In this article, we consider the problem of stabilizing a class of degenerate stochastic processes, which are constrained to a bounded Euclidean domain or a compact smooth manifold, to a given target probability density. This stabilization problem arises in the field of swarm robotics, for example, in applications where a swarm of robots is required to cover an area according to a target probability density. Most existing works on modeling and control of robotic swarms that use partial differential equation (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 under the action of this feedback law, the semilinear PDE system is globally asymptotically stable about the given target probability density. The stabilization strategies with and without agent interactions are verified numerically for agents that evolve according to the Brockett integrator; the strategy with interactions is additionally verified for agents that evolve according to an underactuated s...
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Optimal Control of Velocity and Nonlocal Interactions in the Mean-Field Kuramoto Model
In this paper, we investigate how the self-synchronization property of a swarm of Kuramoto oscillators can be controlled and exploited to achieve target densities and target phase coherence. In the limit of an infinite number of oscillators, the collective dynamics of the agents’ density is described by a mean-field model in the form of a nonlocal PDE, where the nonlocality arises from the synchronization mechanism. In this mean-field setting, we introduce two space-time dependent control inputs to affect the density of the oscillators: an angular velocity field that corresponds to a state feedback law for individual agents, and a control parameter that modulates the strength of agent interactions over space and time, i.e., a multiplicative control with respect to the integral nonlocal term. We frame the density tracking problem as a PDE-constrained optimization problem. The controlled synchronization and phase-locking are measured with classical polar order metrics. After establishing the mass conservation property of the mean-field model and bounds on its nonlocal term, a system of first-order necessary conditions for optimality is recovered using a Lagrangian method. The optimality system, comprising a nonlocal PDE for the state dynamics equation, the respective nonlocal adjoint dynamics, and the Euler equation, is solved iteratively following a standard Optimize-then-Discretize approach and an efficient numerical solver based on spectral methods. We demonstrate our approach for each of the two control inputs in simulation.
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- Award ID(s):
- 1828010
- PAR ID:
- 10432746
- Date Published:
- Journal Name:
- 2022 American Control Conference (ACC)
- Page Range / eLocation ID:
- 290 to 295
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC53348.2022.9867715
