A Cantilevered Extensible Beam in Axial Flow: Semigroup Well-posedness and Postflutter Regimes

Abstract

Asymptotic-in-time feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [E. Dowell, AIAA, 5 (1967), pp. 1857--1862] 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 [I. Chueshov, I. Lasiecka, and J. T. Webster, Comm. Partial Differential Equations, 39 (2014), pp. 1965--1997]. 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. To accomplish this task, a novel decomposition of the nonlinear plate dynamics is utilized: a smooth component (globally bounded in a higher topology) and a uniformly exponentially decaying component. 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 [I. Chueshov and I. Lasiecka, Springer Mongr. Math., Springer-Verlag, Berlin, 2010; I. Lasiecka and J. T. Webster, Comm. Pure Appl. Math., 13 (2014), pp. 1935--1969; I. Ryzhkova, J. Math. Anal. Appl., 294 (2004), pp. 462--481; I. Ryzhkova, Z. Angew. Math. Phys., 58 (2007), pp. 246--261], this is the first treatment which does not incorporate smoothing effects for the structure.