Asymptotics of aeroelastic modes and basis property of mode shapes for aircraft wing model

In this paper, we announce our recent results on the asymptotic and spectral analysis of the model of an aircraft wing in a subsonic air flow. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. We provide the spectral asymptotics for the eigenfrequencies of the system (or aeroelastic modes) and the asymptotical approximations for the corresponding eigenfunctions (or the mode shapes). Based on the asymptotical results, we (a) state that the set of the mode shapes is complete in the energy space; (b) construct a system which is biorthogonal to the set of the mode shapes in the case when there might be multiple aeroelastic modes; and (c) show that the mode shapes form a Riesz basis in the energy space.