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 ISS literature, is that our system involves a free boundary. More precisely, the fluid is described by the viscous Burgers equation, and the motion of the particle obeys Newton 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 the L2(0, infinity) space. Assuming the input also belongs to the L1(0,infinity) space, 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.