Constrained approximate null controllability of coupled heat equation with periodic impulse controls

This paper is concerned with the constrained approximate null controllability of heat equation coupled by a real matrix $P$, where the controls are impulsive and periodically acted into the system through a series of real matrices $\{Q_k\}_{k=1}^\hbar$. The conclusions are given in two cases. In the case that the controls act globally into the system, we prove that the system is global constrained approximate null controllable under a spectral condition of $P$ together with a rank condition of $P$ and $\{Q_k\}_{k=1}^\hbar$; While in the case that the controls act locally into the system, we prove the global constrained approximate null controllability under a stronger condition for $P$ and the same rank condition as the above case. Moreover, we prove that the above mentioned spectral condition of $P$ is necessary for global constrained approximate null controllability of the control problem considered in this paper.