SIMULTANEOUS APPROXIMATE CONTROLLABILITY OF DOUBLE SYSTEMS WITH A COMMON INPUT FUNCTION
带有相同输入函数双系统的同时近似可控性

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 infinite-dimensional 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 $T_0$ and the system generators have no common eigenvalues, then they are simultaneously approximately controllable in any time $T>T_0$. In addition, for special control operators, if one system $(A_1,B_1)$ is approximately controllable and the other$(A_2,B_2)$ is exactly controllable in time $T_0$, and the spectral sets of the two system generators satisfy the condition $\sigma(A_2)\subset \rho_{\infty}(A_1)$, then they are simultaneously approximately controllablein 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.