Feedback boundary stabilization for the Hirota-Satsuma system with time-delay

This work investigates the boundary stabilization problem of the Hirota-Satsuma system. In the problem under consideration, a boundary feedback law consisting of a linear combination of a damping mechanism and a time-delay term is designed. The study shows that, with time delay feedback and a smallness restriction on the size of the initial data the energy of the Hirota-Satsuma system decays exponentially by employing two approaches: the Lyapunov method and an observability inequality combined with a contradiction argument.