THE EXPONENTIAL STABILITY OF COMPLEX DIFFERENTIAL NETWORKS
复杂微分网络的指数稳定性

In this paper, a tree-shaped network of strings equipped with velocity feedback controllers is considered, and the exponential stability of the closed-loop system is obtained. First, it is shown that the system is well-posed by using the $C_0$ semigroup method. Then it is proven that the spectra of the system locate in a vertical strip under certain conditions by an asymptotic analysis technique, and that there is a sequence of root vectors that forms a Riesz basis with parentheses for the state Hilbert space. Hence the system satisfies the spectrum determined growth assumption. Then, the Riesz basis property together with the distribution of the spectra of ${\mathcal A}$ asserts that the system is exponentially stable. Finally, a new concept --- the efficiency of the controllers is proposed, with which, the efficiency of controllers at each node of the network is compared and the least number of controllers to make the system exponentially stable is given.