Backstepping Control of a Wave PDE With Unstable Source Terms and Dynamic Boundary

This letter presents the design of an exponentially stabilizing controller for a one-dimensional wave partial differential equation. The control is acting on a Robin's boundary condition while the opposite boundary satisfies an unstable dynamic. The wave is also subject to unstable in-domain source terms. Closed-loop exponential stabilization is obtained via a full-state backstepping controller. The existence and uniqueness of this backstepping transformation is proven, using the method of successive approximations.