Stabilization of a hybrid system of elasticity by feedback boundary damping. (English) Zbl 0664.73025
The authors consider the controlled vibration of a uniform beam to which a rigid body is attached at one end. The problem they discuss is to find linear feedback controls guaranteeing uniform asymptotic stability of strong solutions. The hybrid-type system can be reformulated as an Euler- Bernoulli beam equation with one end damped and the other under control. The boundary conditions at the controlled end, where the rigid body is attached to, are, however, not of the common form. First a semigroup generation result is given for feedback system. Then arguments based on the \(\omega\)-limit sets are given to ensure the trend to equilibrium. Furthermore, it is shown that the most natural absorbing boundary conditions do not lead to uniform exponential decay of the energy. As a matter of fact, the decay can be shown to be arbitrarily slow.
[This is a continuation of the authors’ earlier article in: Arch. Ration. Mech. Anal. 103, No.3, 193-236 (1988; Zbl 0656.73029).]
[This is a continuation of the authors’ earlier article in: Arch. Ration. Mech. Anal. 103, No.3, 193-236 (1988; Zbl 0656.73029).]
Reviewer: G.Leugering
MSC:
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 93D15 | Stabilization of systems by feedback |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 70Q05 | Control of mechanical systems |
| 74E30 | Composite and mixture properties |
Keywords:
large-scale space-structure; long flexible mast; rigid antenna; closed- loop controllers; controlled vibration; uniform beam; rigid body is attached; linear feedback controls; uniform asymptotic stability; strong solutions; Euler-Bernoulli beam equation; one end damped; controlled end; semigroup generation; absorbing boundary conditions; decay; arbitrarily slowCitations:
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