Design of Controllers and Exponential Stability of a Triangle Loop Strings Network System

A triangle loop strings network system is conceived,which consists of three elastic strings with the same length and uniform quality density. Its displacement is discontinuous and the tensile force is continuous on a vertex,p 1,and on the other two vertexes,the displacement is continuous and the tensile force is discontinuous. A closed loop system is established by the controllers placed on the three nodes. Then its well-posedness is proved by the semi-group theory. From the spectral analysis,it is shown that the spectrum of the system is composed of isolated finite multiplicity eigenvalues and is located in a strip parallel to the imaginary axis in the left half complex plane if the ratio of the velocities of wave propagation of the two strings connected on p 1is not the reciprocal of the density ratio of the two strings. Hence,the generalized eigenvectors of the system operator form the Riesz basis of the state space and the spectrum-determined growth condition holds. So the system is at least asymptotically stable,and can attain the exponential stability if all ratios of the velocities of wave propagation of any two strings are rational.