  In this paper, we study optimal actuator location of the minimum norm
controls for a multi-dimensional heat equation with control defined in the
space $L^p(0,T;L^2(\Omega))$. The actuator domain $\omega$ is quite general in
the sense that it is required only to have a prescribed Lebesgue measure. A
relaxation problem is formulated and is transformed into a two-person zero-sum
game problem. By the game theory, we develop a necessary and sufficient
condition and the existence of relaxed optimal actuator location for
$p\in[2,+\infty]$, which is characterized by the Nash equilibrium of the
associated game problem. An interesting case is for the case of $p=2$, for
which it is shown that the classical optimal actuator location can be obtained
from the relaxed optimal actuator location without additional condition.
Finally for $p=2$, a sufficient and necessary condition for classical optimal
actuator location is presented.
