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On se propose d’étudier la stabilité d’une poutre flexible homogène, encastrée à une extrémité. À l’autre extrémité est attachée une masse ponctuelle où on applique un moment proportionnel à la vitesse de déplacement angulaire. On montre par une analyse spectrale que le taux optimal de décroissance de l’énergie est déterminé par l’abscisse spectrale du générateur infinitésimal du semi-groupe associé au problème.
We study the stability of a flexible beam clamped at one end. A
mass is attached at the other end, where a control moment is
applied. The boundary control is proportional to the angular velocity
at the end. By spectral analysis, we prove that the optimal decay rate
of the energy is given by the spectrum of the generator of the
semigroup associated to the system.
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