Global Exponential Stabilization for a Simplified Fluid-Particle Interaction System

TL;DR

This paper proves global exponential stability results for a coupled fluid-particle system modeled by a viscous Burgers equation and a point mass, with and without feedback control, using Lyapunov functions.

Contribution

It establishes the first global exponential stability results for a simplified fluid-particle interaction system with feedback control.

Findings

Velocity field is globally exponentially stable in free dynamics.

With feedback control, both velocity and distance to target decay exponentially.

The proofs use a special test function and perturbed Lyapunov functions.

Abstract

This work considers a system coupling a viscous Burgers equation (aimed to describe a simplified model of $1D$ fluid flow) with the ODE describing the motion of a point mass moving inside the fluid. The point mass is possibly under the action of a feedback control. Our main contributions are that we prove two global exponential stability results. More precisely, we first show that the velocity field corresponding to the free dynamics case is globally exponentially stable. We next show that, in the presence of the feedback control both the velocity field and the distance from the mass point to a prescribed target position decay exponentially. The proofs of these results heavily rely on the use of a special test function allowing both to prove that the mass point stays away from the boundary and to construct a perturbed Lyapunov function.