Global Exponential Stabilization for a Simplified Fluid–Particle Interaction System

In this talk, we present a stabilization result for a simplified one-dimensional fluid-particle interaction system. First, without any smallness assumptions, we establish a non-collision property: the particle never reaches the fluid boundary in finite time, which in turn yields global well-posedness of the interaction system. Next, we study a stabilization problem of this model. In the absence of feedback control, the particle converges to an unknown limit position $h^\ast$ that cannot be described solely by the initial data. To regulate the final position of the particle to an arbitrary target $h_1\in(-1,1)$, we employ a proportional feedback control on the particle, $u(t)=K(h_1-h(t))$ with K>0, and prove that both the position error $h(t)-h_1$ and the total kinetic energy of the feedback system decay exponentially to zero. Finally, we show that the proposed feedback is robust with respect to external disturbances.