the Wave Equation via Internal Control

Abstract

In this work, we study the stabilization of the wave equation using an internal delayed potential. Interestingly, the stabilization mechanism is entirely induced by the delay, since exponential stabilization cannot be achieved in its absence. We first prove the well-posedness of the associated initial--boundary value problem. Then, thanks to the parametric analysis of the corresponding quasipolynomial, we design a delayed po tential feedback law which, together with appropriate initial conditions, ensures the exponential decay rate for the resulting closed-loop system. The control of the transverse vibration of a string illustrates the effectiveness of the result.

Figures (6)

• Figure 1: Regions of roots in the right half plane on the $\left(\tilde{\beta},\tilde{\alpha}\right)$ parameters plane. The numbers in each region denote the number of roots with positive real parts.
• Figure 2: Without control.
• Figure 3: Closed-loop: case 1 ($\tau=\frac{3}{2}$, $\alpha=5$, $k=1$).
• Figure 4: Closed-loop : case 2 ($\tau=\frac{3}{2}$, $\alpha=3$, $k=1$).
• Figure 5: Closed-loop: case 3 ($\tau=\frac{5}{2}$, $\alpha=-1.7766$, $k=2$).

Theorems & Definitions (14)

• Theorem 1
• proof
• Remark 2
• Lemma 3
• Remark 4
• Lemma 5
• Remark 6
• Remark 7
• Theorem 8
• Remark 9