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

TL;DR

This paper investigates the fastest synchronization of a nonlinear oscillator to its limit cycle under realistic bounded control forces, revealing complex optimal control structures through analytical and numerical methods.

Contribution

It introduces a novel approach combining Pontryagin's Maximum Principle with analytical and numerical tools to solve a complex nonlinear optimal control problem with force bounds.

Findings

Optimal control develops complex phase space structures as force bounds decrease.

Trajectories from extreme points of the limit cycle are key to maximum control bangs.

The van der Pol oscillator exemplifies intricate control features.

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.