In this talk, we present two results on fluid-solid interaction systems.

First, we discuss an infinite time horizon LQR optimal control problem for a system modeling the vertical oscillations of a floating solid coupled with a free boundary fluid, where the fluid fills the whole space. Using analytic semigroup theory and interpolation space theorem, we get the well-posedness result of the open loop interaction system. Applying the standard LQR theorem for infinite-dimensional systems, we prove there exists a unique optimal control for the LQR problem, moreover, the optimal control can be written by feedback form.

Second, we discuss global exponential stabilization for a simplified fluid-particle interaction system, the fluid is governed by a viscous Burgers equation (representing 1D fluid flow) and particle motion obeys Newton’s second law. We design a PD control on the particle and prove global well-posedness results for the closed-loop system. Specifically, we show that the fluid, particle velocities and the distant from particle to the target point are globally exponentially stable. Our proofs rely on a special test function.

This works joint work with Prof. Marius Tucsnak, University of Bordeaux.