Title: Mean-Field Controllability and Decentralized Stabilization of Markov Chains, Part II: Asymptotic Controllability and Polynomial Feedbacks,
Abstract: This paper, the second of a two-part series, presents a method for mean-field
feedback stabilization of a swarm of agents on a finite state space whose time
evolution is modeled as a continuous time Markov chain (CTMC). The resulting
(mean-field) control problem is that of controlling a nonlinear system with
desired global stability properties. We first prove that any probability
distribution with a strongly connected support can be stabilized using
time-invariant inputs. Secondly, we show the asymptotic controllability of all
possible probability distributions, including distributions that assign zero
density to some states and which do not necessarily have a strongly connected
support. Lastly, we demonstrate that there always exists a globally
asymptotically stabilizing decentralized density feedback law with the
additional property that the control inputs are zero at equilibrium, whenever
the graph is strongly connected and bidirected. Then the problem of
synthesizing closed-loop polynomial feedback is framed as a optimization
problem using state-of-the-art sum-of-squares optimization tools. The
optimization problem searches for polynomial feedback laws that make the
candidate Lyapunov function a stability certificate for the resulting
closed-loop system. Our methodology is tested for two cases on a five vertex
graph, and the stabilization properties of the constructed control laws are
validated with numerical simulations of the corresponding system of ordinary
differential equations.