On the hierarchical optimal control of a chain of distributed systems

TL;DR

This paper develops a hierarchical control framework for a chain of distributed systems governed by degenerate parabolic equations, using Stackelberg optimization to ensure controllability and trajectory tracking.

Contribution

It introduces a novel hierarchical optimal control approach for distributed systems under degenerate conditions, employing Stackelberg optimization for leader-follower strategy coordination.

Findings

Existence of optimal strategies under hierarchical control framework.

Effective response of follower to leader's strategy in system control.

Insights into influence of target set on follower's optimal strategy.

Abstract

In this paper, we consider a chain of distributed systems governed by a degenerate parabolic equation, which satisfies a weak H\"{o}rmander 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.