Article Contents
Article Contents

# Simultaneous observability of infinitely many strings and beams

• * Corresponding author: Paola Loreti
The first author was supported by the Visiting Professor Programme, Sapienza Università di Roma
• We investigate the simultaneous observability of infinite systems of vibrating strings or beams having a common endpoint where the observation is taking place. Our results are new even for finite systems because we allow the vibrations to take place in independent directions. Our main tool is a vectorial generalization of some classical theorems of Ingham, Beurling and Kahane in nonharmonic analysis.

Mathematics Subject Classification: Primary: 93B07; Secondary: 35L05, 74K10, 42A99.

 Citation:

• Figure 1.  A system of three strings with vibration planes $\sf{p_j}$ spanned by $d_j: = (\ell_j,\phi_j,\theta_j)$ and $v_j\perp d_j$, $j = 1,2,3$. In (i) $\ell_1 = \ell_2 = \ell_3 = 1$, the $v_j$'s are pairwise orthogonal, and $v_1 = d_3$, $v_2 = d_1$, $v_3 = d_2$. We have $T_0 = 2\max\left\lbrace {\ell_1,\ell_2,\ell_3}\right\rbrace = 2$. In (ii), we have $\ell_1 = \ell_3 = 1$, $\ell_2 = 2/(2+\sqrt{2})$ and $v_1 = d_3\perp d_1 = v_2 = v_3$. Then $T_0 = 2\max\left\lbrace {\ell_1+\ell_2,\ell_3}\right\rbrace\approx 3.1715$. In the planar case (iii) we have $\ell_1 = 1$, $\ell_2 = 2/(2+\sqrt{2})$ and $\ell_3 = 2/(4+\sqrt{2})$, so that $T_0 = 2\max\left\lbrace {\ell_1+\ell_2+\ell_3}\right\rbrace\approx 3.9103$

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