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 for this model. In the absence of feedback control, the particle converges to an \emph{a priori} 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 acting on the particle, u(t) = K\bigl(h_1 - h(t)\bigr), \qquad K>0, and prove that both the position error $h(t)-h_1$ and the total kinetic energy of the closed-loop system decay exponentially to zero.