Optimal control for phase locking of synchronized oscillator populations via dynamical reduction techniques

Technical Summary: Optimal Control for Phase Locking of Synchronized Oscillator Populations via Dynamical Reduction Techniques

Problem Statement
The paper addresses the challenge of controlling the collective phase of high-dimensional populations of coupled oscillators, specifically those modeled by the Kuramoto model under a periodic external input. In biological systems (e.g., circadian rhythms) and engineering applications (e.g., power grids), the loss of synchronization due to disturbances—such as a sudden phase shift in the external input analogous to "jet lag"—impairs functionality. The core difficulty lies in the high dimensionality of these systems ($N \gg 1$), which makes direct control of individual oscillators impossible and renders standard nonlinear optimal control computationally intractable. Furthermore, control inputs must be weak to avoid disrupting the mutual synchrony essential for the system's function. The authors aim to develop a framework to derive an optimal control input that facilitates the rapid resynchronization (phase locking) of the population with a periodic input following a sudden phase shift, while maintaining the degree of mutual synchrony.

Methodology
The authors propose a framework combining dynamical reduction techniques with nonlinear optimal control theory. The methodology proceeds in three main stages:

1. Dynamical Reduction:

   • Continuum Limit & Ott-Antonsen (OA) Ansatz: The authors first consider the continuum limit ($N \to \infty$) of the Kuramoto model with a periodic input. They employ the OA ansatz to reduce the infinite-dimensional continuity equation (describing the probability density of phases and frequencies) to a low-dimensional dynamics of a complex order parameter $A(t) = R(t)e^{i\Phi(t)}$. This results in a two-dimensional Stuart-Landau equation for $A(t)$.
   • Phase-Amplitude Reduction: Assuming the external input is weak, they apply phase-amplitude reduction theory to the Stuart-Landau equation. This yields a pair of one-dimensional equations:
     • A phase equation describing the evolution of the collective phase $\phi(t)$.
     • An amplitude equation describing the deviation $r(t)$ of the system's state from the limit cycle (representing the degree of mutual synchrony).
   • These reduced equations explicitly incorporate the collective phase sensitivity function $\zeta(\phi)$ and the amplitude sensitivity function $\kappa(\phi)$.

2. Optimal Control Formulation:

   • The control problem is formulated using the reduced phase equation rather than the original high-dimensional model. The objective is to minimize a cost function $J$ consisting of a terminal cost (deviation of the phase difference $\hat{\Psi} = \phi - \Theta$ from a target $\Psi_0$) and a stage cost (penalizing the magnitude of the control input $u(t)$ and the phase deviation during the process).
   • Constraints are imposed to ensure the net input $p(t) = p_0 \sin \Theta(t) + u(t)$ remains within a specific range (weak input assumption).
   • The problem is solved numerically as a two-point boundary value problem using the shooting method and the penalty function method to handle inequality constraints.

3. Evaluation of Synchrony:

   • To ensure the control input does not degrade mutual synchrony, the authors utilize the derived amplitude equation. They establish a sufficient condition on the amplitude deviation $r(t)$ such that the decay term dominates the dynamics, ensuring the system state remains close to the limit cycle.

Key Contributions

• Framework Integration: The paper presents a novel framework that bridges dynamical reduction (OA ansatz and phase-amplitude reduction) with optimal control theory to manage high-dimensional oscillator populations.
• Low-Dimensional Control Design: By reducing the system to a pair of phase and amplitude equations, the authors enable the design of optimal control inputs that would be computationally prohibitive in the original high-dimensional space.
• Synchrony Preservation Analysis: Unlike many control strategies that focus solely on phase, this work explicitly evaluates the impact of the control input on the degree of mutual synchrony (amplitude $R$) using the amplitude equation, providing a theoretical condition to ensure synchrony is not disrupted.
• Jet Lag Analogy: The study specifically models the "jet lag" scenario—a sudden phase shift in the periodic input—and demonstrates how to accelerate the recovery of phase locking.

Results
Numerical simulations were conducted on a Kuramoto model with $N=10^4$ oscillators.

• Recovery Time: Under optimal control, the time required for the system to resynchronize with the periodic input after a phase shift was significantly reduced. On average, the recovery time was approximately 50.28% of the time required in the uncontrolled case.
• Phase Locking: The control input successfully re-established phase locking for various initial phase shifts ($\Delta\Theta$). The direction of the control (advancing or delaying the phase) depended on the sign of the phase sensitivity function relative to the phase shift.
• Constraint Satisfaction: The derived control inputs satisfied the constraint that the net external input magnitude did not exceed the original periodic input amplitude.
• Synchrony Maintenance: The simulations showed that the collective amplitude $R(t)$ remained within the theoretically derived bounds ($R_+$ and $R_-$) and stayed close to the unperturbed limit cycle value $R_0$. This confirms that the control input effectively accelerated resynchronization without significantly degrading the degree of mutual synchrony.

Significance and Claims
The authors claim that their framework provides a valid and effective method for controlling high-dimensional synchronized oscillator populations. By leveraging dynamical reduction, they demonstrate that complex synchronization control problems can be solved using low-dimensional reduced dynamics. The paper asserts that this approach is particularly suitable for scenarios like jet lag recovery, where rapid resynchronization is critical. Furthermore, the authors note that while the current study focuses on the simple Kuramoto model, the framework is theoretically extensible to more complex systems (e.g., those with time-varying coupling, higher-order interactions, or time delays). They also suggest that the computational efficiency of the reduced dynamics makes the approach well-suited for real-time control applications, such as Model Predictive Control (MPC), although the current implementation is feedforward. The work emphasizes the importance of jointly considering phase and amplitude dynamics to ensure that control actions do not inadvertently destroy the collective synchronization they aim to restore.