Boundary stabilization of coupled wave system with spatially-varying coefficients and internal anti-damping

In this paper, we consider the exponential stabilization of coupled wave equations with spatially-varying coefficients and internal anti-damping. In contrast to the previous work in the literature, the kernel equations of this system are much more complicated. Using the backstepping method, we verify the kernel equations, which is a system of coupled linear wave equations with spatially-varying coefficients. Then by designing a new characteristic line, the well-posedness of the kernel equations is obtained. Finally we use a Lyapunov function to get the exponential stabilization. A numerical example is presented to illustrate the result.