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

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.