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.