Abstract. 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. In this paper we are concerned with the stabilization problem of a 1-D (onedimensional) wave equation with an internal destabilizing term. Because of this term, the system is antistable in the sense that the eigenvalues of the open-loop system can all be in the right half of the complex plane, which produces an exponential growth of the norm of the solutions. The objective in this paper is to design a boundary control law to stabilize the system. We consider a wave equation on a unit interval as its spatial domain and assume the availability of a Dirichlet actuator at the right end point of the unit interval. We assume that the string is pinned (homogeneous Dirichlet boundary condition) at the left end point. Our approach is based on the backstepping method which uses a Volterra transformation to map an unstable system into a stable "target" PDE (partial differential equation). This method allows us to achieve an arbitrarily large exponential decay rate for the closed-loop system. In the framework of infinite-dimensional systems, the backstepping method has been mainly used for parabolic and first-order hyperbolic equations [14] , [17] , [18] .
The most usual boundary condition is of the Dirichlet type, as considered in REF where the exponential stabilization problem for a wave PDE with an in-domain spacedependent term has been solved.
Boundary Stabilization of a 1-D Wave Equation with In-Domain Antidamping
SIAM J. Control and Optimization
SIAM J. Control and Optimization
Mathematics Computer Science