Optimal actuator location for time and norm optimal control of null controllable heat equation

We consider minimal time and minimal norm of the optimal controls for a multi-dimensional internally controlled heat equation with control domain varying over a class of open sets. Twoproblems are formulated separately into different types of shape optimization problems over this open set class. The governing equation with any given initial value and admissible control are considered as constraints, and minimal time or minimal norm of the optimal controls is considered as cost for related shape optimization. The solution of the shape optimization leads to the optimal actuator location for optimal controls. The existence of such an optimal location domain for both minimal time and minimal norm controls is presented. A different problem that relates the balance betweenminimal time andminimal norm controls is also discussed. This study builds a link between optimal control and shape optimization for this multidimensional heat equation. This work was carried out with the support of the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa. B.-Z. Guo Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, People’s Republic of China e-mail: [email protected] B.-Z. Guo School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa B.-Z. Guo Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia D.-H. Yang (B) School of Mathematics and Statistics, Central South University, Changsha 410075, People’s Republic of China e-mail: [email protected]