三角形回路弦网络系统控制器设计及指数稳定

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

研究了一个由3根等长质量均匀的弹性弦构成三角形连接的网络系统,它在一个顶点p1处张力连续而位移不连续,在其他2个顶点位移连续而张力不连续.通过在网络结点处设计控制器形成一个闭环系统.利用半群理论证明了这个闭环系统的适定性.通过算子谱分析,证明了系统的谱由孤立的有限重本征值构成,并且当连接p1的2根弦波速之比不等于它们的质量密度之比的倒数时,系统的谱分布在左半复平面平行于虚轴的一个带域内.并证明了系统算子的广义本征向量构成状态空间的Riesz基,从而系统满足谱确定增长条件.于是闭环系统至少是渐近稳定的,并且当任意2根弦的波速之比均为有理数时,系统可达到指数稳定.

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, P1 ,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 P1 is 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.