Density Stabilization Strategies for Nonholonomic Agents on Compact Manifolds

TL;DR

The paper advances density stabilization for swarms with nonholonomic, driftless control-affine dynamics on bounded domains and compact manifolds by extending mean-field density control from holonomic, elliptic settings to hypoelliptic generators. It presents two complementary modeling approaches: (i) a linear hypoelliptic PDE with reflective boundary conditions yielding exponential convergence to a target density $f$ via a semigroup generated by $-A_a^b$, and (ii) a semilinear, hybrid switching PDE enabling stabilization to densities with disconnected supports through density-dependent transitions governed by mean-field feedback. The authors prove well-posedness and stability (including a spectral gap and mass conservation) and validate the methods numerically on Brockett-type dynamics, $SO(3)$, and $S^2$, demonstrating both density tracking and energy-efficient, interaction-enabled stabilization. These results broaden practical density control for robotic swarms under nonholonomic constraints, with potential impact on coverage, task allocation, and cooperative transport in complex geometries.

Abstract

In this article, we consider the problem of stabilizing stochastic processes, which are constrained to a bounded Euclidean domain or a compact smooth manifold, to a given target probability density. Most existing works on modeling and control of robotic swarms that use PDE models assume that the robots' dynamics are holonomic, and hence, the associated stochastic processes have generators that are elliptic. We relax this assumption on the ellipticity of the generator of the stochastic processes, and consider the more practical case of the stabilization problem for a swarm of agents whose dynamics are given by a controllable driftless control-affine system. We construct state-feedback control laws that exponentially stabilize a swarm of nonholonomic agents to a target probability density that is sufficiently regular. State-feedback laws can stabilize a swarm only to target probability densities that are positive everywhere. To stabilize the swarm to probability densities that possibly have disconnected supports, we introduce a semilinear PDE model of a collection of interacting agents governed by a hybrid switching diffusion process. The interaction between the agents is modeled using a (mean-field) feedback law that is a function of the local density of the swarm, with the switching parameters as the control inputs. We show that the semilinear PDE system is globally asymptotically stable about the given target probability density. The stabilization strategies are verified without inter-agent interactions is verified numerically for agents that evolve according to the Brockett integrator and a nonholonomic system on the special orthogonal group of 3-dimensional rotations $SO(3)$. The stabilization strategy with inter-agent interactions is verified numerically for agents that evolve according to the Brockett integrator and a holonomic system on the sphere $S^2$.

Figures (7)

• Figure 1: (Brockett integrator without agent interactions) Stochastic coverage by $N_p= 10,000$ agents (in red) at three times $t$, following the linear diffusion model \ref{['eq:Mainsysan']}.
• Figure 2: (Brockett integrator without agent interactions) Time evolution of the $L_1$ norm of the difference between the target distribution and the agent distribution that evolves according to the linear diffusion model \ref{['eq:Mainsysan']}.
• Figure 3: (Nonholonomic system on SO(3)) Stochastic coverage of $SO(3)$ by $N= 5,000$ agents (in red) at different times $t$, following the linear diffusion model \ref{['eq:Mainsysan']}. This plot shows the time evolution of the action of the agents' matrices $\mathbf{Z}_j(t)$ on $-\mathbf{e}_2 = [0 ~~-1~~0]^T$ on the sphere $S^2$.
• Figure 4: (Nonholonomic system on SO(3)) Stochastic coverage of $SO(3)$ by $N= 5,000$ agents (in red) at different times $t$, following the linear diffusion model \ref{['eq:Mainsysan']}. This plot shows the time evolution of the action of the agents' matrices $\mathbf{Z}_j(t)$ on $\mathbf{e}_3 = [0 ~~0~~1]^T$ on the sphere $S^2$.
• Figure 5: (Brockett integrator with agent interactions) Stochastic coverage of $\mathbb{R}^3$ by $N= 1,000$ agents at three times $t$, following the semilinear PDE model \ref{['eq:clpPDEdsicon']}. Blue agents are in the motion state; red agents are in the motionless state.

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 3.3
• proof
• Proposition 3.4
• Corollary 3.5
• proof
• Theorem 3.6
• proof