观测器具有时滞的树形弦网络的适定性

Well- posedness of the tree- shaped networkwith delays in observers

主要研究的是树形波网络系统,该系统是由三根不同的弦组成.首先,通过选择适当空间,对网络的两个外部节点处施加控制和放置同位观测器,且观测器带有时滞.将系统方程转化为抽象线性系统的形式,利用对偶性原理和能量乘子法分别验证控制算子和观测算子都是允许的.再求出直接传输算子,说明系统的正则性和适定性.最后,利用所得结果还得到了观测有时滞的系统的适定性.

We discuss the tree- shaped network. This network has three different strings. At first,By choosing the suitable state space,in which the controls are imposed on two external vertexes with collocated observers.The system equation is transformed into an abstract linear system. By the methods of energy multiplier and the dual principle,the admissbility of observation operator and control operator are obtained. Then,the feedthrough operator is further got,which shows that the system is regular. At last,using the results in this note,the wellposed- ness of the system with delays in observers is also obtained.