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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.

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