Sur quelques problèmes de stabilisation de systèmes à paramètres distribués

Abstract : This work is devoted to feedback stabilization of some distributed parameters systems. The first part deals with the boundary feedback stabilization of a rotating body-beam system. This one consists of a disk with a beam attached to its center and perpendicular to its plan. We propose a local boundary feedback laws, (force and/or moment applied on the free end of the beam, and a torque control applied on the disk). We show that these feedback laws exponentially stabilize the system so that the beam vibrations are suppressed and the whole structure rotates about the axis with a given, sufficiently small, angular velocity. We illustrate this results by some numericals simulations. In the second part, we study the problem of boundary feedback stabilization of a hybrid system. This one is composed of an elastic beam linked to a rigid antenna. The vibrations of this system are governed by a partial differential equation and two ordinary differential equations. In our model, the moment of inertia of the antenna in neglected. Our purpose is to stabilize this system by only a bending moment feedback. We show that this can be achieved by using a simple feedback law. Finally, in the third part, we study the spectre assignment, by bounded linear feedback, for two examples of distributed parameters systems. For the wave equation, we show that the best stability result achievable by continuous feedback is strong stability. For the cantilever beam equation, we show that it is possible to assign the spectrum uniformly by continuous feedback, this garantees the exponential stabilization of the system with an arbitrary decay rate
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Hassan Laousy. Sur quelques problèmes de stabilisation de systèmes à paramètres distribués. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1997. Français. ⟨NNT : 1997METZ056S⟩. ⟨tel-01777234⟩

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