与极限环的最优同步
Optimal synchronisation to a limit cycle
AI总结：
本文研究受时变外力驱动的van der Pol振子在有限时间内到达极限环的最优控制问题，给出连接时间与非保守功之间的速度极限权衡，并推广至Liénard非线性振子及总功最小化。
摘要：
在没有外部驱动的情况下，van der Pol振子相平面上的所有轨迹在无限长时间后都会趋向一条闭合的周期性轨迹——极限环。本文中，我们用一个随时间变化的外力驱动van der Pol振子，使其在给定的有限时间内到达极限环。具体而言，我们关注在将系统从相平面上给定初始点驱动到属于极限环的任意终点时，使功的非保守贡献最小化。文中出现了一个速度极限不等式，它表示连接时间与以非保守功计的代价之间的权衡。我们展示了上述结果如何能够推广到由Liénard方程给出的更广泛的非线性振子族。最后，我们还研究了使外力所做总功最小化的问题。
英文摘要：
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 Liénard equation. Finally, we also look into the problem of minimising the total work done by the external force.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
