Asymptotic representations for root vectors of nonselfadjoint operators and pencils generated by an aircraft wing model in subsonic air flow

This paper is the second in a series of several works devoted to the asympt otic and spectral analysis 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 A. V. Balakrishnan. The model is governed by a system of two coupled integrodifferential. equations and a two parameter family of boundary conditions modeling the action of the self-straining act uators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The sys tem of equations of motion is equivalent to a single operator evolution-con volution equation in the energy space. The Laplace transform of the solutio n of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral p arameter. This generalized resolvent operator is a finite-meromorphic funct ion on the complex plane having the branch cut along the negative real semi -axis. Its poles are precisely the aeroelastic modes and the residues at th ese poles are the projectors on the generalized eigenspaces. In the first p aper and in the present one, our main object of interest is the dynamics ge nerator of the differential parts of the system. It is a nonselfadjoint ope rator in the energy space with a purely discrete spectrum. In the first pap er, we have shown that the spectrum consists of two branches and have deriv ed their precise spectral asymptotics. In the present paper, we derive the asymptotical. approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, a nd the Riesz basis property in the energy space. (C) 2001 Academic Press.