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.