ABSTRACT
In this chapter we announce a series of results on the asymptotic and spectral analysis of an aircraft wing model 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 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 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 the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the state space of the system equipped with the 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, we present the following results: (i) spectral properties of an analytic operator-valued function associated with the generalized resolvent operator; (ii) asymptotic distribution of the aeroelastic modes; (iii) the Riesz basis property of the system of mode shapes in the energy space.
