Uniform Stabilization of 1-d Wave Equation with Anti-damping and Delayed Control

Abstract In this paper, we consider the uniform stabilization problem of a 1-d wave equation with variable coefficients, anti-damping and delayed boundary control. We design a new kind of state feedback controller to stabilize the system exponentially. The designed controller is taken as the integral form, whose kernel functions will be regarded as the selectable parameters. Our goal is to show that one can select appropriate parameter functions so that the closed-loop system is exponentially stable. Herein we mainly give an approach of selecting parameter functions, including the differential equations satisfied the kernel functions and initial conditions. To show the exponential stability of the closed-loop system, as a trick, we construct some function transformations and establish the equivalence between the closed-loop system and a known stable system. As a result, the designed controller eliminates the negative effects of time-delay in input and avoids the traditional complicated stability analysis of the closed-loop system. Finally, a numerical simulation of a 1-d wave equation with variable coefficients and delay input is carried out. The result demonstrates the effectiveness of the presented control law.