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On the solvability of trigonometric moment problems arising in the problem of controllability of rotating beams. (English) Zbl 1028.93005

Hoffmann, K.-H. (ed.) et al., Optimal control of complex structures. Proceedings of the international conference, Oberwolfach, Germany, June 4-10, 2000. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 139, 145-156 (2002).
The very important and useful paper is concerned with the problem of controllability of a slowly rotating Timoshenko beam in a horizontal plane from the position of rest into an arbitrary position at some given time. The equations of motion in a dimension-free formulation are given by \[ \begin{cases}\ddot w(x,t)- w''(x,t)-\xi'(x,t)= -\ddot\theta (t)(r+x)\\ \ddot\xi(x,t)-\gamma^2\xi'' (x,t)+\xi (x,t)+w'(x,t)= \ddot\theta(t) \end{cases}\tag{1} \] with the boundary conditions \[ \begin{cases} w(0,t)=\xi(0,t) =0\\ w'(1,t)+\xi(1,t)= \xi'(1,t)=0 \quad\text{for }t\geq 0\end{cases} \tag{2} \] and the initial conditions, \[ \begin{cases} w(x,0)= \dot w(x,0)=\xi (x,0)= \dot\xi (x,0)=0,\;x\in[0,1]\\ \theta(0)= \theta' (0)= 0 \end{cases}\tag{3} \] The problem of controllability is solved with the aid of a theorem by D. Ullrich on a trigonometric moment problem which generalized a classical result of Paley and Wiener.
The main goal of this paper is to prove the existence of a weak solution to the problem of controllability (1), (2). The authors propose the precise proof of the solvability for the initial-boundary value problem.
For the entire collection see [Zbl 0997.00022].

MSC:

93B05 Controllability
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74M05 Control, switches and devices (“smart materials”) in solid mechanics
42A70 Trigonometric moment problems in one variable harmonic analysis
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