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On the self-stabilization of damped flexible structures. (English) Zbl 0755.49007
Stabilization of flexible structures, Proc. ComCon Workshop, Montpellier/Fr. 1987, 273-281 (1988).
[For the entire collection see Zbl 0745.00028.] The objective of this note is to describe some results concerning the self stabilization of small wavelengths oscillations in damped flexible structures, thus supplementing our results [“Infinite-dimensional dynamical systems in mechanics and physics” (1988; Zbl 0662.35001)] with some more specific and quantitative results. In particular we decompose the spectrum of natural (linear) oscillations of the system into small and large wave lengths. We write the equations governing the interactions between these two groups of oscillations. Finally we show that the small wavelengths oscillations tend exponentially fast to a small (negligible) value without the need of a control mechanism on these modes. Hence, as indicated above, the control mechanism is only necessary for a finite number of modes, the large wavelengths group. In particular this opens the possibility of applying to such systems the methods of finite dimensional control.
49K20Optimal control problems with PDE (optimality conditions)
93D15Stabilization of systems by feedback
70J25Stability of linear oscillatory motions