On optimal location of diffusion and related optimal control for null controllable heat equation

We consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi-dimensional heat equations. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa. We formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems. One is seeking the optimal domain of heat diffusion with a fixed interior actuator domain. This can be considered as a domain perturbation problem in shape optimization. Another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain. The existences of these two types of shape optimization problems over some class of open sets in general RNRN space have been proved separately.