Optimal control of a damped harmonic oscillator through a lossy, dispersive medium

Say a damped harmonic oscillator is at the far end of a dispersive, lossy medium governed by, say, Debye relaxation. I would like to determine what the optimal arbitrary input waveform is to obtain the maximum instantaneous (not steady-state) amplitude on the oscillator.

If there wasn’t dispersion, this would be relatively easy – can’t do much better than the solution to the oscillator DE. However, if group velocity is taken into account, you can do much better.

If one wants to determine the response of an oscillator to some arbitrary known input, it seems like you can build your input with delta functions, and then use Green functions to get the response, maybe with numerical integration in there somewhere.

But I would like to do the reverse; determine the "optimal control", subject to some transfer function. Are there some search terms I should be using – or does anyone know of any examples of analogous problems in other fields that might be of use?

Very sorry if this is too open-ended – frankly, I don’t understand what I’m doing!