## Riesz basis property of Timoshenko beams with boundary feedback control.(English)Zbl 1017.35065

This paper concerns a Timoshenko beam model with a tip mass, i.e. a system of two linear second-order hyperbolic equations with dynamic boundary conditions. This system generates a $$C_0$$-semigroup in an appropriate Hilbert space. The generator of this semigroup is dissipative, has compact resolvent, almost all of its eigenvalues are simple, and the system of its eigenfunctions and generalized eigenfunctions forms a Riesz basis in the Hilbert space. Hence, the rate of the exponential decay of the semigroup is determined by the upper bound of the real parts of the spectrum of the generator.

### MSC:

 35L55 Higher-order hyperbolic systems 47D06 One-parameter semigroups and linear evolution equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces 47B44 Linear accretive operators, dissipative operators, etc. 35L20 Initial-boundary value problems for second-order hyperbolic equations 93B52 Feedback control 93C20 Control/observation systems governed by partial differential equations

### Keywords:

$$C_0$$-semigroups; exponential decay
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