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This paper is concerned with the constrained approximate null controllabilityof heat equation coupled by a real matrix $P$, where the controls are impulsiveand 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 thatthe controls act globally into the system, we prove that the system is globalconstrained approximate null controllable under a spectral condition of $P$together with a rank condition of $P$ and $\{Q_k\}_{k=1}^\hbar$; While in thecase that the controls act locally into the system, we prove the globalconstrained approximate null controllability under a stronger condition for $P$and the same rank condition as the above case. Moreover, we prove that theabove mentioned spectral condition of $P$ is necessary for global constrainedapproximate null controllability of the control problem considered in thispaper.

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