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

Optimal Actuator Location of the Minimum Norm Controls for Heat Equation with General Controlled Domain