On the Connectivity Properties of Feasible Regions of Optimal Decentralized Control Problems

The optimal decentralized control (ODC) is an NP-hard problem with many applications in real-world systems. There is a recent trend of using local search algorithms for solving optimal control problems. However, the effectiveness of these methods depends on the connectivity property of the feasible region of the underlying optimization problem. In this article, for ODC problems with static controllers, we develop a novel criterion for certifying the connectivity of the feasible region in the case where the input and output matrices of the system dynamics are identity matrices. This criterion can be checked via an efficient algorithm, and it is used to prove that the number of communication networks leading to connected feasible regions is greater than a square root of the exponential number of possible communication networks (named patterns). For ODC problems with dynamic controllers, we prove that under certain mild conditions, the closure of the feasible region is always connected after some parameterization, for general communication networks and system dynamics.