For the problem of oscillator control, given by Equation 4.32, consider the specific governing equation

\[ \ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=\frac{A}{m} \cos \omega t+F_{\text {control }}(t) \]

where \(\zeta, \omega_{n}\), and \(F_{\text {control }}(t)\) must be determined so that the maximum amplitude of the response is \(x_{\max }<\beta A\), where \(\beta=0.5\). Since there is no single answer, describe how to proceed and what considerations must be made during the analysis. Provide one solution.