In this paper, we consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak Hörmander type condition, with a control distributed over an open subdomain. In particular, we consider two objectives that we would like to accomplish. The first one being of a controllability type that consists of guaranteeing the terminal state to reach a target set starting from an initial condition; while the second one is keeping the state trajectory of the overall system close to a given reference trajectory on a finite, compact time intervals. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains that are compatible with the strategy subspaces of the {\it leader} and that of the {\it follower}, respectively. Then, using the notion of Stackelberg's optimization (which is a hierarchical optimization framework), we provide a new result on the existence of optimal strategies for such an optimization problem -- where the {\it follower} (which corresponds to the second criterion) is required to respond optimally, in the sense of {\it best-response correspondence} to the strategy of the {\it leader} (that is associated to the controllability-type criterion) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the reachable target set on the optimal strategy of the {\it follower} in relation to the direction of {\it leader-follower} and {\it follower-leader} information flows.
Optimized Hierarchical Power Oscillations Control for Distributed Generation Under Unbalanced Conditions
State Grid Customer Service Center（国家电网客户服务中心）；School of Electrical Engineering, Northeast Dianli University（东北电力大学电气工程学院）；Institute of Energy Technology, Aalborg University（奥尔堡大学能源技术研究所）
Improved Hierarchical ADMM for Nonconvex Cooperative Distributed Model Predictive Control
National University of Singapore（新加坡国立大学）；University of California, Berkeley（加州大学伯克利分校）；Temasek Laboratories, National University of Singapore（新加坡国立大学淡马锡实验室）；University of British Columbia（不列颠哥伦比亚大学）
Hierarchical Plug-and-Play Voltage/Current Controller of DC Microgrid Clusters with Grid-Forming/Feeding Converters: Line-independent Primary Stabilization and Leader-based Distributed Secondary Regulation
