Is there a strategy for discrete control of a system with dynamics near sample rate? I'm trying to control a system where the controller sample rate is physically fixed and the plant has significant dynamics on the same order as the sample rate. I understand that one would prefer to have the sample rate considerably faster than the plant dynamics, but the physics of this system are such that this is inherently impossible. I can get some control of the system by hand-tuning a PID, but it seems considerably sub-optimal.

Is there a strategy for developing a controller like this?

Edit: I should add that the plant is this situation is a relatively complex LTI plant with reverberations caused by pure time delays. The delay cycles are a bit slower than the Nyquist frequency. The plant is stable.
Depending on your plant you might be able to do something. If the plant is unstable, but dominantly linear, you should be able to get a very accurate model using exact discretization (zero-order hold). Either you will be able to stabilize your plant using PID, or PID will be of too low order, in which case you need to switch to a control law matching you plant order. The H-infinity synthesis framework (robust control) will do that for you. Here you can also include uncertainty weights to limit control law gain, which might help esp. if your system is not linear. Depending on the uncertainty you might not be able to find a feasible control law for stabilization, however...

If your plant is stable, or stabilizable, you can further improve performance using feed-forward control. Here the sampling time should not be of much concern, and you can use something like zero phase error tracking.