In this paper, the problem of finite-time boundary stabilization of two strings connected by point mass is investigated. Based on the so-called Riemann invariant transformation, the vibrating strings are transformed in two hybridhyperbolic systems, and leads to the posedness of our system. In order to act in the system, it is desirable to choose boundary feedbacks, in this case, Hölderien stabilizing feedback laws to vanish in finite-time the right and the left of the solutions are considered.

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In this article, the problem of finite-time stabilization of two strings connected by point mass is discussed. We use the so-called Riemann coordinates to convert the study system into four transport equations coupled with the dynamic of the charge. We act by Bhat-Bernstein feedbacks in various positions (two extremities, the point mass and one of boundaries, only on the point mass,...) and we show that in some cases the nature of the stability depends sensitively on the physical parameters of the system.

In this paper, the problem of internal finite-time stabilization for 1-D coupled wave equations with interior point mass is handled. The nonlinear stabilizing feedback law leads, in closed-loop, to nonlinear evolution equations where Kato theory is used to prove the well-posedness. In addition, it is showed that in some cases, the solution of this hybrid system is constant in finite-time if we use Neumann boundary conditions. This result can be improved (in complete finite-time stability sense) if we change the above feedback.