Input-to-state type stability for simplified fluid–particle interaction system

In this paper, we study the well-posedness and the input-to-state type stability of a one-dimensional fluid–particle interaction system. A distinctive feature, not yet considered in the input-to-state stability literature, is that our system involves a free boundary. More precisely, the fluid is described by the viscous Burgers equation and the motion of particle obeys Newton’s second law. The point mass is subject to both a feedback control and an open-loop control. We first establish the well-posedness of the system for any open-loop input in L²(0,∞ ). Then, assuming the input also belongs to L¹(0,∞ ), we prove that the particle’s position remains uniformly bounded and that the system is input-to-state type stable. The proof is based on the construction of a Lyapunov functional derived from a special test function.