Optimal synchronisation to a limit cycle

TL;DR

This paper develops an optimal control framework for driving the van der Pol oscillator to its limit cycle efficiently, minimizing work and revealing a speed-cost trade-off, with generalizations to broader nonlinear oscillators.

Contribution

It introduces a method to reach the limit cycle in finite time with minimal work, establishing a speed-work inequality and generalizing results to Liénard systems.

Findings

Derived a speed-work trade-off inequality.

Formulated optimal control strategies for minimal work.

Extended results to Liénard oscillators.

Abstract

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic, trajectory -- the limit cycle -- after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimising the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed limit inequality, which expresses a trade-off between the connection time and cost -- in terms of the non-conservative work. We show how the above results can be { generalized to the broader family of non-linear oscillators given by} the Li\'enard equation. Finally, we also look into the problem of minimising the total work done by the external force.