  In this paper, we study the controllability and stabilizability properties of
the Kolmogorov forward equation of a continuous time Markov chain (CTMC)
evolving on a finite state space, using the transition rates as the control
parameters. Firstly, we prove small-time local and global controllability from
and to strictly positive equilibrium configurations when the underlying graph
is strongly connected. Secondly, we show that there always exists a locally
exponentially stabilizing decentralized linear (density-)feedback law that
takes zero valu at equilibrium and respects the graph structure, provided that
the transition rates are allowed to be negative and the desired target density
lies in the interior of the set of probability densities. For bidirected
graphs, that is, graphs where a directed edge in one direction implies an edge
in the opposite direction, we show that this linear control law can be realized
using 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 using
Linear Matrix Inequality (LMI) based tools to algorithmically construct
decentralized density feedback controllers for stabilization of a robotic swarm
to a target task distribution with no task-switching at equilibrium, as we
demonstrate with several numerical examples.
