Stabilization for a star-shaped network of strings with joint anti-damping and non-collocated observation

This paper addresses the stabilization problem of a star-shaped network of strings with joint anti-damping, where each string may have different wave speeds and lengths. By designing invertible transformations, the wave system is converted into an equivalent system with conservative linkage. We design feedback controllers with collocated and non-collocated observations to cancel the instabilities at the control ends and obtain the target system. Through the Riesz basis method and the PDE approach, the well-posedness and exponential stability of the non-dissipative closed-loop system are established. The effectiveness of the proposed feedback control law is verified by numerical simulations.