Minimal Time Control of Underdamped Parametric Oscillators

Controlling a damped oscillator is crucial in various technological and scientific fields, such as structural engineering, aerospace, and noise reduction device design. This article considers a classical underdamped harmonic oscillator, focusing on its minimal-time control by modulating its time-dependent frequency. The goal is to connect in minimal time two states with zero kinetic energy but different displacements. We provide a comprehensive theoretical analysis of this problem, characterizing the set of reachable states and detailing the structure of the optimal trajectories, which we precisely describe in phase space. These trajectories correspond to bang-bang controls, and the number of commutations depends on the ratio between the final and initial states. Our approach is based on combining the Pontryagin principle with a suitable choice of “energy-angle” variables and a Bellman-like optimality principle.