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On the stabilizability of a slowly rotating Timoshenko beam. (English) Zbl 0971.93038
The aim of this very important and useful paper is to study a slowly rotating Timoshenko beam in a horizontal plane whose movement is controlled by the angular acceleration of the disk of a driving motor into which the beam is clamped. The problem of strong stabilizability is characterized by: (the linear model) \left.\begin{aligned}\ddot w(x,t) &- w''(x,t)- \xi'(x, t)= -\ddot\theta(t)(r+ x)\\ \ddot\xi(x,t) &- \xi''(x, t)+ \xi(x,t)+ w'(x,t)= \ddot\theta(t)\end{aligned}\right\} (x\in (0,1),t> 0)\tag{1} the boundary conditions: \left.\begin{aligned} w(0,t)= \xi(0,t) &= 0\\ w'(1,t)+ \xi(1,t) & =0\\ \xi(1,t) &=0\end{aligned}\right\} (t\geq 0)\tag{2} ($$w(x,t)$$ – the deflection of the center line of the beam, $$x\in [0,1]$$, time $$t\geq 0$$, $$\xi(x,t)$$ – the rotation angle of the cross section area at $$x$$ and $$t$$, $$\dot w= w_t$$, $$\dot\xi= \xi_t$$, $$w'= w_x$$, $$\xi'= \xi_x$$, $$\theta$$ – the rotation angle of the motor disk $$\dot\theta= d\theta/dt$$).
The authors consider the following problem of strong stability: find a linear continuous functional: $$p(w,\dot w,\xi,\dot\xi, \theta,\dot\theta)$$ such that every solution of problem (1), (2) with feedback control: $$\ddot\theta(t)= p(w(.,t), \dot w(.,t),\xi(.,t), \dot\xi(.,t),\theta(t),\dot\theta(t))$$ tends to $$0$$ when $$t\to+\infty$$ in the following sense: $\int^1_0 w'(x,t)^2 dx\to 0,\quad \int^1_0 \xi'(x,t)^2 dx\to 0,\quad \int^1_0 \dot w(x,t)^2 dx\to 0,$
$\int^1_0\xi(x, t)^2 dx\to 0,\quad \theta(t)\to 0,\quad \dot\theta(t)\to 0\quad\text{as}\quad t\to+\infty.$ Main result: The authors present a precise proof for choosing a feedback control allowing to stabilize a given system (the beam plus the disk) in a preassigned position of rest.

MSC:
 93C20 Control/observation systems governed by partial differential equations 93D15 Stabilization of systems by feedback 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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References:
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