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Riesz basis property and exponential stability of controlled Euler-Bernoulli beam equations with variable coefficients. (English) Zbl 1015.93025

The Riesz basis is generally used to express properties of a system in terms of eigenvector decomposition. In beam theory, even following a uniform Euler-Bernoulli beam model, this is not quite routine, but a reasonable analysis of control theory exists provided that the coefficients are constant functions (this means that we do have a uniform beam). This allows some researchers to use units such that \(\rho= EI= 1\), thus equating all coefficients to one. The corresponding dynamic version of the Euler-Bernoulli equation is: \(y_{tt}(x, t)+ y_{xxxx}(x,t)= 0\), \(0 < x < 1\), \(t > 0\), with boundary conditions: \(y(0,t)= y_x(0, t)= 0\), with the moment and sheer feedback: \(y_{xx}(1,t)=-k_1 y_{xt}(1,t)\), \(k_1\geq 0\), \(y_{xxx}(1,t)= -k_2 y_t(1, t)\), \(k_2\geq 0\). The author extends this type of study to a beam with variable coefficients, that is to a nonuniform beam. In this case the specific shape, or variation of other properties of the beam, will affect both the characteristic equation and the description of the eigenfunctions.
It is well known that the Euler-Bernoulli system with the indicated feedback is dissipative in the Hilbert space with energy norm, and it can be rewritten in the standard evolutionary form: \({d\over dt} Y(t)= AY(t)\) where \(A^{-1}\) exists and is a compact operator. Thus the spectrum of \(A\) consists of isolated eigenvalues only, the eigenvalues appearing in conjugate complex pairs.
The author shows how the asymptotic distribution of eigenvalues, spectrum growth, and exponential stability are determined. Now they add viscous damping replacing the operator \(A\) by \(A+B\), and obtain an analog of the uniform beam case.
The reviewer comments that the first-order differential term for viscous friction is not realistic according to several engineering studies. Fractional-order differential operators have been introduced in engineering publications, better modelling the observed internal dissipative processes in steel beams.

MSC:

93C20 Control/observation systems governed by partial differential equations
93B60 Eigenvalue problems
93D15 Stabilization of systems by feedback
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
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