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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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