Spectral analysis of Timoshenko beam with time delay in interior damping

Abstract

In this paper, we study the spectrum of the Timoshenko beam with time delay in interior damping. At first, we prove the well-posedness of the delayed equations by the semigroup theory. To obtain the detailed spectral information of operator $$\\mathcal {A}$$A determined by the equations, we decompose $$\\mathcal {A}$$A into a sequence of unbounded operators $$\\{\\Lambda _n, n\\in \\mathbb {N}\\}$${Λn,n∈N} in an appropriate Hilbert space, and then we give the spectrum of each operator $$\\Lambda _n$$Λn, including its asymptotic value of the spectrum. The result shows that $$\\sigma (\\Lambda _n)$$σ(Λn) consists of all isolated eigenvalues of $$\\Lambda _n$$Λn of multiplicity one, and distributes symmetrically with respect to the real axis. Furthermore, $$\\sigma (\\mathcal {A})=\\overline{\\bigcup _{n=1}\\sigma (\\Lambda _n)}$$σ(A)=⋃n=1σ(Λn)¯, and there is a sequence of real eigenvalues of $$\\mathcal {A}$$A that diverges to negative infinity. In each strip paralleling to the imaginary axis, there are infinitely many eigenvalues of $$\\mathcal {A}$$A.