ИСТИНА

We investigate the time-optimal control of a harmonic oscillator described by the second-order differential equation: 𝑥′′(𝑡)+𝜔2(𝑡)𝑥(𝑡)=0, where 𝑥(𝑡) represents the system’s position, and control function 𝜔(𝑡) is the time-varying frequency constrained by 0<𝜔0 ≤𝜔(𝑡)≤1. The objective is to determine the minimal time T > 0 for transitioning the system from an initial state 𝑥(0)=𝑥0,𝑥̇(0)=𝑣0 to a final state 𝑥(𝑇)=𝑥𝑇,𝑥̇(𝑇)=𝑣𝑇, while satisfying boundary conditions and ensuring continuity of the solution.