Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow

TL;DR

This paper demonstrates that strong feedback control can stabilize a fluttering panel in inviscid subsonic flow, leading to convergence to a stationary state without smoothing effects, by establishing the existence of a smooth exponential attractor.

Contribution

It proves the stabilization of a flow-plate system using feedback control without relying on structural smoothing effects, a novel result in the field.

Findings

Flow-plate system converges to a stationary set under feedback control.

Existence of a smooth exponential attractor for the system.

Flutter can be eliminated with sufficient structural damping.

Abstract

Asymptotic-in-time feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [22] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of imposed energy dissipation the plate dynamics converge to a compact and finite dimensional set [16,17]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the stationary set in the natural energy topology. In doing so, we demonstrate the existence of an exponential attractor for the plate dynamics. That the exponential attractor exhibits additional smoothness is the technical crux of the main result. This property cannot be taken for granted, as exponential attractors are often compact but not necessarily smooth (in contrast with global maximal attractors). Our result implies that flutter (a periodic or chaotic end behavior) can be eliminated (in subsonic flows) with sufficient frictional damping in the structure. While such a result has been proved in the past for regularized plate models (with rotational inertia terms or thermal considerations [14,33,38,39]), this is the first treatment which does not incorporate smoothing effects for the structure.