Asymptotic analysis of aircraft wing model in subsonic airflow

In this paper we announce a series of results on the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow and provide a brief outline of the proofs of these results. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. 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 unknown functions (the bending and torsion angle) depend on time and one spatial variable. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution type equation in the state space of the system equipped with the so-called energy metric. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator. The generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. The poles of the generalized resolvent are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In this paper, our main object of interest is the dynamics generator of the differential parts of the system, which is a nonselfadjoint operator in the state space with a purely discrete spectrum. We show that this operator has two branches of discrete spectrum, present precise asymptotic formulas for both branches, and outline the main steps of their derivation. The full proof will be given in another work. Based on these results in the subsequent papers, we will derive the asymptotics of the aeroelastic modes and approximations for the mode shapes.