  Controlling nonlinear dynamics arise in various engineering fields. We
present efforts to model the forced van der Pol system control using
physics-informed neural networks (PINN) compared to benchmark methods,
including idealized nonlinear feedforward (FF) control, linearized feedback
control (FB), and feedforward-plus-feedback combined (C). The aim is to
implement circular trajectories in the state space of the van der Pol system. A
designed benchmark problem is used for testing the behavioral differences of
the disparate controllers and then investigating controlled schemes and systems
of various extents of nonlinearities. All methods exhibit a short
initialization accompanying arbitrary initialization points. FF control
successfully converges to the desired trajectory, and PINN executes good
controls with higher stochasticity observed for higher-order terms based on the
phase portraits. In contrast, combined control failed. Varying trajectory
amplitudes revealed that FF, FB, and combined control all fail for unity
nonlinear damping gain. Traditional control methods display a robust
fluctuation for higher-order terms. For some various nonlinearities, PINN
failed to implement the desired trajectory instead of becoming "trapped" in the
phase of a small radius, yet FB successfully implemented controls. PINN
generally exhibits lower relative errors for varying targeted trajectories.
However, PINN also shows evidently higher computational burden compared with
traditional control theory methods, with at least more than 30 times longer
control time compared with benchmark idealized nonlinear feed-forward control.
This manuscript proposes a comprehensive comparative study for future
controller employment considering deterministic and machine learning
approaches.
