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

TL;DR

This paper develops a boundary feedback control method with time-delay to stabilize the Hirota-Satsuma system, proving exponential energy decay under certain initial conditions using Lyapunov and observability techniques.

Contribution

It introduces a novel boundary feedback law incorporating time-delay for the Hirota-Satsuma system and demonstrates exponential stabilization with small initial data.

Findings

Exponential decay of system energy achieved

Boundary feedback law with damping and delay effective

Stability proven using Lyapunov and observability methods

Abstract

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.