We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping. © 2010 Societ y for Industrial and Applied Mathematics.
Smyshlyaev, A., Cerpa, E., & Krstic, M. (2010). Boundary stabilization of a 1-D wave equation with in-domain antidamping. SIAM Journal on Control and Optimization, 4014-4031. https://doi.org/10.1137/080742646