Control the Coefficient of a Differential Equation as an Inverse Problem in Time

Abstract

There are many problems based on solving nonautonomous 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 also unknown, but the initial and final values of the phase variables are given. The variation principle of the minimum time for the entire dynamic process allows for the determination of the optimal solution. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made.