Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s and 0s. I want to find a matrix $A$ that will dampen or eliminate the signal, such that $$ \int_0^T \| x(t) \|^2 dt $$ is as small as possible.
Does any analytical method exist to find such an $A$?
Well this is by no means a complete solution but maybe it takes you somewhere. If you take the Laplace transform of both sides and assume $x(0)$ to be the zero vector, then you get $$X(s) = (sI-A)^{-1}B\frac{\alpha}{s^2+\alpha^2}$$
Then I invoke Parseval's Theorem and write $$\int_0^{\infty}x(t)^{\top}x(t)dt=\frac{1}{2\pi}\int_{-\infty}^{\infty}\left(\frac{\alpha}{-\omega^2+\alpha^2}\right)^2\|(j\omega I-A)^{-1}B\|^2d\omega$$
This clearly has something to do with the controllability properties of $(A,B)$ but I haven't figured it out yet. I'll take another look at it tomorrow with a fresh set of eyes.