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Riesz basis approach to the stabilization of a flexible beam with a tip mass. (English) Zbl 1183.93111

Summary: Using an abstract condition of Riesz basis generation of discrete operators in the Hilbert spaces, we show, in this paper, that a sequence of generalized eigenfunctions of an Euler-Bernoulli beam equation with a tip mass under boundary linear feedback control forms a Riesz basis for the state Hilbert space. In the meanwhile, an asymptotic expression of eigenvalues and the exponential stability are readily obtained. The main results of F. Conrad, Ö. Morgül [SIAM J. Control Optimization 36, No. 6, 1962–1986 (1998; Zbl 0927.93031)] are concluded as a special case, and the additional conditions imposed there are removed.

MSC:

93D15 Stabilization of systems by feedback
35B35 Stability in context of PDEs
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
93C20 Control/observation systems governed by partial differential equations

Citations:

Zbl 0927.93031
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