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  • Open Access

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

C. Ríos-Monje1, C. A. Plata1, D. Guéry-Odelin2, and A. Prados1,*

  • 1Física Teórica, Multidisciplinary Unit for Energy Science, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
  • 2Laboratoire de Collisions Agrégats Réactivité, Université de Toulouse, CNRS, 118 Route de Narbonne, 31062 Toulouse CEDEX 4, France

  • *Contact author: prados@us.es
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Phys. Rev. Research 7, 043187 – Published 19 November, 2025

DOI: https://doi.org/10.1103/ydzs-l1zd

Abstract

Incorporating force bounds is crucial for realistic control implementations in physical systems. Here, we investigate the fastest possible synchronization of a Liénard system to its limit cycle using a bounded external force. To tackle this challenging nonlinear 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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Phys. Rev. Research 7, 043187– Published 19 November, 2025
  • Received 29 July 2025
  • Accepted 22 October 2025
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