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**Asymptotic and spectral properties of operator-valued functions generated by aircraft wing model.**
*(English)*
Zbl 1128.74312

Summary: The present paper is devoted to the asymptotic and spectral analysis of an aircraft wing model in a subsonic air flow. The model is governed by a system of two coupled integro-differential equations and a two parameter family of boundary conditions modelling 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. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter.

More precisely, the generalized resolvent is a finite-meromorphic function on the complex plane having a branch-cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotic approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space.

In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.

More precisely, the generalized resolvent is a finite-meromorphic function on the complex plane having a branch-cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. The dynamics generator of the differential part of the system has been systematically studied in a series of works by the second author. This generator is a non-selfadjoint operator in the energy space with a purely discrete spectrum. In the aforementioned series of papers, it has been shown that the set of aeroelastic modes is asymptotically close to the spectrum of the dynamics generator, that this spectrum consists of two branches, and a precise spectral asymptotics with respect to the eigenvalue number has been derived. The asymptotic approximations for the mode shapes have also been obtained. It has also been proven that the set of the generalized eigenvectors of the dynamics generator forms a Riesz basis in the energy space.

In the present paper, we consider the entire integro-differential system which governs the model. Namely, we investigate the properties of the integral convolution-type part of the original system. We show, in particular, that the set of poles of the adjoint generalized resolvent is asymptotically close to the discrete spectrum of the operator that is adjoint to the dynamics generator corresponding to the differential part. The results of this paper will be important for the reconstruction of the solution of the original initial boundary-value problem from its Laplace transform and for the analysis of the flutter phenomenon in the forthcoming work.

### MSC:

74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |

47G20 | Integro-differential operators |

34K10 | Boundary value problems for functional-differential equations |

34L99 | Ordinary differential operators |

45J05 | Integro-ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |