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**Riesz basis property of root vectors of non-self-adjoint operators generated by aircraft wing model in subsonic airflow.**
*(English)*
Zbl 0972.47039

Author’s abstract: This paper is the third in a series of several works devoted to the asymptotic and spectral analysis of a 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. 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. This generalized resolvent operator is a finite-meromorphic function 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 these poles are the projectors on the generalized eigenspaces. In the first two papers [see “Mathematical analysis of problem arising in modelling of flutter phenomenon in aircraft wing in subsonic airflow”, IMA J. Appl. Math., submitted for publication; and “Asymptotic representations for root vectors of nonselfadjoint operators and pencils generated by aircraft wing model in subsonic air flow”, J. Math. Anal. Appl., submitted for publication] and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. This generator is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches, and have derived their precise spectral asymptotics with respect to the eigenvalue number. In the second paper, we have derived the asymptotical approximations for the mode shapes. Based on the asymptotical results of the first two papers, in the present paper, we

(a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space,

(b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when there might be not only eigenvectors but associate vectors as well; and

(c) prove that the set of the generalized eigenvectors forms a Riesz basis in the energy space.

To prove the main result of the paper, we made use of the Nagy-Foias functional model for nonselfadjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform in the forthcoming papers.

(a) prove that the set of the generalized eigenvectors of the aforementioned differential operator is complete in the energy space,

(b) construct the set of vectors which is biorthogonal to the set of the generalized eigenvectors in the case when there might be not only eigenvectors but associate vectors as well; and

(c) prove that the set of the generalized eigenvectors forms a Riesz basis in the energy space.

To prove the main result of the paper, we made use of the Nagy-Foias functional model for nonselfadjoint operators. The results of all three papers will be important for the reconstruction of the solution of the original initial-boundary-value problem from its Laplace transform in the forthcoming papers.

Reviewer: D.Przeworska-Rolewicz (Warszawa)

### MSC:

47G20 | Integro-differential operators |

74J10 | Bulk waves in solid mechanics |

34K05 | General theory of functional-differential equations |

34K30 | Functional-differential equations in abstract spaces |

47A45 | Canonical models for contractions and nonselfadjoint linear operators |

47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |

### Keywords:

flutter; nonselfadjoint differential operator; nonselfadjoint polynomial pencil; discrete spectrum; completeness; asymptotic and spectral analysis; aircraft wing; subsonic air flow; system of two coupled integro-differential equations; two-parameter family of boundary conditions; coupled linear hyperbolic system; single operator evolution-convolution equation in the energy space; Laplace transform; generalized resolvent operator; finite-meromorphic function; dynamics generator of the differential parts; generalized eigenvectors; Riesz basis in the energy space; Nagy-Foias functional model for nonselfadjoint operators; initial-boundary-value problem
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\textit{M. A. Shubov}, Math. Methods Appl. Sci. 23, No. 18, 1585--1615 (2000; Zbl 0972.47039)

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