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In this paper, we study the controllability and stabilizability properties ofthe Kolmogorov forward equation of a continuous time Markov chain (CTMC)evolving on a finite state space, using the transition rates as the controlparameters. Firstly, we prove small-time local and global controllability fromand to strictly positive equilibrium configurations when the underlying graphis strongly connected. Secondly, we show that there always exists a locallyexponentially stabilizing decentralized linear (density-)feedback law thattakes zero valu at equilibrium and respects the graph structure, provided thatthe transition rates are allowed to be negative and the desired target densitylies in the interior of the set of probability densities. For bidirectedgraphs, that is, graphs where a directed edge in one direction implies an edgein the opposite direction, we show that this linear control law can be realizedusing a decentralized rational feedback law of the form k(x) = a(x) +b(x)f(x)/g(x) that also respects the graph structure and control constraints(positivity and zero at equilibrium). This enables the possibility of usingLinear Matrix Inequality (LMI) based tools to algorithmically constructdecentralized density feedback controllers for stabilization of a robotic swarmto a target task distribution with no task-switching at equilibrium, as wedemonstrate with several numerical examples.

Mean-Field Controllability and Decentralized Stabilization of Markov Chains, Part I: Global Controllability and Rational Feedbacks