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Stabilization of a nonlinear flow-plate interaction via component-wise decomposition. (English) Zbl 1355.74036

Summary: Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [E. Dowell, “Nonlinear oscillations of a fluttering plate. I.”, AIAA J. 4, No. 7, 1267–1275 (1966; doi:10.2514/3.3658); “Nonlinear oscillations of a fluttering plate. II.”, AIAA J. 5, No. 10, 1857–1862 (1967; doi:10.2514/3.4316)] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [I. Chueshov et al., Commun. Partial Differ. Equations 39, No. 11, 1965–1997 (2014; Zbl 1299.74053); Discrete Contin. Dyn. Syst., Ser. S 7, No. 5, 925–965 (2014; Zbl 1304.35114)]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.

MSC:

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
35M32 Boundary value problems for mixed-type systems of PDEs
74K20 Plates
76G25 General aerodynamics and subsonic flows
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