On the hierarchical optimal control of a chain of distributed systems

In this paper, we consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak H"ormander 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.