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In this article, we consider the problem of stabilizing a class of degeneratestochastic processes, which are constrained to a bounded Euclidean domain or acompact smooth manifold, to a given target probability density. Most existingworks on modeling and control of robotic swarms that use partial differentialequation (PDE) models assume that the robots' dynamics are holonomic, andhence, the associated stochastic processes have generators that are elliptic.We relax this assumption on the ellipticity of the generator of the stochasticprocesses, and consider the more practical case of the stabilization problemfor a swarm of agents whose dynamics are given by a controllable driftlesscontrol-affine system. We construct state-feedback control laws thatexponentially stabilize a swarm of nonholonomic agents to a target probabilitydensity that is sufficiently regular. State-feedback laws can stabilize a swarmonly to target probability densities that are positive everywhere. To stabilizethe swarm to probability densities that possibly have disconnected supports, weintroduce a semilinear PDE model of a collection of interacting agents governedby a hybrid switching diffusion process. The interaction between the agents ismodeled using a (mean-field) feedback law that is a function of the localdensity of the swarm, with the switching parameters as the control inputs. Weshow that under the action of this feedback law, the semilinear PDE system isglobally asymptotically stable about the given target probability density. Thestabilization strategies with and without agent interactions are verifiednumerically for agents that evolve according to the Brockett integrator; thestrategy with interactions is additionally verified for agents that evolveaccording to an underactuated system on the sphere $S^2$.

Density Stabilization Strategies for Nonholonomic Agents on Compact Manifolds