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arXiv · 2507.19560

Time-optimal synchronisation to self-sustained oscillations under bounded control

C. Ríos-Monje·C. A. Plata·D. Guéry-Odelin·A. Prados
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Abstract

Incorporating force bounds is crucial for realistic control implementations in physical systems. Here, we investigate the fastest possible synchronisation of a Li\'enard system to its limit cycle using a bounded external force. To tackle this challenging non-linear optimal control problem, our approach involves applying Pontryagin's Maximum Principle with a combination of analytical and numerical tools. We show that the optimal control develops a remarkably complex structure in phase space as the force bound is lowered. Trajectories rewound from the limit cycle's extreme points turn out to play a key role in determining the maximum number of control bangs for optimal connection. We illustrate these intricate features using the paradigmatic van der Pol oscillator model.

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C. Ríos-Monje, C. A. Plata, D. Guéry-Odelin, A. Prados. 2025-07-25. Time-optimal synchronisation to self-sustained oscillations under bounded control. https://doi.org/10.1103/ydzs-l1zd

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