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On the hierarchical optimal control of a chain of distributed systems

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  • 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 over a finite time interval. 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 leader and that of the 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 control strategies for such an optimization problem, where the follower (which corresponds to the second criterion) is required to respond optimally, in the sense of best-response correspondence to the strategy of the leader (which is associated to the controllability-type problem) so as to achieve the overall objectives. Finally, we remark on the implication of our result in assessing the influence of the target set on the strategy of the follower with respect to the direction of leader-follower (and vice-versa) information flow.
    Mathematics Subject Classification: Primary: 35K10, 35K65, 93A13, 93E20, 91A35.


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