Identifying the Connectivity of Feasible Regions for Optimal Decentralized Control Problems

Abstract

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 paper, we develop the notion of stable expandability and use it to obtain a novel criterion for certifying the connectivity of the feasible region for ODC problems with state feedback and an identity input matrix. This criterion can be checked via an efficient algorithm. Based on the developed mathematical technique, we prove that the feasible region is guaranteed to be connected in presence of only a small number of communication constraints. We also show that among the exponential number of possible communication networks (named patterns), a square root of them lead to connected feasible regions. A by-product of this result is that a high-complexity ODC problem may be approximated with a simpler one by replacing its pattern with a favorable pattern that makes the feasible region connected.