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Stabilization of a rotating body beam without damping. (English) Zbl 0908.93055
The beam is attached perpendicularly to the center of a disk and moves in a plane perpendicular to the disk and rotating with it. This system is described by the equation \[ \rho {\partial^2 u(t, x) \over \partial t^2} + EI{\partial^4 u(t, x) \over \partial x^4} + \rho B{\partial u(t, x) \over \partial t} = \rho \omega(t)^2 u(t, x) \] in the interval \(0 \leq x \leq L =\) length of the beam, where \(u(t, x)\) is the beam’s displacement from equilibrium in the rotating plane, \(\omega(t) = \dot \theta(t)\) is the angular velocity, \(B\) represents damping and \(\rho, EI\) are the mass (resp. the flexural rigidity) per unit length of the beam. The boundary conditions are \[ u(t, 0) = {\partial u(t, 0) \over \partial x} = 0, \qquad {\partial^2 u(t, L) \over \partial x^2} = {\partial^3 u(t, L) \over \partial x^3} = 0 \] (the beam is free at the right endpoint). A torque control \(\Gamma(t)\) is exerted on the disk through \[ {d \over dt} \bigg \{ \omega(t) \bigg( I_d + \rho \int_0^L u(t, x)^2 dx \bigg) \bigg\} = \Gamma(t), \] where \(I_d\) is the disk’s moment of inertia. The goal is to stabilize the beam when \(B = 0\) (no damping) with no control on the free boundary. Stabilization is effected by means of a nonlinear feedback torque control law. The equilibrium point \((0, \omega)\) can be globally asymptotically stabilized if \(| \omega| < \omega_c,\) where \(\omega_c\) is a critical angular velocity depending on the various parameters of the system. This generalizes previous results that assume damping.

MSC:
93D15 Stabilization of systems by feedback
92C20 Neural biology
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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