带有相同输入函数双系统的同时近似可控性

SIMULTANEOUS APPROXIMATE CONTROLLABILITY OF DOUBLE SYSTEMS WITH A COMMON INPUT FUNCTION

研究了一类具有相同输入函数抽象双线性系统的同时近似可控性.在此,我们考虑双系统都是无穷维的而且其中一个为Riesz-Spectral系统,证明了如果两个系统在T_0时刻都是精确可控的而且系统的生成元没有共同特征值,那么对于任何时刻T>T_0这两个系统是同时近似可控的.此外,对于特殊的控制算子,如果系统(A_1,B_1)在时刻T_0是近似可控的而系统(A_2,B_2)在时刻T_0是精确可控的,并且生成元算子A_1和A_2的谱集满足σ(A_2)ρ_∞(A_1),那么一定存在某个时刻T>0,使得双系统在时刻T是同时近似可控的.最后,给出定理的一些应用以及说明同时近似可控时刻T是最优的.

This paper is concerned with the approximate controllability of two systems by means of a common input function. In the paper, the two systems are considered to be infinitedimensional and one of them to be a Riesz-Spectral system. In this case, it is shown that if both systems are exactly controllable in time To and the system generators have no common eigenvalues, then they are simultaneously approximately controllable in any time T 〉 To. In addition, for special control operators, if one system （A1, B1） is approximately controllable and the other（A2, B2） is exactly controllable in time To, and the spectral sets of the two system generators satisfy the condition σ（A2） ρ∞（A1）, then they are simultaneously approximately controllable in some time T 〉 0. Finally, some applications of the obtained results are given and it is proved that the time at which the systems are simultaneously approximately controllable is optimal.