设计了两类控制器同时作用于椭圆-抛物系统：一类是传统的内部控制(分布控制器); 另一类是以一个时变系数形式描述的内部控制(块控制器, $\kappa (t)$只是和时间有关, 而与空间位置$x$没有关系, 所以称块控制器, 是相对于分布控制而言的). 由于这里讨论的系统是非线性的, 也只能得到全局近似能控性, 不能得到精确零能控性. 设计这样的控制器的目的是为了克服抛物型偏微分方程描述的系统只在内部控制器驱动下只是全局近似能控的, 采用渐近分析技巧将双线性控制的作用充分分割、组合, 最终得到椭圆-抛物系统的全局近似能控性.

This paper studies the global approximate controllability of the parabolic-elliptic system, which has caught a great deal of attention, governed in a bounded interval by a pair of controls:  i) the traditional internal either locally distributed or lumped control and  ii) the lumped control entering the system as a time-dependent coefficient. The motivation for the latter is due to the well-known lack of global null-controllability properties for the class of systems governed by the parabolic partial differential equations when they are only steered by the former controls. However, only approximate controllability can be obtained because of the nonlinearity of the parabolic-elliptic system. This approach involves an asymptotic technique allowing us to separate and combine the impacts generated by the above-mentioned two types of controls.