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Input-to-state type Stability for Simplified Fluid-Particle Interaction System

Zhuo Xu

TL;DR

The paper addresses well-posedness and input-to-state stability for a one-dimensional fluid–particle interaction with a moving free boundary, where the fluid is governed by the viscous Burgers equation and the particle position $h(t)$ evolves by Newton's law under external inputs. The authors prove global existence and uniqueness of finite-energy solutions for $u\in L^2(0,\infty)$ and establish uniform bounds on $h(t)$ when $u\in L^2(0,\infty)\cap L^1(0,\infty)$, via a Lyapunov-based energy analysis. An ISS-type estimate is derived for $\mathcal{K}>0$ using a Lyapunov functional, yielding an inequality of the form $E(t)\le 16 e^{-\eta t}E(0)+\frac{3}{2}\|u\|^2_{L^2(0,\infty)}$, thereby quantifying robustness to disturbances; for $\mathcal{K}=0$ a decay bound persists but with a rate that may depend on the input. Overall, the work extends ISS techniques to a nonlinear PDE–ODE coupling with a free boundary and provides a rigorous stability framework for open-loop fluid–structure interactions with moving interfaces.

Abstract

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.

Input-to-state type Stability for Simplified Fluid-Particle Interaction System

TL;DR

The paper addresses well-posedness and input-to-state stability for a one-dimensional fluid–particle interaction with a moving free boundary, where the fluid is governed by the viscous Burgers equation and the particle position evolves by Newton's law under external inputs. The authors prove global existence and uniqueness of finite-energy solutions for and establish uniform bounds on when , via a Lyapunov-based energy analysis. An ISS-type estimate is derived for using a Lyapunov functional, yielding an inequality of the form , thereby quantifying robustness to disturbances; for a decay bound persists but with a rate that may depend on the input. Overall, the work extends ISS techniques to a nonlinear PDE–ODE coupling with a free boundary and provides a rigorous stability framework for open-loop fluid–structure interactions with moving interfaces.

Abstract

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.
Paper Structure (5 sections, 7 theorems, 76 equations)

This paper contains 5 sections, 7 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Background on Input-to-State Stability
  3. Proof of Theorem \ref{['global_exists']}
  4. Proof of Theorem \ref{['unforimboundtheorem']}
  5. Proof of Theorem \ref{['theoreminputtostate']}

Key Result

Theorem 1.2

Let $v_0\in L^2(-1,1)$, $g_0\in\mathbb{R}$, $h_0,\, h_1\in(-1,1)$, $\mathcal{K}\geqslant 0$ and $u\in L^2(0, \infty)$, then system conversition_of_mom_of_solid_without_control–inlitial_date_of_solid_without_control admits a unique global finite energy solution as defined above. Moreover, for every $

Theorems & Definitions (15)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Proposition 3.3
  • ...and 5 more