Parametric Resonance: Bridging optimal control theory and dynamical system stability

The time-optimal control of a classical damped harmonic oscillator takes the form of a periodic step‑like modulation of the oscillator stiffness, as established in [1]. Building on this work, we show that its emergent periodic structure admits a direct dynamical interpretation in terms of parametric instability. We further derive new weak-damping asymptotics showing that the optimal frequency associated to a unit amplification approaches twice the natural frequency of the oscillator,revealing a direct and unexpected link between optimality and the condition for parametric resonance. This observation enables a direct mapping between optimal solutions and the stability structure of dynamical systems, as captured by the Pontryagin maximum principle and Floquet theory. Our results demonstrate that time‑optimal control strategies naturally exploit the most unstable regions of parametric excitation.