Beyond Phase Reduction: Amplitude Collapse in Optimal Control of Coupled Oscillators

This paper solves the four-dimensional Hamilton-Jacobi-Bellman equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. By sweeping coupling strength, the numerical solution reveals a sharp transition in the optimal control strategy: below the threshold, the controller steers the phase difference toward anti-phase while maintaining both oscillators near the limit cycle, consistent with reduced-order models; above the threshold, the HJB solution shifts to an amplitude collapse strategy. This finding challenges the traditional phase reduction approach commonly used for weakly coupled oscillators.