Parallel probabilistic swarm guidance by exploiting Kronecker product structures in discrete-time Markov chains

Discrete-time Markov chain has been a popular modeling paradigm for a wide range of applications. In this paper, we study the class of Markov chains that can be characterized by Kronecker products of transition matrices. Such Markov chains exhibit self-replicating property, which is common in both natural and artificial systems. We show that such chains can be decomposed into a system of parallel atomic Markov chains with fewer states. This feature enables analyzing these Markov chains by studying only the properties of smaller chains, that is useful for simplifying the overall analysis and even improving the computational efficiency. We propose an application of the Kronecker-structured Markov chains to probabilistic guidance problem, which is recently developed for decentralized swarm guidance. We then introduce the parallel probabilistic guidance algorithm, which serves as an improved version of the existing algorithms. We demonstrate that our algorithm is scalable for the class of target configurations having Kronecker product structures. Simulations are presented to demonstrate the effectiveness of the proposed method. Our work provides a guideline for the design of swarm configuration by exploiting the Kronecker structure to greatly enhance the computational efficiency.