Non-Collocated Control of Coupled Strings with Different Wave Speeds and Joint Anti-Damping Terms

In this paper, the stabilization of coupled wave equations with different speeds and anti-damping term at each joint points is studied. By designing invertible transformations, the wave system is converted into an equivalent system with both dissipative linkage and 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 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.