Exponential Stability of a Wave Equation with Boundary Delay Control

In this paper, we investigate the stability of a 1-d wave equation with boundary difference-type delay control. We utilize the idea of system equivalence to find a system with known stability characteristics and select an appropriate regulation mechanism, ensuring that the original system becomes equivalent to the stable one. In this method, we adopt integral-type feedback control, utilizing integral kernel functions as parameters, and determine appropriate parameter functions. The specific steps are as follows: To begin with, an exponentially convergent system is selected as the desired target reference model. Next, we construct a bounded, reversible linear mapping to equate the studied system with the target model. During this process, we derive the expressions for the integral controller and the corresponding kernel function. Subsequently, we prove the solvability of the kernel function. By establishing equations for the kernel function and linear transformations, we find that the initial system exhibits equivalence to the desired model. Ultimately, based on the equivalence between the two systems, we conclude that the original system attains exponential stability under the integral-type feedback controller.