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**Spectral operators generated by Timoshenko beam model.**
*(English)*
Zbl 0985.93012

Summary: We announce a series of results on the spectral analysis for a class of nonselfadjoint operators which are the dynamics generators for systems governed by the equations of the Timoshenko beam model with a 2-parameter family of dissipative boundary conditions. Our results split into three groups.

(1) We present asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues) and for their generalized eigenvectors.

(2) We show that these operators are Riesz spectral. This result follows from the fact that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces.

(3) We give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators.

Our results, on the one hand, provide a class of nontrivial examples of spectral operators (nonselfadjoint operators that admit an analog of spectral decomposition). On the other hand, these results give a key to the solutions of various control and stabilization problems for the Timoshenko beam model using the spectral decomposition method.

(1) We present asymptotic formulas for the spectra of the aforementioned operators (the spectrum of each operator consists of two branches of discrete complex eigenvalues) and for their generalized eigenvectors.

(2) We show that these operators are Riesz spectral. This result follows from the fact that the systems of generalized eigenvectors form Riesz bases in the corresponding energy spaces.

(3) We give the asymptotics of the spectra and the eigenfunctions for the nonselfadjoint polynomial operator pencils associated with these operators.

Our results, on the one hand, provide a class of nontrivial examples of spectral operators (nonselfadjoint operators that admit an analog of spectral decomposition). On the other hand, these results give a key to the solutions of various control and stabilization problems for the Timoshenko beam model using the spectral decomposition method.

### MSC:

93C20 | Control/observation systems governed by partial differential equations |

47F05 | General theory of partial differential operators |

35L05 | Wave equation |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

93B28 | Operator-theoretic methods |

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

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |

74H45 | Vibrations in dynamical problems in solid mechanics |

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

### Keywords:

nonselfadjoint operators; operator pencil; spectral asymptotics; generalized eigenvectors; Riesz basis in a Hilbert space; Timoshenko beam; spectral operators
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\textit{M. A. Shubov}, Syst. Control Lett. 38, No. 4--5, 249--258 (1999; Zbl 0985.93012)

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### References:

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