Controllability of tree-shaped networks of vibrating strings (Q2747426)

The aim of this very important paper is to drive the network to rest in finite time by means of an appropriate choice of the control. A tree-shaped network of vibrating strings is considered. The motion of the network is described by the system NEWLINE\[NEWLINEu_{m,tt}- u_{m,xx}= 0\qquad\text{in }[0,\ell_m]\times \mathbb{R},NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_m(x, 0)= u^0_m(x),\quad u_{m,t}(x, 0)= u^1_m(x)\qquad\text{in }[0, \ell_m],NEWLINE\]NEWLINE where \(u^0_m(x)\) and \(u^1_m(x)\) are the initial deformation and velocity of the \(m\)th string, respectively, and boundary conditions at the multiple nodes NEWLINE\[NEWLINEu_m(\varepsilon_{m,n}\ell_{m,t})= u_{m'}(\varepsilon_{m'n}\ell_{m',t}),\quad t\in \mathbb{R}\quad\text{for}\quad m,m'\in J_nNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{m\in J_n} (-1)^{\varepsilon_{m,n}} u_{m,x} (\varepsilon_{m,n}\ell_{m,t})= 0,\quad t\in\mathbb{R}\quad\text{for every }n\in J_nNEWLINE\]NEWLINE and, at the simple ones, NEWLINE\[NEWLINEu_1(0, t)= v(t),\quad u_m(\varepsilon_{m,n}\ell_{m,t})= 0,\quad t\in \mathbb{R} \quad\text{for}\quad m\geq 2\quad\text{and}\quad n\in I_s.NEWLINE\]NEWLINE The problem of controllability when the control acts on the root of the tree is analyzed. Main result: The authors give a necessary and sufficient condition for the approximate controllability of the system in time 2 \((\ell_1+\cdots+ \ell_M)\), where \(\ell_1,\dots, \ell_M\) denote the lengths of the strings of the network. This is a nondegeneracy condition guaranteeing that the spectra of any two subtrees of the network connected at a multiple node are mutually disjoint.