Remarks on the hierarchical control problems with model uncertainty

Abstract: In this paper, we consider a hierarchical control problem with model uncertainty. Specifically, we consider the following 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 system close to a given reference trajectory over a finite time interval. We introduce the following framework. First, we partition the control subdomain into two disjoint open subdomains, with smooth boundaries, that are compatible with the strategy subspaces of the {\it leader} (which is responsible for the controllability-type criterion) and that of the {\it follower} (which is associated with the second criterion), respectively. Moreover, we account at the optimization stage for model uncertainty by allowing the {\it leader} to choose its control strategy based on a class of alternative models about the system, whereas the {\it follower} makes use of an approximate model about the system. Using the notion of Stackelberg's optimization, we provide conditions on the existence of optimal control strategies for such a hierarchical control problem, under which the {\it follower} is required to respond optimally to the strategy of the {\it leader} so as to achieve the overall objectives. Apart from the issue of modeling and uncertainty, this paper is a companion to our previous work.