Optimal actuator location of 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 L2( × (0, T )). The actuator domain is time-varying in the sense that it is only required to have a prescribed Lebesgue measure for any moment. We select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains. We build a framework of finding the Nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open-loop system. The existence and uniqueness of the optimal classical solutions are therefore concluded. As a result, we synthesize both optimal actuator location and corresponding optimal control into a time-varying feedbacks. © 2016 Elsevier Inc. All rights reserved. MSC: 35K05; 49J20; 65K210; 90C47; 93C20 ✩ This work was carried out with the support of the National Natural Science Foundation of China under grants 11371375, 11471080, 11371095, China Postdoctoral Science Foundation and Central South University Postdoctoral Science Foundation, and the National Research Foundation of South Africa. * Corresponding author. E-mail address: [email protected] (Y. Xu). http://dx.doi.org/10.1016/j.jde.2016.05.037 0022-0396/© 2016 Elsevier Inc. All rights reserved. B.-Z. Guo et al. / J. Differential Equations 261 (2016) 3588–3614 3589