波网络系统的适定性和正则性

Well-posedness and Regularity of Wave Network Equations

研究了一波网络系统,在此网络的两个外部节点处施加控制并放置同位观测器,通过选择合适状态空间,将系统方程转化为抽象线性系统的形式.利用能量乘子法验证观测算子是允许的,通过对偶性原理得到控制算子也是允许的.求出传递函数后并验证传递函数在某个右半平面一致有界,从而说明系统是适定的,再求出传递函数在实轴上正方向的无穷远处的极限,找到直接传输算子,进而说明系统的正则性.

In this paper, we discuss a wave network, in which the controls are imposed on two external vertexes with two collocated observations. By choosing the suitable state space, the system equation is transformed into an abstract linear system. The admissbility of observation operator is obtained by the method of energy multiplier and the admissbility of control operator is also obtained by the dual principle. The transfer function of the system is derived and its uniform bounded-ness in some right half-plane implies the wellposed-ness of the system. The limit of the transfer function at the positive infinity of real axis and the feedthrough operator are further got, which shows that the system is regular.