Abstract
This paper deals with feedback stabilization of a flexible beam clamped at a rigid body and free at the other end. We assume that there is no damping and the rigid body rotates with a nonconstant angular velocity. To stabilize this system, we propose a feedback law which consists of a control torque applied on the rigid body and either a dynamic boundary control moment or a dynamic boundary control force or both of them applied at the free end of the beam. Then it is shown that the closed loop system is well posed and exponentially stable provided that the actuators, which generate the boundary controls, satisfy some classical assumptions and the angular velocity is smaller than a critical one.
Highlights
We propose a feedback law composed of either a dynamic boundary control force or a dynamic boundary control moment applied at the free end of the beam while a control torque is present on the disk
We have proposed a feedback law which stabilizes a body-beam system in the case where the rigid body is rotating with a nonconstant angular velocity
We have shown that if the angular velocity is smaller than (1/l2) 12EI/ρ, the system is exponentially stable as soon as a control torque is applied to the rigid body and either a dynamic boundary control moment or a dynamic boundary control force or both of them act on the free end of the beam
Summary
In [20], the author considered a linear rotating body-beam subsystem, which is a reduced model of (1.1), by assuming that the angular velocity of the disk is constant, and the angular momentum equation of (1.1) is omitted In this case, the author proposed dynamic boundary controls at the free end of the beam to obtain an exponential stabilization result. The main contribution of this paper is to show that the body-beam system is exponentially stabilized by means of a control torque on the disk and dynamic boundary controls (force and/or moment) applied at the free end of the beam To prove this main result, we first consider a decoupled subsystem and use LaSalle’s principle together with Ingham’s inequality [12] to show the strong stability of the subsystem.
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