There are many problems based on solving non-autonomous differential equations of the form x¨(t)+ω2(t)x(t)=0, where x(t) represents the coordinate of a material point and ω is the angular frequency. The inverse problem involves finding the bounded coefficient ω. Continuity of the function ω(t) is not required. The trajectory x(t) is unknown, but the initial and final values of the phase variables are given. The variation principle of minimum time for the entire dynamic process allows for the determination of the optimal solution {x(t),ω(t)}. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made.

Optimal control has emerged as an indispensable tool in the domain of mechanical robotic systems. The dynamic processes under consideration in this paper are characterized by differential equations with an unknown coefficient. The problem addressed is time-optimal and exhibits bilinear characteristics. To investigate this inverse optimal problem, the classical method has been employed alongside Pontryagin’s Maximum Principle (PMP). This article aims to provide an exact piecewise function for controlling trajectories, specifically accounting for viscous friction. The goal is to determine the reachability set and to find the minimal process time. Notably, no simplifying assumptions were made during the analytical transformations.